Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1087 Wtadysfaw Narkiewicz
Uniform Distribution of Sequences of Integers in Residue Classes
Springer-Verlag Berlin Heidelberg New York Tokyo 1984
Author
Wtadys{aw Narkiewicz Wroc{aw University, Department of Mathematics Plac Grunwaldzki 2-4, 50-384 Wroc~'aw, Poland
AMS Subject Classification (1980): 10A35, 10D23, 10H20, 10H25, 10L20, 10M05 ISBN 3-540-13872-2 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13872-2 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding : Beltz Offsetdruck, Hemsbach / Bergstr. 2146/3140-543210
To
my
teacher
Professor
Stanis~aw
on his s e v e n t i e t h
Hartman
anniversary
INTRODUCTION
The aim of these notes, given by the author
which
at various
form an e x t e n d e d
places,
is k n o w n
about u n i f o r m d i s t r i b u t i o n
classes.
Such
when
sequences
L.E.Dickson
tional
i.e.
with respect
We shall
standard
weakly
uniformly
After shall
example
uniform distribution
sequences
and sequences
shall
functions,in
polynomials arithmetical ~-function
f(pk)
in chapter
those,
= Pk(p)
II-IV.
which
are
for primes
functions,
like the number T-function.
to the classical
of the theory of a l g e b r a i c
also
defined
p
and
lead
numbers.
theory
of P.Deiigne,
In such cases we shall
a proper
cular we shall denote
we
i.e.
satisfy
suitable
classical Euler's
to certain
and include
In c e r t a i n
wh i c h will
needed with
func-
ques-
of polynomials.
like the theorems
We shall use n o t a t i o n
with
or sum of divisors,
on m o d u l a r
result
star-
recurrent
by m u l t i p l i c a t i v e
consider
H.P.F.Swinnerton-Dyer
function.
we
arithmetical
k ~I
This will
to
is
of sequences, linear
use more r e c e n t work,
of R a m a n u j a n ' s
results
"polynomial-like",
number
forms,
which
In the fast two c h a p t e r
we shall
the v a l u e d i s t r i b u t i o n
Our t o o l s ' b e l o n g
types
mea-
prime
N.
general
of a d d i t i v e
of sequences
In p a r t i c u l a r
and R a m a n u j a n ' s
of r e s i d u e
(mod N),
integer
certain
of c e r t a i n
P1,P2, . . . .
tions c o n c e r n i n g
mentals
for every
by v a l u e s
distribution
particular
the c o n d i t i o n
of p e r m u t a -
of sequences,
of all primes,
and c o n s i d e r i n g
defined
This will be done study u n i f o r m
a permutation
classes
sequence
(mod N)
sequences
in residue
of this century,
study
distribution
in the first chapter,
polynomial
a thorough
in r e s i d u e
is the
distributed
proving,
consider
weak u n i f o r m
here
of integers
the b e g i n n i n g
inducing
of lectures
a survey of what
prime.
distribution
ting with
tions.
polynomials
also c o n s i d e r
N. The
since
thesis m a d e
to a fixed
ning by that u n i f o r m
of sequences
studied
in his Ph.D.
polynomials,
classes
were
version
is to p r e s e n t
funda-
places we shall J.P.Serre
be used
explicitly
and
in the study state
the
reference.
which
the number
is standard of d i v i s o r s
in number of
n
by
theory.
In parti-
d(n) , o(n)
will
Vl
denote
the
powers,
sum of d i v i s o r s
only p o s i t i v e
of a set
A
will
for primes
(except when
by
Z and
residue
classes
factor
ring
the text,
(mod N),
of W r o c ~ a w
lemmas
wroc%aw,
February
1984
into account. letter
The ring
group
through
to Mrs for
problems
k-th cardinality
be r e s e r v e d
of integers
of invertible
open
The
p will
and p r o p o s i t i o n s
Certain
University
of the typescript.
the sum of their
the
the group
my gratitude
paration
taken and
a word).
consecutively
to express
of M a t h e m a t i c s
i.e.
Theorems,
~!A
ok(n)
be the m u l t i p l i c a t i v e
in each chapter.
numbered
I wish
will
and
being
by
inside
G(N)
Z/NZ.
n
divisors
be denotes
denoted
vely n u m b e r e d
of
will be
of r e s t r i c t e d
elements
of the
will be c o n s e c u t i will
be stated
in
all chapters.
Dambiec the
from the D e p a r t m e n t
patient
and careful
pre-
CONTENTS
I.
GENERAL
RESULTS
I.
Uniform
2.
The
3.
Weak
4.
Uniform
distribution
sets
I
Permutation Generators
3
Hermite's Examples
Consequences
LINEAR
N)
. . . . . . . . . . .
of
sequences
. . . . . .
. . . . . . . . . . . . . . . . . . .
the
group
of
and
Fried's
polynomials
(mod
. . . . . . . . . . . . N)
of
polynomials
....
. . . . . . . . . . . . . . . . . . .
properties
9 11
12
14
polynomials
permutation
theorem
SEQUENCES
8
12
permutation
. . . . . . . . . . . . . . . . . . . . . . . . . .
RECURRENT
I 4
. . . . . . . . . . . . . . . .
of
distribution
comments
I
of
. . . . . . . . . . . . . . . . . . . . . . . .
uniform
15 18 21 23 25 26
. . . . . . . . . . . . . . . .
28
. . . . . . . . . . . . . . . . . .
28
I.
Principal
2.
Uniform distribution ( m o d p) o f s e c o n d - o r d e r linear recurrences . . . . . . . . . . . . . . . . . . . . . .
32
3.
General
. . . . . . . . . . . . . . . . . . . .
38
4.
Notes
. . . . . . . . . . . . . . . . . . .
48
modulus and
Exercises
IV.
. . . . . . . . . . . . . .
(mod
characterization
4
Weak
N)
systems
polynomials for
5
Notes
of
SEQUENCES
2
6
distribution
distribution
POLYNOMIAL
7
(mod
. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
Exercises
III.
M(f)
uniform
Exercises
II.
. . . . . . . . . . . . . . . . . . . . . .
ADDITIVE
comments
. . . . . . . . . . . . . . . . . . . . . . . . .
FUNCTIONS
I~
The
criterion
2.
Application
3.
The
4.
Notes
sets
Exercises
and
of of
M(f)
. . . . . . . . . . . . . . . . . . . .
51
Delange
. . . . . . . . . . . . . . . . .
51
tauberian
54
Delange's for
comments
50
additive
functions
theorem
. . . . . . .
. . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
58 59 60
VIII
V.
MULTIPLICATIVE I
Decent
3
The
number
functions
4
The
vanishing equality
5
The
6
Ramanujan's
7
Notes
and
of
2.
An
3.
Applications
4.
The
5.
Notes
ADDENDA
of
and
the
Am(N)
sum
=
Euler's (5.2)
G(N)
comments
Generating
INDEX
62
~-function
71
. . . . . .
. . . . . . . . . . . . .
algorithm
85 88
. . . . . . . . . . . . . . . . . . .
94 95
. . . . . . . . . . . . . . . . .
the
set
of
values
of
a
polynomial
96 .
. . . . . . . . . . . . . . . . . . . . . . ot
functions and
by
the
ok
comments
for
79
. . . . . . . . . . . . . . . .
FUNCTIONS G(N)
77
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
I.
REFERENCES
62
. . . . . . . . . . . . . . . . . . . .
divisors
T-function
POLYNOMIAL-LIKE
Exercises
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
2
Exercises
VI.
FUNCTIONS
Dirichlet-~D
study k z3
of
M*(f)
. . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
96 102 104 107 112
. . . . . . . . . . . . . . . . . . . . . . . . .
113
. . . . . . . . . . . . . . . . . . . . . . . . . .
114
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122
125
CHAPTER GENERAL
§ I. U n i f o r m
I. If then
N
the distribution
the modulus
F(k)
N
all
limits
is c o n s t a n t ,
here
{an}
this notion
is
in c o m p a c t
however
problems
most
situation
and
In the
that
sequel
the
tion
for
f(n)
has
(mod N).
{an}
of t h i s
f
PROPOSITION
equal
here
to
have
nothing
more
certain N)
says
shortly
in t h e r a t h e r
respect
to
function
that
UD(mod
the N).
of u n i f o r m
easy
finite
to d o w i t h
convenient
the
case,
abstract
which
to c o n s i d e r of course
advantages.
if t h e
the
of n o help.
of s e q u e n c e s ,
UD(mod
one
of t h e n o t i o n
is u s u a l l y
sometimes
case when
I/N,
(mod N), case
groups
in p l a c e
criterion
however
and only if for
with
integers,
(k=O,1,...,N-l)
In the p a r t i c u l a r
but presents is
sequence
of
formula
approach be
a sequence
We
sequence
arith-
does
shall
hence
f ( 1 ) , f ( 2 ) ....
this property.
a sequence
lim x -I X -~°°
arising
essence
From Weyl's
mediately,
abelian
it w i l l
a function
its v a l u e s
exist.
the g e n e r a l
functions
change
say
F(k)
by the
is a p a r t i c u l a r
distribution
of
and
uniformlEdistributed
sequence
not
integer
thus necessarily
Obviously
metical
distribution
function
is d e f i n e d
RESULTS
= lira x -I # { n -<x: a n ---k(mod N) } x-~oo
provided F(k)
is a p o s i t i v e
I
the
following
to b e u n i f o r m l y we
1.1.
prefer
A sequence
Z exp(2Zianr/N) n -<X
distributed
to g i v e
r=1 ,2 .... ,N-!
necessary
a simple
{a n }
and
sufficient
(mod N)
direct
of integers is
results
condiim-
proof:
UD(mod
N)
if
one has = 0
•
(1.1)
Proof.
Put
N-I fr(t)
for
-I
=
if
hence
N-I Z r:O
fr (J)
UD(mod
then
of
for
=
N-I ~2 r=1
I/N + (Nx)-I
r=1,2,...,N-1
of t h e
sequence
r = I , 2 .... ,N-I
I : n~x an: j (rood N)
N-I ~ fr(J) + 0 ( x ) j=O
-I
fr (an-J)
=
for
for
N-I ~ fr (J) j =0
n<-x
since
(1.1) (mod N)
N)
=
N-I ~ ~ n-<x r : O
~ fr (an) n_<x
the c o n d i t i o n
fr (an)
= xm
In v i e w
Nlt
(Nx)-I
distribution
is
r = 0 , I , 2 ..... N-I.
otherwise
~ I = n-<x an- j (rood N)
uniform
for
j=O,I,...,N-1
(mx)-1
{a n }
~ N ( O
get
x
= exp(2~itr/N)
:
r=O we
f~(t)
N-I ~ j=O
fr (J)
~ fr (an) n-<x
is s u f f i c i e n t {an}.
we
for
Conversely
the
if
get
fr(J) (x/N + 0 ( x ) )
=
= 0(x)
r=1,2,...,N-1
we have
obviously
N-I fr (J) = O . j=O
This since
it i n v o l v e s
awkward, used
criterion
however
to d e d u c e
2. T h e done
by I.NIVEN
the
last
sequence
shall
systematic [61]
implies
[61].
that
R.SATTLER for all
N
zero.
Note
1.1
sums,
which
sequences
may
cases
{anx}
(mod N) UD(mod
gave
the
be very
it c a n b e
first
for all
I)
was
UD(mod
S.UCHIY/iMA
(mod N)
N
3)
[68]
only
in
of a
for almost
all
and
found which
was
time
contained
a counter-example
{a n }
can be
In L . K U I P E R S ,
for
the assertion,
is
[70]
distribution
occurs
that
[72] a s e q u e n c e but
to particular
in c e r t a i n
of u n i f o r m
{anx}
H.G.MEIJER, N)
that
distribution
In fact, H . G . M E I J E R
a set of m e a s u r e
to a p p l y
conditions.
study
is i n e x a c t .
UD(mod
later
and Proposition
that uniform
{a n }
easy
of e x p o n e n t i a l
see
convenient
in S . U C H I Y A M A
paper
always
evaluations we
more
first
explicitly
is n o t
x
in
is
for
the p r o p e r
x
from
version
of the d e b a t e d by a weaker
statement
Connections was
considered
distributed
large
in w h i c h
then
UD(mod
for
I)
was replaced
if
on an interval
I
that
every
N a I
the
holds.
then
The
the use
implying
fm-fn
[74]
if
same w a s
of r e s u l t s UD(mod
N)
the
is m o n o t o n e
5).
is u n i f o r m l y [Na n ]
is
also proved
by
of J . K O K S M A
[35]
for a l l
for a
N
of d i f f e r e n t i a b l e
following and
concepts
H.G.~IJER
(Ch.
{a n }
sequence
is a s e a u e n c e
satisfying !
and A . D I J K S M A ,
H.NIEDERREITER
fn(X)
!
measure-theoretical
[62]
shown
also
result,
sequences:
m #n,
certain
it w a s
[68] w h o w i t h
following of
(a) If
with
S.UCHIYAMA
[62]
the converse
[65],
the
class
N)
of L . K U I P E R S ,
DEN EYNDEN
and
functions
UD(mod
(mod I)
J.CHAUVINEAU obtained
of
the book
In C . L . V A N
N)
proved,
in M . U C H I Y A M A ,
[69]. Cf. a l s o
UD(mod
was
concept.
two conditions:
of c o n s t a n t
sign on
I,
and (b) T h e ~ e
exists
has
IfL-f~l
then
for a l m o s t
all
where
all
x eI
the
I
is n o d i f f i c u l t y is a n
the G a u s s i a n J.R.BURKE,
(tsj)
method. for an
lim s-~o
A
ideal
field
C
sequence
such
that
{[fn(x) ]}
in g e n e r a l i z i n g
of an a l g e b r a i c
by L.KUIPERS,
L.KUIPERS
3. A n o t h e r
[7~
generalization
sequence
integer
N,
Cl,C2,...
~ tsj j~s cj ~k (rood N)
for
m ~n
one
is
UD(mod
N)
for
7 for
a necessary
It f o l l o w s
immediately for a l l
number
r
which
J.S.SHIUE
matrix
integers
to
This
UD(mod
I)
was done
J.S.SHIUE
[753.
for Cf.
[803.
by A.F.DOWIDAR
defining
[72]:
a regular
is s a i d to b e
k=O,1,...,N-1
and
sufficient
from Proposition
integers
is n o t a n
exp{2~ianr}
field.
considered
of
for
number
N)
let
summation
T-UD(mod
N)
o n e has
I
=
UD(mod
N)
was
infinite
provided
UD(mod
H.NIEDERREITER,
, L.KUIPERS,
be a triangular
(see e x e r c i s e
is
constant
N. There
T =
a positive
aC,
N a 2
integer
condition
1.1.that
if a n d
only
for T - U D ( m o d
a sequence
if for e v e r y
{a n } rational
one has
= 0 (x) .
n~x W.A.VEECH to b e a n y r e a l ting property
[71]
considered
number of t h e
which
a stronger
is n o t
sequence
{a n }
an
condition,
integer. Veech-UD.
allowing
L e t us c a l l He c a l l e d
here
r
the resul-
further
N)).
the
sequence has,
{a n }
well-distributed,
uniformly
in
provided
for e v e r y
real
r ~Z
one
k a 0,
exp{2~ian+kr}
= 0(x)
nKx and p r o v e d
that
for
sequences
a polynomial
the c o n c e p t s
and m o r e o v e r
such
is l i n e a r or
P
with
sequence
leading
is n o n - l i n e a r
(except
the
is w e l l
equal
additive
One
shall
of p r o b l e m s set
M(f)
now prove
positive
if
by
here
sets M(f)
is to d e t e r m i n e
integers which have
N
such
that
characterizes
the
form
for a g i v e n f
is
those
X = M(f)
function
UD(mod
subsets
N).
X
for a s u i t a b l e
f.
such that
(A.ZAME
has the following n ~X
and
The
divides
"only
if" p a r t
enjoys
let
of
n,
then
stated
in the
be the
sequence
denote
by
is an a d d i t i v e
an a d d i t i v e
multiple
=
and o b s e r v e
character
character of
fn(x)
(XNj(X)
theorem.
the
residue Let ~s
Write
integers,
assume
X = { ~ = I , M 2 .... }, inte-
exp(2~ix/Nj),
trivial let
class thus
positive
function
n=I,2,..,
MI,...,Mn,NI,...,Nn,
n I + ~1 j~l
that
on the
every
(mod N).
of the r e m a i n i n g
XNj(X)
(mod Nj) . For
since
classes
j=I,2,..,
which
d eX .
is evident,
residue
N I
For
fn(X)
N/d
the p r o p e r t y
gers.
inte-
if and only
X = M(f)
property:
d
is a u n i o n
X
[72]).
We
of the
is a subset of the positive
1.2.
P
its c o e f f i c i e n t s
f
Proof.
over
if e i t h e r
of an i n t e g e r
X
(mod d)
common
generated
is
If
If
and
which
group
only
inverse
P
coincide
then there exists a function
X
that
a result
integers
THEOREM
gers,
arising
of all
if and
to the
where
distribution
is n o n - c y c l i c .
§ 2. T h e
f, the
an = [P(n)]
and w e l l
distributed
coefficient
term)
form
Veech-UD
and the
constant
of the
of
K~
on
NjZ, b e the
thus least
put
+-XNj (x))2-j
is p e r i o d i c
(mod Kn),
non-negative
and m o r e -
K n
_~1 ~ Kn x=1
fn (X) = I
Assigning
to an e l e m e n t
u
we o b t a i n t h u s a n o n - n e g a t i v e at the full g r o u p the v a l u e is p o s s i b l e
F(x) I
lira
N
~
~
N÷~
on
F(xl ~)
Z/KnZ
in p a r t i c u l a r
Kn ~ u=1
m n. T h i s m e a n s
measure.
Z/KnZ
which
attains
Thus
it
is uni-
t h a t for e v e r y
func-
(I .2)
F(u) fn (U) .
1
XNi
(u) +
t h a t for
i=1,2,..,
Kn = K~ 1 U=l~ X N i ( U ) f n(U)
N
N÷~lim ~ k ~ 1 XMi(xln))
= K~ 1
to
= fn(u) K[ I
which
u= I
It f o l l o w s I
is a p r o b a b i l i t y in
mn(U)
Z/KnZ
one has
Xn I K. ~
=
k= I
the m e a s u r e
on the group
x#n) ,x~ n) ....
with respect
defined
Z/KnZ
I, h e n c e
to find a s e q u e n c e
formly distributed tion
of
measure
n
K~ I
2
~ i=I
2 -i
one has
=
Kn ~ ( (U)×N i(u) + u=l XNj XNj (u) XN i (u))
S i nce
Kn
[ Kn
E u=1
(u) XNj
if
N i =Nj =2
(u) = ] ×Ni
0
otherwise
and Kn u=1
[ Kn ×Nj_ (u) ×Ni (u) =
0
N i =Nj otherwise
w e get I N (n) e lim ~ ~ XNi( X k ) = 1 N+~ k I 2 ~I
where
ei =
I 2 1
If m o r e o v e r on
MiZ
if N i = 2 otherwise X(x)
for a c e r t a i n
lim 1 N ÷ ~ ~"
(1 .3)
is an a d d i t i v e i ~n
N (n) ~ X(Xk ) = 0 k=1
.
character
then proceeding
of the i n t e g e r s
trivial
in the same w a y we get (I .4)
~ite
now
consecutively
trivial
on
forming
a sequence
and
(1.4)
and
for
M2Z,
we N
can
non-trivial
trivial
Xl,X2,X3,... choose
, (n)
E
all
those
No(n)
exceeding
N
1
IF
then
xj [x k
of in
No(n)
)
on
one
additive
M3Z,
on
non-trivial
such has
way
for
characters
M4Z
and
so
characters.
that
No(1)
j=1,2,...,n
By
of
(1.3) < ...
inequalities
2-n
<
Z
forth,
(1.5)
k=1 and
11
and
N ~ , In) k ~ I XNj tx k
define We
•
• .
T(n)
claim
X
has
x(y m) :
m=1
where
s = s(N)
s Z j=1
and
T(j)
(l+n). o sequence
the
s
T (n)
Z
Z
n=l
k=1
character
(mod
M j)
for
a certain
j,
then
N-R
x(x~ n~) +
Z
X(Xk s+1~
k:1
is d e f i n e d
s+l ~ j=1
(I .6)
,
x~ I) ..... x (I) (1),x ~2) ..... x (2) (2) ..... x #n) ,.. n-th term. be its properties. Let Yn
desired
additive
<_ N <
-n I < 2
by
T(j)
s ~ T(j) . j=1 (].5) w e o b t a i n
R : NUsing
view
the
is a n y
N
Z
= 2nN
that
,x~n)" (~) ....
If n o w
2--I-j -sj
of
T(n)
now
for
n
large
enough,
say
> N O(n)
T(n) (n) 2-n I ~ X(x k ) I < T (n) k=1
and
N-R
I~ k=1
] N O (S+ I )
x(x~1+s') I < [
if
N-R
2_s_ I (N-R)
otherwise
o
(s+1)
n ->N 1 (j) , in
Observe n i t y and
that
the
N o (1+s)
(N-R) 2 - s - 1 / N
same
_
applies
T (s)
to
tends
to z e r o
No(S+1)/N
for
because
N
tending
to
infi-
of
2-s
<
2SN It f o l l o w s
that
N
[1
and
s
X(Ym) [ < ~1
~ m=1
since
N _> T(1)
+T(2)
+ 0(I)
+ ... + T ( s )
we
get
N
I
lira ~ ~ N÷~ m=1
The
~ T(n) 2 - n n=1
same
X(Ym ) = O .
approach
shows,
with
the use
of
(1.6)
that
for
j=I,2,...
the r a t i o
N m~l
does
not
having
XN'j (Ym)
tend
to zero,
in m i n d
trivial
on
not divisible
every
of sum
later
integer
a sequence
1),
now
it s u f f i c e s
non-trivial form
to a p p l y
additive
exp(2~ir/N)
Proposition
character
with
1.I,
of the
a certain
integers,
integer
r,
N.
shown
positive
(i.e.
every
is of the by
2. It w a s
mN
that
NZ
and
by H . N I E D E R R E I T E R
N
one
prescribes
([75],
a positive,
a o ( N ) , a 1 ( N ) , .... aN_I(N)
and a s s u m e s
that
if
M
th.4)
divides
N
that
normed
of n o n n e g a t i v e then
for
if for measure reals
j=O,I,...,M-I
one has
aj (M) =
then N ~1
~ ai(N) imodN i~j(modM)
one c a n and 1
find
a sequence
j=O,I,...,N-I
one
lim x # { n -<x: Un - j ( m o d X-~co
Ul,U2,...
with
has
N) } = aj(N)
.
the p r o p e r t y
that
for all
If,
1.2.
It is w o r t h
by N i e d e r r e i t e r
I. We
shall
tribution
~D
uniform
also
(i)
study
set
the
{n:
following
prime
terms
to
then
one o b t a i n s
constructed integers.
(mod N)
notion
of u n i f o r m
of a g i v e n
sequence
N. To be p r e c i s e ,
distributed
two c o n d i t i o n s
= 1}
N, {u n}
set of all
weaker
those
following
(an,N)
and
of the
weakly uniformly
{a n }
j
sequence
distribution
only
are r e l a t i v e l y
provided
The
the
in w h i c h
all
the
§ 3. W e a k
which
N)
for
that
a permutation
a sequence mod
noting,
is a l w a y s
(mod N)
considered call
a~ (N) =I/N
in p a r t i c u l a r ,
Theorem
we
(mod N)
are
disare
shall (shortly
satisfied:
is i n f i n i t e ,
and (ii)
For
every
~ { n ~x: lim ~ {n ~x:
j
prime
to
an ~ j ( m o d N) } (an,N) = I }
N
one has
I ~(N)
X ~
The ling
first
condition
a sequence
buted
(mod
The that
like
of w e a k
of u n i f o r m
groups
E n~x
where The 1.1
and the
to the
1.3.
of integers
Dirichlet
XO
general
following
If
distribution
in the g r o u p
N
x(mod
theory
analogue
the p o s s i b i l i t y
weakly
uniformly
of cal-
distri-
coincides
of r e s t r i c t e d
of u n i f o r m
distribution
of P r o p o s i t i o n
integer,
with residue in
1.1:
then the sequence
if and only if for every non-principal
N)
N).
(mod N) G(N)
is a positive
~D(mod
one has
= 0( ~ Xo(an)) n~x
is the principal
proof
and we
is
character
X(an)
uniform
distribution
(mod N) leads
PROPOSITION {a n }
to e x c l u d e
3).
notion
classes
is i n t r o d u c e d
1,2,0,0,O, .... ,O,0,...
can be c a r r i e d
leave
character out
along
it to the reader.
0
(mod N).
the
lines
of t h a t
of P r o p o s i t i o n
2. F o r between
another
uniform
It is s h o w n
proof
namely,
integral
distributed
(mod N).
L e t us d e n o t e is
WUD(mod
c
N).
the p o s s i b l e case
we propose
the
is
UD(mod
may
in t h e
those
style
is e s t a b l i s h e d .
N)
if a n d o n l y uniformly
obvious).
integers
be posed,
here
a relation
is w e a k l y
is o f c o u r s e
set of a l l
M*(f)
also
distribution
{un + c }
if" p a r t
the
distribution,
N
how one
of T h e o r e m
the a n s w e r
for w h & c h
can charac-
1.2.
Contrary
is u n k n o w n
and by
following
Prove
I.
{u n]
the q u e s t i o n
sets
of u n i f o r m
[68], w h e r e
uniform
sequence
"only
M*(f)
Again
to t h e
PROBLEM
the
(The
by
terize
analogy
S.UCHIYAMA and weak
that a sequence
if for e v e r y
f
cf.
distribution
that if
X
then there exists a function
is a subset of the positive
f
such that
X =M*(f)
integers,
if and only if
X
has the f o l l o w i n g p r o p e r t y ~ If
n eX
and
prime divisors
One ver
sees
of
d
is a divisor of
n,
then
immediately,
the proof
of
b y dr
I.RUZSA,
divides
the
set of a l l
then
it is p o s s i b l e
bers
of
A
but
in p r o v i n g
subset
of o d d
To
the
I. T h e n o t i o n
say t h a t
we
NI,...,N r
let
system
provided
lira x -1 #{n<_x: X-~
of
shall
f
is n e c e s s a r y ,
difficulties.
following
case,
is
result:
Mrs.
WUD
X
in-
if o n e sets
for
A,B
all mem-
E.ROSOCHOWICZ
when
howe-
I was
into two disjoint which
B. R e c e n t l y in t h e
return
distribution
of uniform
Let
and
the
numbers
of c h a r a c t e r i z i n g
distribution
of i n t e g e r s
the
a function
conjecture
condition
to present
suc-
is a n a r b i t r a r y
integers.
of f u n c t i o n s
of s e q u e n c e s .
stated
seems
square-free
§ 4. U n i f o r m
uniform
the
t h a t he c a n p r o v e
to f i n d
the question
classes
that
for n o m e m b e r
ceeded
which is d~v~sible by all
d e X.
its n e c e s s i t y
formed
n
M(f)
NI,...,N r S
be given
is u n i f o r m l y
for a l l c h o i c e s
- (j ~n
_--cj(mod Nj)
be
and
generalized
such a system
positive
distributed of i n t e g e r s
for
for
certain
of sequences.
(mod N)
can be easily
S = < a ~ ~) , . . . , a n(r) >
M*(f)
sequel.
of systems
distribution
(mod N)
and
in the
that of weak to c o v e r of
r
integers.
We
with
respect
cj,...,c r
systems
sequences shall to
one has
j = I , 2 ..... r} = I / N I . . . N r .
10
Similarly, ted is
with
infinite
(cj,Nj)
the
respect and
=I
It
will
every
integers
for = 1
that
UD
S
is
weakly
uniformly
distribu-
(a~ I)
a n(r) ,N) = I }
Cl,...,c r
satisfying
has
(a n~I) . . . a n(rJ ,N)
if
called
t h e. s e t . {n:
of
one
an(j} - c j (rood N i)~
clear
be
if .
choice
j = 1 , 2 ..... r
~,{ n < x:
is
S
N I . . . , N. r
for
for
lira #{n-<x: X÷oo
system
to
I
j = 1 , 2 ..... r}
~ ( N I) . . . ~ ( N r)
with
respect
to
N~,...,N
r
then
m
the
sequence
2. of
The
a n~i)
first
of
is
related
introduced
to
by
these
the
case
N i)
notions
two
(mod
if
N)
notion
[74a] a n d of
evident:
the
L.KUIPERS,
H.NIEDERREITER
two
for
of
for
is
i=I • 2 s . . . .,r
a particular
J.S.SHIUE
r72c]
the
an,
bn
and
passage
i,j=O,1,...,N-1
all
to
a n -i(mod
N) } :
ai ,
l i m x -I # { n x-~oo
_<x:
b n _=i(mod
N) } :
Bi
a n -i(mod
N) , b n _= j ( m o d
and
moreover
UD
with
UD(mod
N)
is
The sition
and
following
to
N,N,...,N
furthermore
criterion
can
the be
of later
define
by
the
the
ceneral
are
said
[77]
seqnences, L.KUIPERS,
last
case
to be
notion
being
independent
,
Yij =aiBj"
has
notion
N)}
One if
system proved
sees
and
easily
only
is in
= Yij
if
all
independent the
same
way
that its
a system members
(mod as
our
N). Propo-
1.1:
PROPOSITION N I , N 2 ..... N r
satisfying
2.4.
(i]
The system
S
is
UD
with respect to
if and only if for every choice of integers b l , . . . , b r ( i = 1 , 2 ..... r) and not all vanishing one has
O sb i
exp{2~i n_<x
one
respect
the
limits
-<x:
S
of
N)
studied We
integers
lira x -I # { n x-~
exist
(mod
F77].
of
case
by M.B.NATHANSON
independence
M.B.NATHANSON
sequences,
sequences
l i m x -I # { n - < x : x+oo
are
UD(mod
strong asymptotic independence c o n s i d e r e d
and
in
is
r ~ j=l
cj)
an
b~N~-
I
] =
0 (x)
v
11
The system
(ii)
only 6f for every
S
is
choice
with respect
WUD
of characters
all of them being principal,
xi(mod
if and
NI,...,N
Ni),
not
(i:1,2,...,r)
one has
E x 1 ( a J l ) ) x 2 ( a J 2) ) ... × r ( a n(r)) n_<x
and further
to
: O(x)
this sum with all characters
principal
×i
is unbounded.
Exercises
I. (A.DIJKSMA, of
integers
x ÷~)
is
2. is n o t
H.G.MEIJER
which
UD(mod
N)
(L.KUIPERS, UD(mod
3. Show,
for
for
that
]
all
Prove
(i.e.
that
an i n c r e a s i n g
x-]#{an
~x}
tends
sequence
to u n i t y
an for
N.
S.UCHIY~'[A
N)
its c o n s e c u t i v e
[69]).
has density
[68]
Prove
that
the
sequence
[log n)
N ~I.
if
P(x)
values
is
is a p o l y n o m i a l UD(mod
N)
such
for a l l
that
N,
the
then
sequence
P(x)
of
must be
linear. 4. P r o v e , for
all
N.
that
if
O < c < I, t h e n
(The s a m e h o l d s
the
for all
sequence
c >O,
c {Z
[n c)
is
UD(mod
N)
; see J . M . D E S H O U I L L E R S
[73]). 5. real
(C.L.VAN
numbers
[N~ n)
is
UD(mod
6. P r o v e coefficients, sequence 7. is
DEN EYNDEN
is u n i f o r m l y
if
P(x)
least
one
of w h i c h
is U D ( m o d
(A.F.DOWIDAR
[72]].
N]
that (mod
if t h e I),
is a n o n - c o n s t a n t
[ P(n)]
T-UD(mod
Show,
sequence
then
the
an
of
sequence
N).
that at
[62]).
distributed
if and
N)
polynomial
is i r r a t i o n a l ,
with
and P(O)-O,
real then
the
for all N.
Prove,
that
only
~f for
ts.j e x p { 2 ~ i c j k / N }
= O .
a sequence
{cj}
k=~,2,...,N-]
of
integers
one has
s
lim s÷~
~ j=1
8. G i v e [P(n) ] 9. all
N
is
an e x a m p l e
of a p o l y n o m i a l
UD(mod
for a l l
(I.NIVEN
[6]]).
if and o n l y
a non-zero
N)
integer.
Prove
if e i t h e r
P
N, w i t h o u t
that the a
such that being
sequence
is i r r a t i o n a l ,
the
sequence
well-distributed. [n~]
or
~
is is an
UD(mod
N)
inverse
for of
C H A P T E R II POLYNOMIAL
SEQUENCES
~. P e r m u t a t i o n p o l y n o m i a l s
I. The simplest class of sequences which comes to mind is formed by sequences rational,
{P(n)}
of c o n s e c u t i v e ~alues of a p o l y n o m i a l
integral coefficients.
(mod N)
for every integer
(mod N)
if and only if the finite sequence
P(N)mod N
P(1)mod N, P(2)mod N,...,
1,2,...,N.
is called a permutation polynomial
P
with
N, hence it will be u n i f o r m l y d i s t r i b u t e d
is a p e r m u t a t i o n of the set
polynomial
P
Since such a sequence is periodic
the
(mod N) . An extensive
study of such p o l y n o m i a l s
in the case of prime
L.E.DICKSON
([971) where he sanplified and v a s t l y extended
in his thesis
previous o b s e r v a t i o n s of E.BETTI MITE
N
If this happens,
was inititated by
[51],[52], [55], E . M A T H I E U
[61], C.HER-
[633 and others. 2. Dickson considered, m o r e generally,
arbitrary
finite fields, h o w e v e r he was
in p e r m u t a t i o n p o l y n o m i a l s for that.
In fact,
(mod N)
p e r m u t a t i o n polynQr0ials in never
for
N
s e r i o u s l y interested
composed.
He had reasons
the next simple result shows that in studying permu-
tation p o l y n o m i a l s one can r e s t r i c t a t t e n t i o n to prime moduli~ P R O P O S I T I O N 2.1.
(W.NOBAUER
[65]).
(i) A polynomial
is a permutation polynomial
(rood N), for
is a permutation polynomial
(mod pap)
(ii) A polynomial
with a prime
p
a prime
p,
is a permutation polynomial
if and only if it is a permutation polynomial
and the congruence (iii) If
P(x) £Z[x]
P(~)
P' (x) =-O(mod p)
P(x) cZFx]
over
Z
N = K pap if and only if it p for every prime p dividing N. (mod p2) (mod p)
has no solutions.
is a permutation polynomial
(mod p2)
with
then it is also a permutation polynomial for all powers of p.
13
Proof. Theorem P(x) M
(i) The "only if" part is a c o n s e q u e n c e of the easy part of
1.2. To prove the "if" part it is sufficient to show that if
is a p e r m u t a t i o n p o l y n o m i a l
and
N
(mod M)
and
(mod N)
then it is also such w i t h r e s p e c t to
w i t h copXime
MN. However if
then the C h i n e s e R e m a i n d e r T h e o r e m implies that if the maps and
Z/NZ ~ Z/NZ
map
Z/(MN) Z + Z/(MN) Z
induced by
P
(M,N) =I,
Z/MZ ÷ Z/MZ
are both surjective then the induced
is also surjective and hence m u s t be a permu-
tation. (ii) If
P'(x) ~ O ( m o d p)
tion p o l y n o m i a l ~m(mod p2) mial
has a u n i q u e solution,
(mod p2).
If however
P(x) ~P(Xo)
(mod p2)
polynomial
(rood p2).
(iii)
is insolvable, and
(rood p), then for every
has
thus
p
COROLLARY.
polynomial
For a given polynomial p
such that
(mod p)
but not
pl,...,pr ~T(P);
Proof.
P
the set of all primes
the set of all integers
s ~O;
P
is not a p e r m u t a t i o n
(ii) that
P' (x) ~ Q ( m o d p)
the unique selution of
P(x)
P(x) ~ m ( m o d pk)
D
set of all primes T(P)
P(x)
is a p e r m u t a t i o n polyno-
can be lifted to a unique solution of
k=3,4,...
with
m
is a permuta-
then the c o n g r u e n c e
thus
From the a s s u m p t i o n s it follows by
~m(mod p2)
and by
P(x)
solutions,
P(x)
the congruence
P' (x o) aO(mod p)
has no solutions and thus for every
for
m
denote
P cZ[X]
is a permutation p
such that
by
polynomial
P
ql .... ,qs kS(P);
(mod p2)
is a permutation
(mod p2). Then
of the form
the
S(P)
the set M(P) al ~s pl...prql ...q
coincides with
r zO,
aj >O.
Follows i m m e d i a t e l y from the proposition.
The T h e o r e m 2.8 b e l o w will show that for
S(P)
and
T(P)
one can
take a r b i t r a r y finite sets of primes. The following q u e s t i o n arises thus: PROBLEM II~ For which di~sjoint sets
a polynomial
P
such
that
S(P) = S
and
S,T
one can find
T(P) = T ?
It seems that one should be able to deduce it using Fried's
of primes
an
answer to
solution of a p r o b l e m of Schur w h i c h we shall quote
in section 4 below.
14
§ 2. G e n e r a t o r s
1. If
p
x p ~x(mod
p)
for
is a p r i m e holding
to p o l y n o m i a l s
group
number,
to o b t a i n
not
of p e r m u t a t i o n
then
identically
of d e g r e e
not d i f f i c u l t
the
in v i e w
in
x,
exceeding
one
of the c o n g r u e n c e can r e s t r i c t
p-%.
a description
polynomials
Having
the
this
attention
in m i n d
of all p e r m u t a t i o n
it is
polynomials
(mod p):
THEOREM
2.2.
(L.CARLITZ
permutation polynomial such that for all
x
we have
is a composition
Q(x)
a fO(mod
It
=Qt(x)
the
inverse
by the
P(X)
of the
to show set
satisfying of the
is a prime and
Q(x)
of the form
(with
ax ÷ b
x p-2
that
every
transposition
{0,I,...,p-I}
the a s s e r t i o n
element
is a
P(x)
and the p o l y n o m i a l
--Q(xl (rood p)
of polynomials
suffices
t=1,2,...,p-1) Q(x)
p
then there exists a p o l y n o m i a l
and the polynomial
p))
Proof.
If
[53]).
(rood p),
u(mod
is i n d u c e d
of the
p] # O
(Ot)
(for
by a p o l y n o m i a l
theorem.
Writing
we can d e f i n e
such
u'
for
a polynomial
formula
Qt (x) = -t 2 ((x-t) p-2 +t') p-2 -t) p-2
Clearly mial
x P-2
element
proving
Qt
that
assertion
If
showed
by the pair quadratic
obvious
x #O,t
sa£isfies
this
case
are that
residue
p=5
that
Q
polynomials (mod p)
and
the p o l y n o -
interchanges
the
needs.
holds
finite was
without fields.
considered
change
for
L.CARLITZ
the a n a l o g u e
of
[53]) .
by E . B E T T I
[52]
and
the case
[97].
other one
{x+1,
linear
then
our
argument
in a r b i t r a r y
the
of
' +t') p-2 -t) p-2 ----t2(-t2x') q-2 ---x(mod p)
by L . E . D i c k s o n
2. T h e r e [553
t.
=-t2((x-t)
Earlier p=7
it is a l s o
and
that
(Note the
is a c o m p o s i t i o n
and
0
Qt(x)
Qt
possible
can r e p l a c e
ax p-2] (mod p)
where
variants the
set
a is any
in c a s e
of this {ax+b: fixed
p ~l(mod
4)
theorem.
So K . D . F R Y E R
a iO(mod
p)} u {x p-2]
integer and
which
is a
a quadratic
non-
15
residue in case
p a 3 ( m o d 4). C . ~ L L S
may be taken by the triple root
[68] proved that the same role
{x+1, x P-2, gx}
where
g
is a p r i m i t i v e
(mod p). C h o o s i n g a d e q u a t e l y sets of p e r m u t a t i o n p o l y n o m i a l s
can often give a simple set of g e n e r a t o r s full symmetric group on
p
letters.
(mod p)
E x a m p l e s of this procedure can be
found already in the c l a s s i c a l work of L . E . D I C E S O N
[97],[O1]. F r o m
newer papers on this subject the reader may consult H.HULE, [73], H.LAUSCH,
W.B.MULLER,
M U L L E R [763, C . ~ L L S
one
for various subgroups of the
W . N ~ B A U E R [733, R.LIDL
W.B.MULLER
[73], R.LIDL,
W.B.
[67].
§ 3. Hermite's c h a r a c t e r i z a t i o n of p e r m u t a t i o n polynomials
1. The following result of C.HERMITE [63] is often helpful ving that a p a r t i c u l a r p o l y n o m i a l T H E O R E M 2.3. Let
p
polynomial
(mod p)
is not permutational.
be a prime and
with its degree not exceeding
p-l.
Then
P(x) P(x)
if and only if for every
exists a polynomial
Qt(x)
in pro-
of degree
~p-2
a polynomial over
Z
will be a p e r m u t a t i o n t=1,2,...,p-1
there
such that the congruence
pt (a) -= Qt (a) (mod p)
holds for every a and m o r e o v e r
the congruence
P(x) z O ( m o d p)
has
exactly one solution. (A v a r i a n t of this t h e o r e m was proved by L.CARLITZ,
Proof.
Observe
p-1
p) -~I O(mod -1(rood p)
x j
x=O Indeed,
if
g
J.A.LUTZ
F78J).
first that one has
for
j=I,2 ..... p-2
for
j=p-J
is a p r i m i t i v e root
left h a n d - s i d e equals
(2.1)
(mod p) , then the sum on the
18
p-1
gkj
k=1 and
the a s s e r t i o n
immediately. p-1 For a n y p o l y n o m i a l P(x) = ~ aixl and i=0 Pt(x) =Pill=0 ai 'txl" Adding these equalities
obtain
for
every
p-1 pt (b)
P
t=1
2,...,p-2
and u t i l i z i n g
write (2.1)
we
b
p-1 p-1 ~ ai ~ b i - -a (mod p) i=O ,t b ~ O p-l,t
-
b=O
If now
follows
is a p e r m u t a t i o n
polynomial
(2.2)
(mod p),
then
in v i e w
of
I st s p-2
p-1 F t (b) : b =O
thus
p-1 ~ b t - O(mod b =O
ap_1, t ~ O ( m o d Noting
that
p) .
a permutation
we o b t a i n
the
necessity
note
due
to
that
p)
(2.2)
polynomial
of our they
has
conditions.
imply
exactly
To p r o v e
one
zero
their
(mod p)
sufficiency
the c o n g r u e n c e s
p-1 pt (b)
=- O ( m o d
p)
(2.3)
b=O
for
t=1,2,...,p-2
and
p-1 pp-1 (b)
- -I (mod p)
.
b=O
Let of
p
V(x)
elements,
P(p-1) . U s i n g the
= j=0 { Ajx j whose
Newton's
(Ap=I) roots
are
formulas
be the p o l y n o m i a l
over
the r e s i d u e s
p
we obtain,
with
mod
k-1 ~ Ap_iSk_ i +kAp_ k = O i=I
finite
(k:1,2 ..... p)
field
P(O) ,P(]),...,
St = p i t p t ( b ) b=0
equalities
Sk +
of
the
(mod p),
17
thus
in v i e w
sults, P(O)
of
showing
-O(mod
p)
get
If w e w o u l d
have
=- P(1)
result,
finally,
that
As
Proof.
of
xP -x
Consequently
by
If
This M(P)
vide
the
ticular
p
k
Then
leading M(P)
V(x)
-a o we
a n d the
implies with
term
of
contains
results
the
same
assertion
the
same
fact holds
see
later,
[97]
that this
a permuta(mod p).
the c o n d i t i o n was
[]
I ~t ~p-1
later made
polynomials,
P
is a g a i n
+ ... + a l ) m
with
case
that
only
of
x p-1 = x
the
theorem.
true.
P
whose
TM
and
assume
polynomial
of
and
[63] p o s e d
(V(x)-ao)m
of d e g r e e prime
the middle
a finite
for q u a r t i c
polynomial
is in f a c t
V(x) =
= x m ( x re(k-l) + ... + a I)
for a p o l y n o m i a l
S.R.CAVIOR
for e v e r y
so let (mod p)
a permutation
but do not divide
imply
1 ( m o d k)
get
set of i n t e g e r s
in t h i s
to
(mod p)
polynomial
coefficient
that the
is true.
see with
leading coefficient not divi-
monic
we get a contradiction
coincides
forms
polynomial
is a prime congruent
with its
= x m ( x k-1 + a k _ i x k - 2
and w e
coincides
same observation
be a permutation
m = (p-1)/k
corollary
DICKSON's
and
to c o n s i d e r
+a o
k).
m(k-1)
unity,
set
and hence
one c a n r e p l a c e The
p
[65].
of degree
Putting
since
[91]
k ~2
It s u f f i c e s
(V(x)-ao)m
equals
A I =-I
is a p e r m u t a t i o n
I ~ t ~ (p-I)/2. N.G.STARKOV
GF(p)
Thus
cannot be a p e r m u t a t i o n p o l y n o m i a l
p ~1(mod
(mod p).
(rood p)
in P
we have
congruence
our a s s u m p t i o n .
= x k + a k _ 1 xk-1 + . . . that
j=O,1 ..... p-1
P(O)mod p,...,P(p-1)mod
then a p o l y n o m i a l p
the
- P(p-1)
sequence
2. C O R O L L A R Y .
sible by
then
for
reSince
(I + A 1 ) P ( j ) .
contradicting
GF(p).
theorem
=
kAp_ k =0
~(x) = x p + A 1 x + A o.
the
by V.A.KURBATOV,
and
- ...
the equality and
Moreover
+ AIP(j)
shown by L.J.ROGERS
in t h e
=Ap_ I =0
A I #-I
set of a l l r o o t s of
k=1,2,...,p-2
A o =O.
= PP(j)
would
tion
we
for
A 2 =...
0 = V(P(j))
0 - P(O)
the
(2.3) that
number sextic
term.
In p a r -
polynomials
the question,
2.7) ~
di-
of p r i m e s .
of an e v e n d e g ; e e .
(See t h e o r e m
2 the
factors
We
whether shall
18
§ 4. E x a m p l e s
I. N o w w e the
linear
dividing
shall
produce
polynomial a.
(mod p)
p~b.
p
Indeed,
ent
(mod p)
tely,
if
(mod p)
for e v e r y
prime
polynomials
respect
t o at
least
primes.
The reader
2. T o g i v e
the
following:
Tn(X)
cient
for
2.4.
with
arccos
For
n ~I
integral
ax n + b
~6
his
examples
n=I,2,..,
(Ax+B) 2 + C
and
not
ax 2 +bx +c if
pla,
is c o n g r u -
also
image
immedia-
polynomial
L.E.DICKSON
are permutational
also
the
[97] with
appropriate
as an e x e r c i s e .
shall
in v a r i o u s
p
so its
sees
(n,p-]) = I . which
result
we
and
One
is a p e r m u t a t i o n
and d e t e r m i n e d
may rediscover
polynomial if a n d o n l y
(mod p).
satisfying
one prime,
p
Clearly
prime
t h e n our p o l y n o m i a l
form
of d e g r e e s
can b e d e f i n e d
= cos(n
LEMMA
lynomial
p~a
non-trivial
They
a,
polynomials.
for e v e r y
a quadratic
residues
polynomial
characterized
polynomials.
of t h e
I + (p-I)/2
the binomial
that
an odd p r i m e
not divide
to a p o l y n o m i a l
at m o s t
that
for
does
of p e r m u t a t i o n
is p e r m u t a t i o n a l
It is a l s o o b v i o u s
is p e r m u t a t i o n a l
contains
examples
ax + b
now consider manners,
-I < x
the
the C e b y N e v simplest
being
put
x)
the f u n c t i o n
coefficients,
Tn(X )
of degree
can
be
n
and
extended
to a po-
leading
coeffi-
2 n-1 .
Proof. for a l l
Let
n zl
fn(X) = s i n ( n the
following
(i)
Tn(X) , Un(X)
(ii)
deg T n =n,
arccos facts
x) = ( 1 - x 2 ) ~ U n ( X ) .
We c l a i m
that
hold:
c a n be e x t e n d e d
to p o l y n o m i a l s
over
Z,
deg U n =n-1
and
the
(iii)
The
leading
Since
Tl(X)
=x
coefficients
and
U1(x) = I
identity
fn+1 (x) = xf n (x) + ( 1 - x 2 ) ½ T n ( X )
we obtain
,
of
Tn
(i)
-
and
Un
(iii)
are
are e q u a l
true
for
to
2 n~1.
n=1.
Using
19
U n + I (x) = Tn(X)
+X~n(X)
(2.4)
and
T n + I (x) = xT n (x) - f q (x) fn (x) : x T n (x) - X 2 U n ( X )
and
this
leads
to
(i)
9~e s h a l l h a v e
for ~ e b y ~ e v other
which and
In/2]
~ k=1
polynomials
fascinating
either
(iii)
(-I)
k
to u s e
-
neither
the explicit
D
form
-~-
(which c a n b e p r o v e d o f them,
treating
orthogonal
is a s o u r c e
argument.
{ (nkk) + (nk~ 11) } 2 n - 2 k x n - 2 k
properties
to a n y b o o k paper
b y an e a s y r e c u r r e n c e
no opportunity
(x) = 2n-lx n +
T n
-
- U n (x)
of m a n y
by recurrence)
referring
the
polynomials
arithmetical
nor many
interested
reader
o r to I . S C H U R
results
concerning
[73] Tn(x)
Un(X). The
lemma.
only
further
property
It m a y b e u s e d
gonometric
LEMMA v =v(x)
to g i v e
of
Tn
we need
a definition
is g i v e n
of
Tn
in t h e n e x t
independent
on tri-
functions.
2.5.
Let
x
= x - (x2-I) ½
be be
a real
the
number
roots
of
and
u =u(x)
let
y2-2xY
+I.
= x + (x2-I) ½ ,
Then
U n +V n T n (X) --
2
and
u n -v n U n (x) = 2 (x 2- ]
Proof.
Ul (x) = I recurrence
For and
)½
n=1
the
the e q u a l i t i e s
argument.
[97]:
(2.4)
results and
immediately
(2.5)
pave
from
the w a y
Tl(X) = x ,
for an e a s y
D
3. N o w w e c a n p r o ~ e by DICKSON
assertion
the m a i n
result
of this
section
ohtained
first
20
THEOREM
If
2.6.
p
is an odd prime number,
is a permutation polynomial Proof. To p r o v e M.FRIED tivity ting
[70]. than
from
ponding then
(Dickson's
last
residue
t(x)
original
lemma,
of
proof
if
We
with
rather
the
is an i n t e g e r
due
to
on s u r j e c -
observation,
and
of the
Tn
=1.
argument
concentrated
as an e l e m e n t
p = An(X)
(p2-1,n)
an e l e g a n t
start
x
treated
Tn(x)mod
we use
Tn).
that
(mod p)
the r e s i d u e
where
sufficiency
on i n j e c t i v i t y
the
I ( t n <x)
the
then the polynomial
if and only if
(mod p)
x
resul-
the c o r r e s -
field
GF(p),
equals
+ t-n (x))
is for e v e r y
x
an e l e m e n t
of
satisfying
GF(p 2)
t 2 - 2 ~ t +I = 0 . In p a r t i c u l a r
t(x)
Let of the
+ t -I (x) = 2x .
(n,p2-1) solutions
:I,
(2.6)
x I /x2(mod
p),
and d e n o t e
i=I,2
for
t
by
one 1
of
2 x i = y + y-1
in
G F ( p 2) . If
n
t 1
with
-n
+
t 1
n
=
tln =t2-n. elements, (2.6)
From that
Tn
=
c
An(Xl)
=An(X2) , hence
,
This and
shows
it f o l l o w s
that
Xl = x 2 '
the n e c e s s i t y Tn
to T h e o r e m
cannot 2.3.
the d e l i b e r a t i o n s
n
that
group
either
-n
n
-n
t l , t I , t2,t 2
thus w e m u s t
the m u l t i p l i c a t i v e
implies
To prove
t 2
y2 - c y +I
the p o l y n o m i a l Corollary
+
c {GF(p).
Since
then
-n
t 2
a certain
the e q u a t i o n
by
T n ( X I) - T n ( X 2) (rood p) ,
have of
t I =t 2
either
G F ( p 2)
or
all n
satisfy
n
t I =t 2
or
is c y c l i c
t I =t[ I
must
of
p2-I
hold,
which
a contradiction. observe
first
that
he p e r m u t a t i o n a l Thus
assume
above
is not
a permutation
integers
x~,x 2
incongruent
= tn(x2).
Since
(n,p+])
we k e e p
polynomial (mod p)
in the
case
(mod p)
(n,p-1) ~ I
hecause
of the
#~.
in m i n d
that
(mod p)
for w h i c h
in o r d e r
it s u f f i c e s
one w o u l d
have
to s h o w to find tn(xl ) =
21
~rs(X)
= Tr(Ts(X))
I+D°
Expressin~
image
oT
t(x)
we may assume
~
in t e r m s
consists
(uP+l-l)(uP-l-l)=O. all lyin~ in the (p+l)/n
COROLLARY.
If
Proof.
mials
ax n + b
son number, N
for
order,
then
is a permutation
P(x)
p.
j=I,2 ..... s
and
from Dirichlet's
(mj,p-1) =1
prime number
of Fried's
to the T h e o r e m
theorem
polynomials that
any
theorem
2.6 provides
us with polynomials
for infinitely many primes. polynomial
It was
of prime degree which
for infinitely many primes must be a composition and Ceby~ev polynomials.
if the same assertion
giving a sufficient
condition
of finite permutation
groups.
He called an integer
for
N
in a later paper
applications
Since then,
number was commonly
of this result,
the assertion
called Schur's
for every
that every
conjecture.
later,
in terms
(I.SCHUR N
a Dick-
of degree
to publish
to be a Dickson number
Although
[333 )
and promised
he never returned integer
is
of binoN
is true for all polynomials
whose proof he promised
he was able to show that this condition holds to give arithmetical
for
N ~ 2 ( m o d nl...nsml...mr).
and stated also a theorem,
this subject.
of Ceby~ev ~ polynomials and of volwnomials
j=1,2 ..... s
§ 5. Consequences
proved by I.SCHUR [23]
p)
to prove that there are infinitely many primes
but this follows
permutational
are at most
must be a pair x I ~ x 2 ( m o d
is a composition
many primes
applied to the progression
I. The corollary
the
non-zero integral aj's and arbitrary integral bj's
(nj,p2-1) =I
which are permutation
that
u ~ G F ( D 2) ~ i t h u p+I = i
and ~e o b t a i n that there
for
(nj,6) =I
for infinitely
j=1,2,...,r,
immediately
of
[]
P(x) e Z [x]
It suffices
divisor
of G~(~ 2) for ~ h i c h
elements
for u n, thus there
(with odd mj,
satisfying
elements
p+l
j=I,2 .... ,r), ~n an arbitrary
polynomial
p
image of t(x)
with
ajx mj +bj
for
are
t n ( x I) = in(x2).
Tn I ..... Tns
of t(x) we o b t a i n
of those
There
possibilities
such that
that n is an odd prime
to
is a Dkckson
22
After KURBATOV
certain
established These
new classes
[49] a n d U . W E G N E R by M.FRIED
methods
sent a proof
lying
[70],
outs£de
of F r i e d ' s
two corollaries
of it.
which
in § 3:
we quoted
THEOREM
mial over
2.7.
Z
of D i c k s o n
[28] the who the
utilized scope
result, The
and
first
(H.DAVENPOPT,
of even degree
numbers
truth
limit
was
the
of R i e m a n n
surfaces.
lectures
ourselves
If
[63]).
we do not pre-
to a d e d u c t i o n
of
posed
by S . R . C A V I O R
p(x)
is a polyno-
then it can be a p e r m u t a t i o n p o l y n o m i a l
only for a finite number of primes
(mod p)
theory
a question
D.J.LEWIS
f o u n d b y V.A. conjecture
of t h e s e
answers
were
of S c h u r ' s
p,
i.e.
the set
can-
M(P)
not contain infinitely many primes. Proof. Fried's
If the
theorem
set
e a c h of t h e m b e i n g a binomial
that
that
contains
infinitely
is a c o m p o s i t i o n
equal
ax n +b.
it f o l l o w s assume
M(P)
P(x]
either
Since
for all
to a c e r t a i n
~eby~ev
f r o m the d e f i n i t i o n m,n
all p o l y n o m i a l s
many
primes,
of p o l y n o m i a l s
o n e has
which
the d e g r e e
of
polynomial
of C e b y ~ e v
Tmn(X)
Tn(X)
in t h e
or to
polynomials
=Tm(Tn(X))
occur
t h e n by
Pl (x], o..,Pr(x)
hence set
we may
{PI,...,Pr}
r
have
prime
lows
that
either
indices. at l e a s t
Since one
T 2(x) = 2 x 2 -I
it c a n n o t
Pj
must
or
for
with
~ M ( P i) is f i n i t e . H o w e v e r i=I a n d w e a r r i v e at a c o n t r a d i c t i o n .
Certain
special
n=10
was
settled
of t w o w a s
T(P)
disjoint
2.8.
sets of primes. over
Z
consequence
finite
THEOREM
of T h e o r e m
in H . L A U S C H ,
occuring sets
N
many
this
hence
even
primes
known
the
W.NOBAUER
and
of F r i e d ' s
In b o t h so t h e
equals
earlier.
case when
it f o l -
it e q u a l s
n.
intersection
2.7 w e r e
[67] and
in the c o r o l l a r y
n
interM(P)
For
[97],
cases
polyno-
the case
is a p o w e r
[73].
theorem
shows
to P r o p o s i t i o n
that 2.1
the
sets
c a n be
S(P)
arbitrary
of p r i m e s .
(W.NOBAUER
[66]).
Let
S,T
be finit~ and disjoint
Then t~ere exist infinitely many polynomials
with the property,
and only if
i~i d e g Pi
4 or 6 it w a s shown b y L . E . D I C K S O N
by D.R.HAYES
treated
2. A n o t h e r and
cases
n=2,
a certain
infinitely
section
m i a l s of d e g r e e s
equals
be of e v e n d e g r e e ,
ax n +b
be permutational
P
that an integer
has no prime divisors
outside
not divisible by a square of any prime from
N
belongs to the union
T.
SuT
P(x) M(P)
if
and is
23
Proof. Otherwise define
If
S oT
let
for
q
p
is empty,
be any p r i m e w h i c h
in
=
where
ap
nomial
22
integer
of d e g r e e
Po(X)
p r i m e not d i v i d i n g
exceeding
1+ap
which
for all
p cS uT u {q}. By c o n s t r u c t i o n for
by
M(Po)
p e S uT
D
of
the p r o d u c t
satisfies
p cS
Varying
large degree,
a
satisfying
distribution
simple result reduces
pk
P' (x) ~ O ( m o d
polynomial
M(P o)
for hence
2.1 w i l l
contains
pcT.
By T h e o r e m
if we d e n o t e S uT
and put
show that
P(x)
in this way p o l y n o m i a l s D
distribution (mod N)
the p r o b l e m
(mod N),
so let us
of p o l y n o m i a l
sequences.
to the study of p e r m u -
(mod p) :
P(x) c Z [ x 3
is
if and only if it is a permutation
gruence
is an be a p o l y -
(mod N) of p o l y n o m i a l s
P R O P O S I T I O N 2.9. (i) A polynomial P(x) ap for N =~ p if and only if it is W U D ( m o d f dividing N. (ii) A polynomial
p2
our a s s e r t i o n .
distribution
look n o w at w e a k u n i f o r m polynomials
that
do not lie in
we can o b t a i n
only uniform
tation
a
Po(X)
is a p e r m u t a t i o n shows
to P r o p o s i t i o n
As far we c o n s i d e r e d
following
and
Let
f i n i t e l y m a n y primes,
of t h o s e of t h e m w h i c h
§ 6. W e a k u n i f o r m
The
2.1
b u t d o e s not c o n t a i n only
t h e n the C o r o l l a r y
our needs.
of a r b i t r a r y
Po
and P r o p o s i t i o n
can c o n t a i n
P(x~ = D P o ( X )
p(p-1)
peT.
(rood p)
(mod p)
2.7
and
satisfies
for all
a l s o all p o w e r s
S uT
if p = q
2a
- Pp(x)
of
if p c T
is an a r b i t r a r y
arbitrary
all e l e m e n t s
w i l l do.
if p £S
x ap x
exceeds
P(x) = 2 x 2
S uT u {q}
I x Pp(x)
then the p o l y n o m i a l
p)
over
Z
WUD(mod
N)
pap)
for each prime
p
WUD(mod
pk)
polynomial
has no solutions
xo
is
for a prime power (mod p),
satisfying
and the conp~P(Xo).
24
Proof. WUD(mod is
(i)
MN)
It is e n o u g h
if and
WUD(mod
MN)
that
(T,P(x))
=(j,N)
=1,
only
to s h o w
if it is
and we d e n o t e =1,
then
that
if
WUD(mod
by
obviously
(M,N) =I M)
and
then
P(x)
WUD(mod
N).
A(T)
the n u m b e r
of
x(mod
A(MN)
=A(M)A(N)
and
so,
T)
if
is
If
P(x) such
(i,M] =
then
#{I~<x_<MN~ P(x)
=-i(mod M),
P(x)
zj(mod
_
N)}
A (M) A (N) ~(M)~(N)
'
hence
N #{x
rood M:
P(x)
-i(mod
M) } = ~ { x m o d
WiN: P(x)
-i(mod
#{X rood M/q: P(x)
-i(mod
M),
P(x)
- j(mod
#{x m o d MN:
---i(mod M),
P(x)
-= j(mod N)}
M) } :
N) } +
jmodN (j ,N)=I
P(x)
=
jmodN (j ,N) #I
A ( M ) A (N)
+
~(M)
_
and
A(M)A(N) @ (M)
+ #{x m o d
M:
P(x)
-i(mod
M) } (N-A(N))
so
#{x m o d
showing the
#{x rood M: P(x) - i(mod M ) } - ~ { x mod N: P(x)---j (/roodN)} = E jmodN (j ,N)#I
M: P(x)
that
same
P(x)
is
assertion
for
Now assume =(j,N)
#{x
=1,
that
A(M) M) } - ~(~)
WUD(mod
M) . I n t e r c h a n g i n g
M
and
N
we o b t a i n
N.
P(x)
is
WUD(mod
M)
and
WUD(mod
N).
If
N)}
=
then
m o d MN:
= #{x rood M:
_
-i(mod
P(x)
-i(mod
P(x)
A ( M ) A ( N ) _ A(MN) (M) ~ (N) @ (}~)
M)
,
P(x)
_-i(mod M)} # { x
- j (rood
N) }
rood N:
P(x)
=
-j(mod
(i,M) =
25
thus
P (x) (ii)
among must
is
If
~D
(mod MN) .
P(x)
is
~D(mod
P(1) ,...,P(p) also
occur
tion one needs Proposition
one
then
every value time,
only
2.1
pk),
and
a(mod
UD(mod
to r e p e a t
to o b t a i n
I. T o c o n c l u d e
the
story
l e t us m e n t i o n
A polynomial
f
over
Z
distribution
when
(mod p)
results
H.DAVENPORT,
is s u f f i c i e n t l y then
f
large
C.R.MAC
also
CLUER
H.LAUSCH,
S.D.COHEN tional
[70]
In c e r t a i n
has no
This was
For rings
W.NOBAUER
of a l g e b r a i c
(mod p)
Every
mial.
over
Z
provided to
Zn
(th.
component This
n
the map
can be
closure
f eZ[x3, respect
later
to
p,
was
pro[681.
shown by thus excep-
all p r i m e s
p.
in fact,
and
p
with
respect
a proof
of p
by K.S.WILLIAMS
inverted,
(mod p),
absolu-
is
conjecture
It w a s
for
if
f(x)
is s u f f i c i e n t l y
is a l s o
of p e r m u t a t i o n
see H . N I E D E R R E I T E R ,
the notion variables.
variables
to
given
p. in
of a p e r m u t a t i o n
A sequence a
to
[791.
polynomial
f1' .... fn
of
n
permutation vector
[f1(xl .... ,x n) ..... fn(Xl .... ,xn~
mapping
vector
first
+
S.K.LO
polynomials
of a permutation
is c a l l e d
Ix I ..... xv]
was
if
is u n n e c e s s a r y ,
[671 and
a one-to-one
generalization
which
8.81).
several
in
induces
p
of t h e n o t i o n
generalize of
8.31).
on
is e x c e p t i o n a l
by D.R.HAYES
integers
also
th.
polynomial
(f(x) - f ( y ) ) / ( x - y ) ,
with
given
polynomials
result
f
that
(mod p) . T h i s proof was
[731,
then
[731
to t h e c a s e
(mod p), Zn
f),
of p o l y n o m i a l
factor
is e x c e p t i o n a l
Another
Cohen's
a generalization
polynomials
f
permutation
permutation
p aC(deg
2. O n e c a n
of
[66J.
are
established
H.LAUSCH,
and
p)
p a r t of
in t h e a l g e b r a i c
conjectured
polynomial
W.NOBAUER
cases
is a n o n - l i n e a r say
[631
that the restriction
polynomials
large,
prime
is a p e r m u t a t i o n
ved by (Cf.
D.J.LEWIS
O(mod
observa-
and g e n e r a l i z a t i o n s .
bf t h e p o l y n o m i a l
tely irreducible, i.e. r e m a i n s i r r e d u c i b l e GF(p).
so
this
exceptional ((or uirtually-one-
is c a l l e d
irreducibles
thus
D
related
p,
into
p),
and c o m m e n t s
-one) w i t h r e s p e c t to t h e p r i m e factored
once,
After
of the c o r r e s p o n d i n g
of u n i f o r m
certain
~D(mod
occurs
follows.
assertion.
§ 7. N o t e s
sequences
p) # 0
p)
the proof
our
it is a l s o
of
(Z/pZ) n
is c a l l e d
considered
a
onto
itself.
permutation polyno-
by W.NOBAUER
[641,
who
26
dealt by
with
polynomials
several
nomial tic a n d The
authors.
vectors
last
We m e n t i o n
same
result,
without
obtained
trary
finite
WELLS
[72].
Cf.
a permutation
if is a p e r m u t a t i o n rivative
does
solved
polynomial
not vanish
the
finite
4. R e c e n t l y
nal.
They
produced
P(x)
over
several
5. F i n a l l y
all
the
polynomials.
N =
ring with
fields,
such polynomials
K =GF(q)
if a n d o n l y
n)
and
. Later
its d e -
J.V.BRA~EY is r e p l a c e d
unit.
defined
the polynomial
K
GF(q)
C.
L.CARLITZ,
field
over
n/2
an a r b i -
R.LIDL,
a finite
case when
also of Theorem
over
[721,
matrices
where
in t h e
was
analogue
by J . V . B R A W L E Y , over
poly-
[821 c o n s i d e r e d as
those
P(x)+x
complete
permutation
is a l s o p e r m u t a t i o -
of d e g r e e s
smaller
than
of u n i f o r m
distribution
6 and
examples.
let us n o t e
system
of p o l y n o m i a l s
ch.
§ 4
I.
permutation
and K.H.ROBINSON
finite
characterized
GF(q),GF(q2),...,GF(q
G F ( q N)
for w h i c h
determined
n×n
pursued
of o d d c h a r a c t e r i s -
H.NIEDERREITER
of
commutative
was
in t w o v a r i a b l e s
considered
in
who
field
~721
a polynomial
problem
H.NIEDERREITER
mapping polynomials polynomials
in
analogous
b y an a r b i t r a r y
vectors
of the ring
topic
on t h e c h a r a c t e r i s t i c
In R . L I D L
was
that
[71~,
quadratic
R.LIDL,
generalization
[75], w h o p r o v e d
induces
E761
also
This
a finite
restriction [72bi.
rings.
R.LIDL
over
for p e r m u t a t i o n
field.
3. A n o t h e r J.LEVINE
here
case described
by H.NIEDERREITER
2.2 w a s
arbitrary
in t w o v a r i a b l e s
in t h e
obtained
over
that
with
respect
is u n i n t e r e s t i n g ,
"'''Pr
of p o l y n o m i a l s
if for
i=1,2,...,r
is
the
t o a set of m o d u l i
since UD
study
one
with
(mod N i)
and the moduli
NI,...,N r
statement
for w e a k
holds
easily
respect
the p o l y n o m i a l
analogous
sees
to
as d e f i n e d
that a system
NI,...,N r
Pi
is a p e r m u t a t i o n
are
pairwise
uniform
if a n d
of in P it... only
polynomial
relatively
prime.
An
distribution.
Exercises
I. prime
(L.CARLITZ ~ 1(mod
polynomial
[62ai) . P r o v e
3)
(mod p)
then
that
if
the p o l y n o m i a l
for a t l e a s t
one
p
is a s u f f i c i e n t l y
xCP+2)/3+ choice
of
ax a.
large
is a p e r m u t a t i o n
27
2.
(L.CARLITZ
suitable
choice
permutation 3.
[62a3) . P r o v e
of
polynomial
(L.CARLITZ
choice
of
a
that
a / O(mod
p
p)
is a p r i m e
a7,
the p o l y n o m i a l
then with
x(P+1)/~+
ax
a
is a
(mod p).
[63]).
Prove
that
the p o l y n o m i a l
for any
xCP-~)/2+
prime
ax
p
with
a suitable
is a p e r m u t a t i o n
polynomial
(mod p). 4. D e t e r m i n e polynomials 5.
for
which
(mod p)
enlarged -+a = ~ ,
nomial
f
p #2
A, B, C,
elements a
are no s u c h there
nomials
r(x)
with
that
one has
D
[55]).
are p e r m u t a t i o n
then
is a r a t i o n a l
set of r e s i d u e
the
usual
there
function
classes
conventions
exists
(I/0 = ~ ,
a permutation
poly-
mapping
Prove
that
residue
det(~
every
B
D ) / O(mod
permutation
(mod p)
in case
p) .
of the
of the p o l y n o m i a l s
K.H.ROBINSON polynomial
polynomials
polynomials
exist
(loc.
A
satisfying
x+1
field
and
p ~1(mod
of
axP -2
4)
and
otherwise.
(H.NIEDERREITER,
are p e r m u t a t i o n
8.
~
a #O)
is a s u p e r p o s i t i o n
led a c o m p l e t e
p
if
of the
for
is a q u a d r a t i c
a non-residue 7.
which
= f((Ax+B)/(Cx+D))
(K.D.FRYER
where
that
by an e l e m e n t
such
polynomials
primes.
a permutation
a ~ =~
(mod p)
suitable 6.
and q u a r t i c
[62b]) . P r o v e
induces
I/~ = 0 ,
r(x)
cubic
suitable
(L.CARLITZ
(mod p)
with
all
of d e g r e e
Prove 2.
(mod p) . P r o v e
of d e g r e e
some w i t h
cit.).
[82]). (mod p)
degree that
p-1
A polynomial
P(x)
is c a l -
and
x +P(x)
if b o t h
P(x)
that
p #2,3
and
p-2,
do n o t
exist
if but
then
for c e r t a i n
there primes
p-3. there
complete
mapping
poly-
CHAPTER LINEAR
§
I. A n o t h e r studied
class
consists
of
I.
of
III
RECURRENT
Principal
sequences
SEQUENCES
properties
for w h i c h
linear recurrent
uniform
sequences
distribution
{Un} , d e f i n e d
was
by
the
condition
U n + k = a o U n + a l U n + I + ... + a k _ l U n + k _
where
k
{u n}
and The
is a f i x e d
oldest
century
and
is u s u a l l y
(or s o m e t i m e s by
example
Fibonacci
brated
positive
ao,al,...,ak_
I
are
of
such
integer, fixed
associated
F o =O,
F I =I,
(3.1)
(n aO)
called
the
order of the sequence
numbers.
a sequence with
sequence d e f i n e d
by
1
by
goes
rabbit
back
to
breeding.
Fn+ 2 = F n +F~+ I
in w h i c h
case
the
the
thirteenth
It
is
and
indices
the
cele-
F o =F I =I become
shifted
I). To
every
f(x)
sequence the It term
LE~ (3.1)
{u n}
...
values
and assume
there
to
{u n}
corresponds
{un}.
determined
find
an
by
polynomial
It
is o b v i o u s
that
its
associated
polynomial
the
I.
explicit
formula
for
the
general
sequence:
be a sequence
of rational integers s a t i s f y i n g
that its a s s o c i a t e d p o l y n o m i a l
integral coefficients.
the
- a°
Uo,Ul,...,Uk_
recurrent
Let
(3.1)
a s s o c i a t e d with
difficult
a linear
3.1.
-
is c o m p l e t e l y
initial is n o t
of
satisfying
= x k - ak_ixk-1
the p o l y n o m i a l
called
and
sequence
Assume further
that
f(x)
has rational
28
r
f(x)
:
e.
~ j=1
(x - a j)
is its canonical K
0
factorization,
the splitting field of
f
with
al, .... a r
distinct
over the rationals.
and denote by
Then the following
holds: (i) There
with
d e g Pi
exist polynomials ~ ei -I
with coefficients
P1,...,Pr
such that for
(i=I,2 ..... r)
in
K, the
n = 0 , I , 2 ....
equality r U n
(3.2)
Pi (n) a n
=
i=1
holds. If furthermore
(ii)
minant
of
e I =e 2 =...
f, then for
and
= e r =I
D
denotes
the discri-
one has
n = 0 , I , 2 ....
r
=
u n
~
~ja~
j=1
with certain gers of (iii)
k
and
which m u l t i p l i e d
by
D
become
inte-
of the form
(3.2)
satisfies
(3.1)
set of all
solutions
{u n}
with suitable
a o , a l , . . . , a k _ I. (i) O b s e r v e
complex
since
Vl .... 'Vr eK
Every sequence
Proof. with
numbers
K.
terms
everysuch
Uo,...,Uk_1, Now direct
Eij
=
solution
which
shows
that
a solution
of
it has
elements
(3.2) that
already
in
is f u l l y
for
V
over
determined
the c o m p l e x by the
arbitrarily,
the e q u a l i t y
j=I,2, .... r;
V
of
finite is
(3.1)
numbers
and
sequence
k-dimensional.
f(s) (aj) = 0
i=O,1,...,ej-1
(j=1,2,...,r; the
sequence
{nis~}
since
show
space
utilizing
forms
lity
the
c a n be p r e s c r i b e d
evaluation,
s=O,1 .... ,ej)
that
is a l i n e a r
k
with
(3.1).
Pi(x)
in c a s e w h e n K
write
set
{Eij }
it f o r m s
The
a basis
being un
polynomials
are r a t i o n a l
is l i n e a r l y
independent,
of
implies
with
V.
This
complex
numbers
these
the
and equa-
coefficients,
To
coefficients
lie
30
ei-1 E j=o
Pi (x) =
put here
consecutively
equations
resulting
terminant
of t h i s
and
(i=1,2,...,r),
~ijx ~
x=O,1,...,k-1
from
(3.1).
system
and
Since
is n o n - z e r o
the and
solve
the
ai's
are d i s t i n c t ,
lies
in
system
K. T h i s
of
linear the d e -
proves
8ij e K
(i) r e s u l t s . (ii)
system, nant,
whose
(iii)
was
This
Note,
that
2. L e m m a UD(mod
associated
ciated
polynomial
UD(mod
p)
for
Proof. integers of
f
and
k
serve
If
has
of t h e
linear
is for all
which
determi-
choices
of
i,j
obviously
lies
in
K.
f r o m the o b s e r v a t i o n
that
due
to
combination
of the
us to
recurrent
of t h e
coefficients
the p r o o f sequence
Eli'S.
of
(3.2)
D
f(x)
was
used
p
only
only
elementary
splitting
field
its d e g r e e .
p
roots,
divides
theory K
of
It f o l l o w s
integers
recurrent
simple
if
of a necessary
condition
in the c a s e w h e n
the
roots.
is a linear has
f(x)
a prime
suitable
determinant
(i) is a V a n d e r m o n d e
D2Sij
no m u l t i p l e
{u n}
We u s e h e r e
of t h e
that with
now
of a l i n e a r
3.2.
of
(ii).
polynomial
THEOREM
the
DBij,
results
integrality of
2.1 w i l l N)
D. T h u s so is
is a l i n e a r
the
in the p r o o f
for
and
assertion {u n}
= e r =I
in the p r o o f
equals
integer
sequence
only
solved
square
an a l g e b r a i c
the
e I =...
In the c a s e which
of f.
then the
ideals Let
D
from part
AI,...,A k
sequence
its
can
{u n}
asso-
be
discriminant
of
f.
in the r i n g
ZK
of
be t h e
(ii)
from the
and
discriminant
of the
field
K
last
lemma
we have
r
Du
for
=
~ j=1 A j a j
n=1,2,...,r.
Let
p
be a rational
prime
not dividing
D
and
let
t t pZ K = p1 ! ... Ps s
be
the
factorization
ideals. ence
of
the
ideal
generated
by
p
in
ZK
into prime
Observe now that for j=1,2,...,s and i=1,2,...,k the sequth a s its p e r i o d Tij ~in ( m o d PjJ) ~ - e i t h e r e q u a l to I (in c a s e w h e n
31
~i {Pj)
or to the o r d e r
of
~i
in the m u l t i p l i c a t i v e
group
of r e s i d u e s
tj (mod Pj
)
prime
the
numbers
the
sequenc e If n o w
residue must
in this
I,
of this
%K(IJ)
=
for c o p r i m e
CK(I)
shall K
of s=1,
now
same
show
common
that
T
ti=I,
PI=pZ
multiple
of
is a p e r i o d
group
ideal
I
of E u l e r ' s
of times,
this
leads
of r a t i o n a l
since
of
field
i.e.
K #Q.
the n u m b e r
classes
Z K. We n e e d
~-function:
to a c o n t r a -
is a d i v i s o r
i.e.
T
is i m m e d i a t e ,
now that
K,
of r e s i d u e
every hence
numbers,
this
T
p. A s s u m e
of
full p e r i o d
integers
and
of
its
number
that
field
are r a t i o n a l
be a m u l t i p l e
analogue
of Denote of
(mod I),
only
two
prime
simple
pro-
its m u l t i p l i c a t i v i t y
CK(I) CK(J)
ideals
I, J,
~[ PII
the p r o d u c t
and
the
explicit
is t a k e n
over
n o r m of
all p r i m e
the a b s o l u t e
i,j,
is a d i v i s o r of f. N(Pj) = p o (j=1,2, .... s),
Tij
of
K/Q
formula
(I -N(P) -I)
denotes
If n o w
least
(ii)
in e v e r y
the
is the
f
non-zero
= N(I)
the d e g r e e
the
3.1
then
appear
in the m u l t i p l i c a t i v e
for e v e r y
perties
where
case
T
the Euler's function of the
CK(1)
elements to
p. We
it c a n n o t
p)
will
when
al,...,a k
~(p) = p - 1 by
UD(mod
(mod p) by
by
from Lemma
p) }.
is
In the case
roots
Denoting
infer
{Un(mod
be d i v i s i b l e
since
Pj.
we
{u n}
class
diction. all
to
Tij
ideals
dividing
I, and
N(I)
I, i.e. N(I) = # Z K / I . S i n c e for e ~ e r y t. CK(Pj J) w e see that T divides CK(PZK) then
tlf I + ... + t s f s
equals
[K:Q],
thus
s ¢ K ( P Z K ) = p[K:Q] ~
_ (1 - P
E fj)
= P
s
(pfj -1
.
where
E :
s ~ j=1
(tj - 1 ) f j
It f o l l o w s the
integers
K/Q [743,
and
that
p
tl,...,t s
according
cor.2
.
can d i v i d e exceeds
CK(PZK)
I, in o t h e r
to the d i s c r i m i n a n t
to th.4.8.)
every
such
p
only
if at
words
p
theorem must
least
one of
is r a m i f i e d
in
(see e . g . W . N A R K I E W I C Z
divide
the
discriminant
of
32
K,
hence
also
tradicts
the d i s c r i m i n a n t
our choice
COROLLARY.
sequence
is
Proof. since only
The
p)
to t h i s fails
corollary
UD(mod
I, 3 a n d
sible
by
formly who
prime
is a l s o To
see
the
distributed.
gave
equals
UD(mod
true.
consider
L 1 =3.
Here
sequence, terms equals
hence
the
3.5)
this
to t h e
proved
without
last
there
theorem by
is the
is no are
1, 3, 4,
sequence
is d i v i -
to w h i c h
by L.KUIPERS,
the use
converse
defined
(mod 5)
respect
p)
be
polynomial
time
reduced
with
UD(mod
that the
sequence
associated
Ln
is
and
the assertion.
the c o n v e r s e t h e Lucas
x 2 - x -I
it m u s t
4 and no term of our
first
proof,
sequence
to T h e o r e m
however
of
case
then also
N,
However
this
This was
N)
dividing
is n o i n t e g e r
an e l e m e n t a r y
con-
of 5.
in this
5, t h e F i b o n a c c i
p
first
there
f, b u t t h i s
If the Fibonacci
[72a3).
be a power
(see t h e c o r o l l a r y
so the p e r i o d
5. T h u s
must
if it is
+ L n + I, L o = I ,
since
[713,
polynomial
for the F i b o n a c c i 5)
N
equals
But
show later
in g e n e r a l .
as
J.S.SHIUE
associated
p=5.
of t h e p o l y n o m i a l
D
then
N)
for e v e r y
Ln+ 2 = L
2,
UD(mod
its d i s c r i m i n a n t
shall
same
p.
(L.KUIPERS,
in c a s e
UD(mod
We
of
D
Ln
is u n i -
J.S.SHIUE
of algebraic
[72bi
number
the-
ory. From Theorem ble
(mod p),
M.HALL
[38a],[38bl,
distribution and
3.2
k
occurs.
is t h e
in the p e r i o d nificative
follows
that
cannot
have
then we
order
in t h i s
in t h o s e
§ 2. U n i f o r m linear
I. N e c e s s a r y ence of order
k
We
and
shall
treat
here
cases
are very
be able
ralizable
cases
Tp
when
T
f(x)
is i r r e d u c i -
However,
as s h o w n
certain
uniformity
is t h e m i n i m a l
times.
residue
(Of c o u r s e
period a(mod
(mod p) o f
of U n ( m o d p)
appears
this r e s u l t
is c o m p a r a t i v e l y
P
of p)
by
is s i g -
large.)
second-order
recurrences.
UD
conditions
are known
only the case
to f i n d
p).
a given
distribution
sufficient
to b e
in o t h e r should
if
{un} , t h e n
T p / p + 0(p k / 2 - I / 2 )
only
UD(mod
case nevertheless
In fact, of
if the p o l y n o m i a l
cumbersome simpler
to l a r g e r v a l u e s
of
in the c a s e
recurrent k=2,3
k=2,
since
the conditions
and
one h a s
the f e e l i n g
conditions k.
for a l i n e a r
only
which,
may
be,
sequ-
and
4.
obtained
that
would
one be g e n e -
33
~
start with
THEOREM
3.3.
a criterion
(R.T.BUMBY
for
UD(mod
[75],
p)
with
M.B.NATHANSON
a prime
Let
[75]).
r e c u r r e n t sequence of second order with integral
p.
be a
un
terms. Denote by
its a s s o c i a t e d p o l y n o m i a l and let
f(x) = (x-a) (x-b) = x 2 - A x - B
p
be
a prime number. (i) If
p
only if
then
the d i s c r i m i n a n t of
If
(ii)
are rational
a,b
divides
p
divides
B(U~ -AUoU I -Bu~)
Proof. A s s u m e
p
if and only if
p)
then
is
un
the d i s c r i m i n a n t of
f
B(Ul-aUo).
UD(mod
if and
p)
and does not divide
= a b ( u I - a u O) (u I - b u O)
first
the discriminant all c a s e s
UD(mod
and does not divide
are not rational,
a,b
is
un
f
D
must
that
of
f
un
is
UD(mod
vanishes
divide
we
get
D. R e m e m b e r i n g
p).
Since
in c a s e
from Theorem
3.2
a=b
that
in
that
3.3)
Un+ 2 = AUn+ I + Bu
we
see t h a t
implies
if
p
divides
u n ~ uoAn(mod
is d i v i s i b l e
by
p
never divisible
by
Similarly,
p)
for
then p,
p)
for all
so w e c a n n o t
obtaining
(ul-aUo) (ul-bu O) = u ~
(i)
-AUoU 1 -Bu~
of the prime
ideal
integers
of t h e
field
we
a ab(mod
P)
n=I,2,..,
and hence tional
p~u n
2. H a v i n g
quadratic hence have
which
proved and
must
discriminant
cannot is e v e n
expect
u I ~auo(mod
n a2
possibility
f(x)
P)
implies
(nal)
that
lie
dispose over
of o u r
quickly
GF(2),
be congruent
2)
viz.
~x 2 +l(mod
case.
UD(mod 2),
thus
or
un
get
from
and by
if
A
is
p,
(3.3) in the c a s e
then with
in the r i n g
P)
which
ZK
in v i e w
of of
u n ~uoan(mod
P)
either
no
un
P
P
since
by
and
p, r u l i n g
conditions
we p=2.
x2+x
to o n e
of them.
proved
part
If
21)
uo
we obtain
x 2, x2+I,
(mod 2)
in t h i s again
p
of t h e c a s e
I, t h u s b y t h e a l r e a d y
thus
in
be all divisible
the n e c e s s i t y
first
UD(mod
for
un
they must
polynomials
f(x)
get
as b e f o r e
or a l l
integers,
sufficiency,
K(a)
then we as b e f o r e
dividing
Un+ 2 {2aUn+ I -a2un(mod
this
p) .
is d i v i s i b l e
P
and
and otherwise
UD(mod
plul-aUo
choice
for
have
p)
If n o w e i t h e r
n z l
a contradiction
a proper
and
Un+ 2 £ Aun+1(mod
n=1,2,3, ....
PlUn
if in the c a s e
u n ~ Uoan(mod (ii)
B, t h e n
f(x)
=x2(mod
2)
in
they are out
turn
2)
p~.
to t h e i r
and The
ra-
UD(mod
There
of the
is i m p o s s i b l e .
Un+ 2 ~Un(mod
lies
are
four
x2+x+l, last two theorem then
we un+ 2
It r e m a i n s and we
shall
the
34
get
UD(mod
Now
it
stated In
2)
is
easy
in
the
the
if to
and
only
check
if
that
uo
the
and
uI
are
of
obtained
conditions
base
argument
opposite
parity.
coincide
with
those
theorem.
general
case
we
shall
our
on
the
following
lemma:
LEMMA
K, P
a prime
by it and
f(x)
let
r
Let
3.4.
ideal
let
=
a,B,7,6 of
N(P)
=
unique
solution
6r ~l(mod
x mod
P
Proof. becomes
serted
is
(p,r)
(It when
n(mod
p),
=I
and
by
,
of
class
Having consider
~
we
be
is divisible
the
~O(mod
P) .
of
imply
P)
r)
the
the
r
s ~0
values
value
existence
the
6~(mod
of
~ EZ
the
same
of
P)
also
such
that
.
n
the
is
by
~,
rational.
As
rlN(P)
can -1
be =p%
as-1
system
integer
have
f(h)
h =h(~),
=O(mod
P)
uniquely and
determined
since
~(mod
r)
attains
follows.
P) the
case
~(mod
exactly
because
that
first
(mod
settled
the smallest
that for any
=- n (rood p)
noted
the
(i.e.
P)
are
determined
a rational
lemma
6(mod
and assume
uniquely
so
X
of
P))
there
assumptions
is
the
is
nor
define
Then
value
P)
should
P
residue
we
Our
pr) . C l e a r l y
values,
~
field
P)
Z.
f(h)
a fixed
=- ~ (rood r)
satisfied
(mod r
that
n(mod
about
get
in
13n) 6 ~ + y = O ( m o d
Since
X
For
fixed.
(~ +
neither
number
of the congruence
~ O(mod
has a r e p r e s e n t a t i v e such
order
with
~x) 6 s + Y
pr)
that
in an a l g e b r a i c
(e + ~ x ) 6 h + X,
integer
(a +
we
such
Furthermore
pt.
be the m u l t i p l i c a t i v e
positive
h(mod
K
be integers
the
degree, contains
case
when
p=2 f
has
last
condition
i.e.
t=1,
rational before a double
is
since
trivially in
that
satisfied
case
every
integers.) assume root,
now i.e.
p
to b e f(x)
odd.
= (x-a) 2
First
35
with
a eZ
un =
holds and
and
for
n aO
p)
Un
suitable
thus
denotes
congruence
(i) s h o w s
shows
every
for t h i s
rational
that
by
M o =B 0 ,
M 1 =ula
that
in t h i s
purpose
case
a
pr
-u o
we
order
of
that
satisfying
Bo = u o
aa
get
a(mod
is o n e
of the periods
every
pr
class
we
It f o l l o w s
an i n t e g e r
p)
among
residue
Bo,6 I.
denoting
the multiplicative
to s h o w t h a t
sequence and
with
-u o
(M ° + M 1 n ) a n ( m o d
r
remains
3.1
(3.4)
and putting
-
If n o w
Lemma
(B o + B l n ) a n
B I =ul/a
~1(mod
p~a(u I -aUo).
(mod p)
utilize
of
consecutive will
Lemma
3.4.
p)
then
un(mod
elements
appear
last
p) . It of this
exactly
In fact,
the
r
for a n y
times
integer
h
we have
Un+ h - u n - { (Mo + M 1 n + M 1 h ) a h
applying
that
lenmla w i t h
is l e g i t i m a t e , u I - a u o, class among
since
contrary
(mod p)
(mod p) Now
a,b
which
is e i t h e r
being
Our
is a p r i m e
/ O(mod shows
P).
un =
get
n =
has
distinct
ideal
here
this
with
integers
in this p
B = M I, y = - ~ , MI
of
then
obtain un
K,
that every
that
i.e.
then
~1, B2 c K
r
times
residue
class
the
f(x) = (x-a) (x-b)
splitting Q
or
a : b(mod we have
P)
P=p).
field
of
f,
to its q u a d r a t i c
case may be restated
K=Q
divide
residue
exactly every
(which
also
frequency.
roots,
field
P =p
it w o u l d
occurs
implies
the r i g h t
simple
in c a s e
suitable
n=0,1
and
two
dividing
(Of c o u r s e
we
sequence
to the r a t i o n a l
$i an + B2 bn
BI =
u
attained
equal
divide
assumptions)
terms,
f
~ =Mo+MIn,
would
by the
assumptions
that with
Putting
pr
that
with
P
p
to o u r
is a c t u a l l y assume
extension.
6 =a,
if
attained
consecutive
- (M e + M 1 n ) }an (rood p)
as
and Lemma
follows:
if
a b ( u I - a u o) 3.1
(ii)
we have
(n-> O)
and
(ul-uob)/(a-b),
solving B2 =
(Ul-U°b)an + (Uoa-Ul)bn a-b
the r e s u l t i n g
(Uoa-Ul)/(a-b)
= Ul
a n -b n a -b
system
for
BI, ~
we
thus
uoab
an-1 a -b bn-1
(3.5)
36
Due
to
a H b(mod
at _b t a -b
P)
t-1 ~ aJbt-J j=O
-
to the h a n d a b l e
t h i s m a y be s i m p l i f i e d
-= ta t-I (mod P)
in v i e w
of
(t _>I)
form
nulan-1
-
U n
--
u
~ an-1(nul
(n-1)uoan(mod
P)
thus
-
(n-1)Uoa) (mod P)
n
and we
see
that
multiplicative that
u
(mod P)
order
of
of the p r e v i o u s
has
a(mod
case:
pr
for its p e r i o d ,
P) . T h e
we have
for
remainder
with
r
being
o f the a r g u m e n t
the
imitates
h a I
un+ h - u n - a n + h - 1 ( ( n + h ) u I - ( n + h - 1 ) U o a )
- a n-1 (nu I - (n-1)Uoa)
-
= a n-1 ((e + S h ) a h -~) (rood P)
with
e =nu I -(n-1)Uoa, Now we have I.
N(P) = p .
implies
that
~ =u I -Uoa
to d i s t i n g u i s h In this
among
pr
class
(mod P)
which
Since
N(P) = p
we
case Lemma
consecutive is a t t a i n e d
infer
elements
by
that every
this
so e v e r y
class
(mod p)
same
frequency,
proving
uniform
II.
N(P) = p 2 .
(Hence
K ~Q
and
apply
Lemma
K.)
Here we that
shall
in t h e p r o o f ,
where
= x 2 -Ax -B,
D
of
K,
ding
then
the
gree)
rational
p~2
of
have
plD,
p#2,
p~d
of a quadratic
p21D R. T h u s
and
f
number
a =
P)
(A + p R / d ) / 2
with
is u n r a m i f i e d
d
If
f(x) =
the d i s c r i m i n a n t
all prime
ideals
f i e l d a r e of t h e D =p2R2d
and with
to
is t h e o n l y p l a c e
be used.)
(because
{u n}
first we have
(This
and
so w e m a y w r i t e
get a -= (A + pR~/-d)C - A C ( m o d
will
sequence
p
3.4 h o w e v e r
of
times. is a t t a i n a b l e
(mod p).
thus
satisfied.
the assumption
discriminant
integer
are
and
residue
r
(mod P)
in the
p =P
every
occurs
class
appears
applicable
{u n}
distribution
is t h e d i s c r i m i n a n t
in v i e w
we must
also
its a s s u m p t i o n s
of
sequence
residue
the
ascertain
two possibilities:
3.4 is d i r e c t l y
and
in
residue
.
between
C =
with
divi-
first
de-
a certain
(p+I)/2
we
37
Putting
for
- T(mod
which
shortness
P)
,
8 - U(mod
U =u I -UoC
we
obtain
now
P)
gives
(~ + S h ) a s - a
Since
are
~O(mod
satisfied
elements times.
--- (T + U h )
T,U,A,C
(a+Sh)a s -e are
T =nu I -(n-1)uoC,
in
of Now
p) our
{u n} only
presentative every
residue
same
frequency,
case.
every
Z
that
Hence
in
every
UD(mod
p)
linear
recurrences
valent
form:
{U n }
is
are
TO
apply
this
a given
UD(mod
relation
of
is
useful,
of
the
lower which
uI
uI °
.
o
...
u2 o
.
.
°
.
Uk_ I
vanishes The
states
it
that
have
we
r
a re-
obtain
{u n}
is
an
with
characterized
p)
in
of order
necessary Un(mOd old
this
[77]
the
that the
second-
following,
and the sequence
p)
sequence Here
appears
which
sequence
3.4
consecutive
P)
classes
that
equi-
to h a v e p)
result
happens
does
of if
v n =Un(mod
p)
lower than 2. a way
L.KRONECKER
and
of
satisfy
only
if
checking, a recurrence
[813 (§ V I I ) the
determinant
matrix
uo
lowing
order.
Lemma
if and only if its a s s o c i a t e d poly-
p)
(mod
criterion
recurrent
the
such
of
pr
(mod
attained such
UD(mod
does not satisfy a r e c u r r e n c e relation
whether
be
J.S.SHIUE
which
nomial has a multiple root
of
class
p
h eZ
follows.
H.NIEDERREITER,
order
can
in
find
conditions
sequence
are
occurs
may
the
residue
there
p)
one
that
classes
since
(mod
and
shows
residue
p)
integers
this
attainable
and
class
3. N o t e
rational and
those
in
(AC) s - T ( m o d
... .
uk
(mod paper
Uk_ I
.
.
.
.
.
uk .
.
.
...
p). of
.
U2k_ 2
For
another
proof
H.NIEDERREITER,
characterization
of
of
this
J.S.SHIUE
recurrences
of
see M.WILLETT
[77] third
contains degree,
[76]. also
the
which
are
fol-
U D (rood p) : Assume
that
the
sequence
its
associated
un
is
un(mod
p)
polynomial
a third is
not f
order
recurrent
a recurrent
satisfies
sequence,
sequence
of
such order
that ~
2 and
38
f(x)
with or
-= (x-a)2(x-b) (mod p)
p~ab,
then
a ~b(mod
ratic
un
p)
is
and
non-residue
UD(mod
p=2
or
(mod p)
p)
if a n d o n l y
finally,
and
(Cf.
also M.J.KNIGHT,
A similar ences
of order
four
J.S.SHIUE
plicated
refrain
and
so w e
in the [80].
general
case:
THEOREM
settled
3.5.
Let
= x 2 - Ax
If
N
UD(mod
A / 2(rood 4),
41N,
S ~ 3(rood 4) ,
(c)
91N ,
A 2 +B
~ O(mod
In these three cases
In t h e Theorem
3.3
we
shall
turn
[75]
now
for
to the
N =pk).
of order two, whose
and
get
occurs
its discriminant.
D =A 2 +4B
then
is
{u n]
p
UD(mod
N)
dividing
if and only
N, p r o v i d e d none
9).
prove
that
distributed
in the c a s e s
(a),
(mod N).
(b],
(c) o n e
distribution.
(a) w e m a y
(i) w e
in f a c t n e v e r
we
sequence
is not uniformly
{u n}
uniform
case
com-
cases holds:
41N ,
First
modulus C.T.LONG
for all primes
p)
(b)
expect
appears
extremely
- B
(a)
Proof.
its p r o o f
become
let
of the f o l l o w i n g exceptional
cannot
of a p r i m e
is a given integer, is
{u n}
and
sequ-
modulus
be a recurrence
{U n}
be its a s s o c i a t e d polynomial,
if
paper,
recurrent
them here.
[75]; W . A . W E B B ,
terms are rational integers f(x)
distributed
conditions
from quoting
the c a s e
(R.T.BU~Y
same
The
§ 3. G e n e r a l
1. H a v i n g
p)
a is a q u a d -
F80]) .
of uniformly
is s t a t e d
in H . N I E D E R R E I T E R ,
a /b(mod
p#2,
p)
W.A.WEBB
characterization
if e i t h e r p),
the c o n g r u e n c e
(u 2 - 4 a u I + a 2 U o )2 ~ 4 a 2 U o U 2 ( m o d
holds.
a {b(mod
assume
2~a
in t h a t
and
N=4. thus
case.
If
If
f(x) = (x-a)
A =-2a
=2(mod
f
distinct
has
4)
2
then
using
so t h a t roots,
(a)
then again
39
by Theorem implies
3.3 w e
due
Thus
Un+ 2 ~ B U n ( m o d
then
O = U n + 2 j (mod 4)
uniform
distribution
assume
to
(a),
assume
that
A ~ 2(mod
for
n a3
the
If h o w e v e r
this
classes
are d i v i s i b l e
the case.
This
ruling
out
that case tails
Writing
of
un(mod
eight
B ~2(mod
of
un(mod
terms we
2.
proves
the
which
ger and assume
in w h i c h
x,
y,
2x+4y,
"only
of the
2x+4y,
2x+4y, of
"if"
h
terms
in this
case.
first
since
period
UD(mod
impossible. 9).
If b o t h
so a s s u m e
this
[5,2],
[7,5],
with
the worst
4x,
for
three
(mod 9), leave
we
7x+5y,
thus
5x,
in
viz. occurs.
compute
2y,
after
or six,
full d e -
long preperiod 9)
the
patience
since
the
case,
and
[8,8].
a little
either
A
n o t to b e
six possibilities
of 9. We
7y,
terms
o n e o f the r e s i d u e
4)
cases
3x+6y,
the
2x+4y,
a preperiod
first 3x+6y, of
six.
of t h e
be a sequence
theorem.
shall use
{an}
is
with the property,
(n=I,2,3,...)
two easy
of integers,
N~. Let also
sequence
bn = -nh{J~ = a j + h n
4)
all
the
and
y =u1(mod
part we
that the sequence
a positive integer
then
would
Uo,Ul, 2u1+Uo, 2 U o + U l , U o , U l , . . .
only
{an}
not d i v i d i n g
Un+ 2
distribution
length
if" p a r t
un
thus
2 1 u O)
a rather
9),
then
4)
(in c a s e
be a multiple here
can,
4)
one o b t a i n s
Let
3.6.
a prime,
we have
and w e c a n n o t
even
UD(mod
[4,2],
be now established:
p
n
4)
uo z2u I +Uo(mod
n ~2,
will
LE~
A ~2(mod
is e v e n w e
same parity,
is in t h e s e
consider
get a period
In the p r o o f
results,
9):
6x+3y,
for
of uniform
9]
were
following
case
A
A2+B ~O(mod
[2,53,
x =Uo(mod
shortness
7x+5y,
This
and
and
j zO
B ~ 1(mod
in this
and
the
9)
should
B
making
[1,8],
the possibility
9),
and
31u n
us with
in the p r e c e d i n g
the period
for
14 t e r m s 7x+5y,
four
3 then
since
are e q u a l
N =9
to t h e r e a d e r ,
A z4(mod
and
as
for a l l
is no
(in c a s e 4)
twice
B mod9]:
the period
If
so t h e r e
assume
by
4).
assume
are o f t h e
appears
leaves
[A m o d 9 ,
Proceeding
4)
is of p e r i o d
(c) w e
is e v e n
if for a c e r t a i n
and
then either
u n =un(mod
(mod 4)
In c a s e
that
and
u I ~2u o +Ul(mOd
sequence
and holds
N =4
also
Uo,U I
same parity,
sequence
pairs
If
u o / u l (mod 2)
2 1 u I) or of t h e
so w e m a y
4).
A
(mod 4).
again
z 2un+ I +Un(mod
B
4)
(b) w e
have
thus
4)
In c a s e
be even
un
21D :A 2 +4B
41A.
u n ~O(mod have
get
~
N1
auxiliary
a given integer
be a n o n - n e g a t i v e
UD(mod
that for
N1pa) . If there j=O,1,...,h-1
inteexists the
40
(mod NI p~)
is constant
for all large
n,
say
b~ J) ~ y(j) (mod Nlpa) and m o r e o v e r
pl+a
not divisible by
c =c(j)
there is an integer
such
that
bn+ I -b n z c ( m o d
holds for all large
Proof.
We m a y
n,
j ~ [O,h-]].
p~IIc(j)
we o b t a i n (mod p1+a),
that period. which
{ n sx: and if
B
assume
The s e q u e n c e
that
p that
is a r e s i d u e
p1+~)}
U D ( m o d N i p l+a)
is
b~j) (mod p1+e)
hold
y(j) (mod pa)
if
x
occurs
is a r e s i d u e
for
is p e r i o d i c
of it. O b v i o u s l y
to
7 ~y(j) (mod p~),
b n ~ ¥(mod
{a n }
that all a s u m p t i o n s
is a p e r i o d
congruent
If follows
satisfies
then the sequence
certainly
N o w fix class
p1+a)
class
and since
every
exactly
n aO. residue
once
in
(mod p1+e)
then
= x/p +0(x) (mod N Ip1+a)
class
satisfying
8 ~y(j) (mod N1p1+a) , then
# { n s x: b n H S ( m o d
Observe
now that
lows that the number residue
class
N1p1+~) } = x/p + 0(x)
since
the s e q u e n c e
of indices
(mod N1Pa)
j
equals
{an}
such that h/Nip ~
is x(j)
and thus
UD(mo d NlP~) lies
it fol-
in a given
for every
B
#{n -<x: a n z ~ (rood NI p1+~) } = ~ #{m ~<Xh-~: bm(~) -~B (rood N1p1+~) } J (where
the sum is t a k e n
and this
equals
over
those
j's for w h i c h
(x/ph +0(x))
= hx/N1p]+ah
+ 0(x)
= x / N i p 1+a + 0(x)
J which
shows
y(j)
further
that our s e q u e n c e
is
UD(mod
N1p1+~).
~8(mod
N1P~)),
4~
LEM~I~ 3.7. If K, P s
ZK,
is defined by
sp ~ s+w ,
PS+WllxP -7
if
sp > s + w
is defined by
Proof.
If
y =x-l,
I +py +yP(mod ps+wllpy,
COROLLARY such
then
holds with
x,y
x ~I (mod P)
and
x p : (1+y) p = 1 + p y + c p y 2 + y
an i n t e g e r
p
c, and in v i e w of
follows.
are integers
that with a positive
s
of
one has
if
sp ~ s+w ,
pS+WllxP - y P
if
sp > s + w
w
p,
,
pS+WlxP -yP
where
the prime number
y c P s \ P s+1
our assertion
I. If
an integer of
PWllp.
p2S+W)
pspIlyP
x
then
if
w
number field,
containing
PSllx-1
p S + W l x P -I
where
K
and
P
Psllx-y
a prime ideal of
and
P~xy,
then
,
has the same m e a n i n g as in the lemma.
Proof. L e t Then
is an algebraic
a prime ideal of
and
ZK
K
z
be an i n t e g e r
of
x P _ y p ~ y p ( z p _ l ) (mo d pS+W+1)
it s u f f i c e s
to a p p l y the
COROLLARY
2. Under
lemma
K and
satisfying
to the n u m b e r
the assumptions
yz ~ x ( m o d
pS+W+1) .
x - y ~ y ( z - 1 ) (mod pS+W+1) z.
thus
D
of corollary
1 one has for
3=1,2 .... PS+jWlxpJ
-yPJ
if
sp a s+w ,
Ps+~WllxPJ -yPJ
if
sp > s + w
Proof.
Follows
by i n d u c t i o n
.
f r o m the p r e v i o u s
3. N o w we turn to the p r o o f of s u f f i c i e n c y in the theorem.
It is e n o u g h
t h e n the t h e o r e m w i l l of the m o d u l u s ,
to p r o v e
f o l l o w by
counted
of the c o n d i t i o n s
the f o l l o w i n g
induction
according
corollary.
proposition,
in the nunfoer of p r i m e
to t h e i r m u l t i p l i c i t y :
stated since factors
42
PROPOSITION
of Theorem
3.5.
is at least that none
Let N
equal
to the
largest
of the f o l l o w i n g
prime
three
p
N =2 u
(u Z I),
A ~2(mod
4),
N =2 u
(u > I),
B ~3(mod
4),
(c)
p = 3,
N =2u3 v
is
{u n}
if
UD(mod
v > I) ,
A 2 +B
and
p)
of
N
the conditions
Assume
that
and assume
p
further
holds:
p = 2,
(u > O ,
satisfying a prime.
divisor
conditions
p = 2,
UD(mod
=O(mod
9) .
then it is also
N)
pN) .
Proof.
First
polynomial N =N1pk
has
since
un
by
pfN I
treat
root,
and
by Theorem
v n =aUn,
for
shall
the
i.e.
k ~0.
3.3
without
h zO
we
(i) w e m u s t spoiling
and
h = pk(p-1)N1~(N
p k N ] lh
and
If w e w o u l d
have
(3.4)
-c])
Put
.
thus
.
also
p]+kN])
,
then
0 =
(cllc2h]ah
the
with
associated
a eZ.
Write
get
UD-property.
c] = a u o + c 2 n .
pkNj)
when
(a,N1p)
get
a h ~I (mod p 1 + k N I)
Vne h -V n ~ O(mod
we
have
the
I) = N ] ~ ( p 1 + k N 1 )
Vn+ h -v n ~ O(mod
case,
= (x-a) 2
(n=O,1,...)
Vn+ h -v n = an((c]+c2h)ah
c 2 =u I -au o
f(x)
(n=O,1 .... )
(au o + ( u l - a u o ) n ) a n
every
easier
From
(u O + ( u l a - l - u o ) n ) a n
and
vn =
we
a double
with
un =
Then
and
(b)
Then,
with
be a sequence
{u n}
be an integer
(a)
UD(mod
and
3.8.
Let
-c I ~ c2h(mod
p 1 + k N I)
now
=I
we may
Thus
replace
43
however
Theorem
however
pXN I
and
and
thus
p[q(q-1) neither
and
so we
Because n ~j(mod tions
h)
q
p
3.6
that j
which
has
that
simple
that
p 1 + k l h , then
of
divisible an(mod
sequence it we
In this
q
it c a n n o t by
{Un}
case
PINI~(NI),
then
hence
divide
N1
is c o n s t a n t
for
p1+k.
p 1 + k N I)
obtain
the a s s o c i a t e d
roots.
NI,
~(NI) . B u t
is not
the
if
dividing
divide
h
and a p p l y i n g
readily
and
the v a l u e
let us a s s u m e
(x-a) (x-b)
p~c 2
is a p r i m e
cannot
see
it f o l l o w s
of L e m m a
(3.5)
implies if
for any
4. N o w =
3.3
satisfies
UD(mod
the a s s u m p -
p 1 + k N I)
polynomial
for
u n.
f(x) : X 2 - A x - B
we u t i l i z e
the
=
formula
implies
Un+ h : a h ( ~ l a +62bn) +5 2bn (an-b n) = ahun + (Uoa+Ul)bn aha -b-bh
for all
h zO,
hence a h -b h a-b
Un+ h -Un = ( a h - l ) U n + cb n
with
c =Uoa Write
field it), any
of P
K
prime
ideal
of
ideal
q
that
find
with
pXNI,
equals
either
ZK
define
in the n e x t
Observe we
N =N1pk
(hence
utilized
once
- u I.
again f
prime
(3.7)
lying lq
sequence
due
to L e m m a
an i n t e g e r
hk
Q
over
by of
k ~O,
~ O(mod
Nipk )
(ii)
a
hk - b hk a -b
J O(mod
pkp)
(iii)
a
be the
This
w
splitting
extension
by
PWllp
of
and
for
notation
will
be
the p r o p o s i t i o n
will
follow
following
conditions:
lemmas.
3.6
hk _ b h k a a -b
K
p, d e f i n e
qAqNa-b.
and
(3.7)
satisfying
(i)
let
or a q u a d r a t i c
the
three
,
and
Indeed, = Uj+hm
hk
~ b hk
(i) and the
~ 1(mod N1pkp)
(iii)
imply
congruence
that
for
h =hk,
bm+ I - b m ~ O ( m o d
j=0,1 ..... h-1
NIpk)
holds,
i.e.
and
bm :
bm(mod
N1pk)
44
is c o n s t a n t .
Further,
a h -I ~ O ( m o d
pkp)
ference
since
for
(due to
bm+ I -b m
n =j+hm
(iii))
we have
it f o l l o w s
is c o n g r u e n t
(mod pkp)
b n {bJ(mod
from
(3.7)
that
pkp)
and
the d i f -
to
cb j a h - b h a-b '
which
is a c o n s t a n t ,
a rational the Lemma
3.6
Before we
First we
ourselves
reduce
If
for
Proof.
The
the e x c e p t i o n
that
LE~MA
p =3,
and
tisfied
of
3.10.
k =0
(i) a n d
~w
or
If
for
that
k=O
p =3,
p ~2
(i)
either
(iii)
3.7.
bm+ 1 -b m
is
and
thus
in the c a s e
p=2
we get
-
for
The
s =I,
then
a prime
(iii)
N 1 =I
and
p #2
w =2,
also
lp = I ,
s ~I
(i)
-
k =r+l.
p =3,
follows
(with
w ~2,
k=1:
for
or
same holds
p =3,
since
resp.
satisfying
p =2,
is v i o l a t e d
and
k=O
is an i n t e g e r
when
sp > s+w
s(p-1)
= CK(PNI
ho
Since
(mod p1+k)
observe
since
of the c a s e
2 generates
for
(ii). also
to the c a s e
hr
1 to L e m m a
however
p =2
hk
satisfies
except
the a s s u m p t i o n
either
of
k ~I,
and
hr+ I =ph r
truth
then
to
the p r o b l e m
r =0,
from the Corollary
sp ~ s + w ,
holds
obvious.
r al
then
holds
where
to z e r o b y
assertion
the e x i s t e n c e
becomes
3.9.
k =r,
same
same
is a p p l i c a b l e .
can restrict
with
the
we prove
the assertion
LEMMA
not congruent
integer,
w =2,
(iii) The Ip = I .
immediately
for
(ii), w i t h
the only case
s =Ip) . Indeed, and
p ~2
if
it follows
w =2.
with
the
exception
ideal
the
conditions
of the case [i~
-
(iii)
when are
sa-
with
]-~ qlq) qlN I
where
the p r o d u c t
Proof. by Theorem
a
ho
The
is
taken
conditions
3.3 w e h a v e
- b h°
over
all p r i m e
(i) and
(iii)
(ab,PN I) = I
I (rood PN]
N
q!N1
qlq)
and
.
ideals
are
q
clearly
thus
dividing
satisfied,
N 1.
since
45
Moreover
by the
ahO . b h o a -b
-
same
we have
ho-1 • ~ aJbh°-j-1
ho-1
j=o
j--o
and to s h o w t h a t either
theorem
divides
a ~b(mod
a h°-1
h o ~O(mod
P)
p(p-1) (p+1)
observe
or
q11N1. In fact, if for p r i m e ~q by q [INI, t h e n
P),
-- ho ah°-1 (rood P)
that
any p r l m e
p1(P1-1) (pi+1)
ideals
q
hence
(3.8)
divisor
of
for a c e r t a i n
dividing
NI
ho
prlme
we d e f i n e
eq
aq+aq-1 h
=
(N(P) -I)
]--[ N(q) qiN1
o
and
N(P)
all
Pl'S
=p
or
are
in c a s e w h e n plp1+1
and
generates P~h o •
p2,
N(q) = P l
smaller
than
for c e r t a i n since
p~
for a s u i t a b l e
the o n l y
q" Pl
we have
this
possibility piN(q)
is p o s s i b l e
ideal
q
in
Z K, but
lemmas
we
obtain
only
this
P11N1.
of
Since
Plho
arises
-I = (Pi-1) (Pi+I), if
p =3,
c a s e we d i d
Pl = 2
thus
and
exclude.
2
Thus
D
5. F r o m when
these
p =3
we h a v e
and
to d e a l
LEMMA
either with
If
3.11.
2 remains
p =3
then with
r =0.
If
our
prime
the r e m a i n i n g
will be satisfied for then
or
p,
Pl < p
a prime
(N(q) -I)
proposition in
cases.
h O =N I
p =3,
w =2,
{i) - (iii) will be s a t i s f i e d with Proof.
m al (i) -
We
start
since
for
(iii)
get
m
m =0 the
the c a s e
w =2 let
and
hp = I ,
Here
and
for N I =2 m
is e v i d e n t .
except
Ip = I .
Now
p =3.
the conditions
h I =3N I
r =O.
the p r o p o s i t i o n
follwoing
or
p a3,
The
~i) - (iii)
A2+B JO(mod
a)
r =I.
with
a suitable
conditions
form:
2m
a2
-b a -b
a2
_b 2 a-b
m
with
K
First
for all
- O(mod
2 m)
J O(mod
P)
(3.9)
m
(3.10)
and
a 2m ~ b 2m
-= 1(mod
2raP)
(3.11)
48
Let
P2
P~Ila-b. O(mod all
m
and
we
- O(mod
the
first
the
second
For
the
to b e
and
the
third
and
we
I to
define
get
Lentma
u
by
a2-b 2 3.7
that
for
we of
be
that
since
a ~b(mod
P)
we
get
ma2m-1
one
i O(mod
has
to
f(x)
recall,
~ x 2 -I (mod
Corollary
P)
that 2)
2 to Len~na
since thus
3.7
in
un
is
assu-
a 2 ~ b 2 ~1 (mod the
case
of
congruent
mod
(mod
a 2 ~(mod
3
P)
P)
has to
must
a(mod
P)
a rational hold
and
for
its
integer, we
obtain
double
i.e.
a(mod
the
w =2,
take
a -b
the
lepta
Ip = I
in
and
following
the since
case
r =O.
N I =2 m
In
the
(m ~ 0 ) ,
case the
r =I
we
conditions
shape:
_= O ( m o d
3-2 m)
/ O(mod
3"2m.p)
a3-2 m _b3-2 m a-b
3.2 m
If (3.9)
z
m al resp.
P)
immediately
P) .
establishes assume
f(x)
a3-2 m _b3-2 m
a
21
prime
to
so
a 2m = b 2m ~ 1 ( m o d
(iii)
note
~ 2
have
the
polynomial
GF(3),
This
holds.
condition
_- I (rood 2 m)
the
a must in
-
P~II2 2)
2 m)
condition
2)
2 leads
= b 2m
Since
(i)
let
~O(mod
Corollary
- O(mod
condition
application
a 2m
to
2,
za-b
from
2m-I " 2 m "-I ~ a3 b -8 j=O
-
UD(mod
over
have
P~%)
that
med
lies
follows
check
a 2m - b 2m a-b
root,
it
dividing =a+b
- b 2m
showing
ideal
ideal
have
a-b
To
a prime
(a2-b2)/(a-b)
~ u + t ,J ~2
m ~I
a2
be
Since
b3.2 m
then (3.10)
~ 1(mod
the
3-2m-P)
first
and
by Corollary
.
third
conditions
I to L e m m a
3.7.
are
consequences
of
47
In c a s e
m =O
the
Pl]a-b.
Thus
we are
i.e. fices
to c o n s i d e r
3P I ( a 3 - b 3 ) / ( a - b ) We
shall
show
In fact, last
LE~9~
In
in case
B #3(mod
Proof.
Since
by
and
16,
The does
the
then
3a 2 + (b-a) 2 sO(rood 3P) . case
b = (A-D½)/2
A 2 +B sO(mod
we
obtain
that
9). the
thus
A 2 +B
~O(mod
p =2,
in w h i c h
the case
p =2 h I =2,
by
3P)
and
(mod 3 2 ) clearly
when
A 2 +B
is
D
N I =I.
the c o n d i t i o n s except
since
results.
(i)
-
either
(iii)
will
A ~2(mod
be
4)
to s h o w t h a t
=a ÷b =A ~2(mod a 2 ~b 2 { 1(mod
4)
the c o n d i t i o n s
2P)
observe
that
(i)
if
But
+8A 1+ 2+B+
~ 2 +B+
D = 4 ( 4 ( A ~ + A I) + I +B) D½
is d i v i s i b l e
by
2AID2+
is in v i e w 4,
of
D ½ (rood 4)
B ~3(mod
4)
so f i n a l l y
_-- I (rood 4)
obvious
about if
proposition
b
follows
by conjugation
in c a s e
K #Q
and
K =Q.
results
now
irmmediately,
and,
as a l r e a d y
noted,
so
theorem.
6. T o g i v e
an application,
its r e l a t i v e s ,
defined
(In t h e F i b o n a c c i to T h e o r e m N
lp = I ,
it s u f -
to
r =I
thus
the a s s e r t i o n
is a n y w a y
on
which
then
a 2 _-- 2 ÷ B
and
hence
to the e x c l u d e d
(a2-b2)/(a-h)
D =A 2 +4B.
divisible
If it is v i o l a t e d ,
same congruence
a 2 = ( A 2 + D + 2 A D ½) /4 = 8 A
where
time based
condition,
3P)
3P), the
this
second
4).
(ii) h o l d ,
A :4A I +2,
leads
the case
3.12.
satisfied
m =0.
a = (AeD½)/2,
- O(mod
applies,
the
+ (a-b) 2
is e q u i v a l e n t
integer,
It r e m a i n s
and
this
6AD ½ ~O(mod
a rational
or
in t h e c a s e
that
A + B + 6AD ½
however
left with
=3a 2 +3a(b-a)
writing
congruence
same argument
3.2)
is a p o w e r
of
case
that
by
we consider
un+ 2 =Un+ ~ +u n
u o =u I =~.)
the F i b o n a c c i
5 and
the s a m e
We
the F i b o n a c c i with
prescribed
saw already
sequence
applies
also
sequence
can he
uo
and
and
u I.
(in the C O r o l l a r y UD(mod
to s e q u e n c e s
N) with
only
if
another
48
choice will
of
be
of the
initial
UD(mod
5 k)
associated
a =
and
the
5~ab,
51Uo,
this
by
then
if
condition
to
one
sees
the sequence
is a p o w e r
[72a3.
p a11
the
iS
{Un}
3.3
will
and
(ii)
that
the
c a n be
found
M.WARD
[31a],
be
satisfied
{u n}
will
the roots
be
UD(mod5)
2
uO .
for any
u I ~tuo(mod
5)
uI
not d i v i -
we can
transform
5)
if and o n l y
J.S.SHIUE
UD(mod
if
5~Uo,
[71]).
If
and
u I /3Uo(mod
5).
then
un+ 2 = u n + U n + I
if and only if
N)
the F i b o n a c c i
of 5. T h e
sequence
sufficiency
[72a]
Another
first
_
: u 2 -UoU I
putting
and
proof [62],
the
N
is a power of
G.BRUCKNER
first
study
results
in p a p e r s
contain
[31b],
of
which
[70]
and
5
[33] w h e r e
[74]. imply
that
if and
was
Note
only
if
established
and J . S . S H U I E also,
that
the
for no p r i m e
all r e s i d u e s
recurrences
connected
of R . D . C A ~ 4 I C H A E L
N)
(mod p) .
comments
linear are
UD(mod condition
by L . K U I P E R S
P.BUNDSCHUH
sequence
general
is
of t h i s
necessity
gave
§ 4. N o t e s
[78]
Since
{u n}
U
the F i b o n a c c i
I. T h e
5).
that
following
(L.KUIPERS,
of A . P . S H A H
can
then
it h o l d s
by H . N I E D E R R E I T E R
results
now
(I-5"2)/2
condition
claim
5~u I - 3 u O -
[71],
UD(mod
~ u o2 ( t - 3 ) 2 ( m o d
In p a r t i c u l a r , N
if it is
shows
are
from Theorem
5~Uo,
that
we can
COROLLARY.
and
theorem
((2u I_uO) 2 _ 5 u 2)/4
this
5 and
Finally
last
if
0 / u~(t2-t-1)
and
b =
we o b t a i n
5~N(u1_aUo ) =
sible
The
if and o n l y
polynomial
(I + 5 ½ )/2 ,
if and o n l y
If
elements.
[203,
the p r o b l e m
with
was done uniform
H.T.ENGSTRGM
of d e t e r m i n i n g
by E . L U C A S
distribution [31] the
and
least
49
period rent
v(p)
of
sequence)
powers
pt
in the voted
un(mod
one has
first
N)
quoted
paper
showed
concerning
the d i s t r i b u t i o n paper
(mod p) newer
applies
[81], w h o u t i l i z e d The k n o w l e d g e bution,
since
divisible
by
(mod N)
mes
and
up to
neral. p by
p
of
that
[6~I
p2.
From
cf. A . V I N C E
them.
proved
out u n i f o r m d i s t r i (mod N)
m u s t be
[72a3
and then
sufficient
[75] and W.A.WEBB, and Webb
J.S.SHIUE
3. G . J . R I E G E R for recurrences,
was
a given
conditions
[751.
for
(Nathanson
another
[773 o b t a i n e d
a sufficient
(x-1) 2 (x_m I ) ... (x_ms)
proof
polynomial
primes divisible
sufficient
condition
to other
[743 proved and
for r e c u r r e n t
sequences J.S.
of c h a r a c t e r i -
recurrence
with
se-
the p #2.
by L.KUIPERS,
is
and P . B U N D S C H U H , UD
in ge-
sequence
recurrences
by R . T . B U M B Y
and L o n g d e a l t w i t h prime
associated
this h o l d s
The p r o b l e m
[72d3
[77] p r o v i d e d
whose
and
obtained
second-order
J.SHIUE
then for
for all those
J.S.SHIUE
[72a3.
sequence
that for all pri-
generalized
U D ( m o d N)
and were
independently
C.T.LONG
later
second-order
concerning sequence
also,
of the F i b o n a c c i
P.BUNDSCHUH,
by L . K U I P E R S ,
solved
k(p 2) #k(p)
also a n e c e s s a r y
for a r b i t r a r y
for w h i c h
and
that this holds
and H . N I E D E R R E I T E R
N
of the F i b o n a c c i
and asked w h e t h e r
which
[663.
k(N)
He c h e c k e d
an e l e m e n t
He gave
first r e s u l t s
[733 who o b t a i n e d
of primes
is a p e r i o d
polynomial.
the period
is a prime
k(p 2) #k(p)
k(p 2) ~k(p)
first tackled
REITER,
if
exists
of Wall
zing integers
powers,
study of the
a number w h i c h
for r u l i n g
then
k(p ~) = p J - l k ( p ) .
the F i b o n a c c i
[713,
N),
showed
there
first r e s u l t
was
UD(mod
the p e r i o d
by D . W . R O B I N S O N
2. The
to the
of r e c u r r e n c e s
to d e t e r m i n e
is useful
[603 c o n s i d e r e d
but not by
concerned
(mod N)
is de-
of r e s u l t s
of such a recurrence.
theory Joe.
a wealth
the same a s s o c i a t e d
theory
if we have
104 one has
for v a l i d i t y
SHIUE
matrix
S.E.MAMANGAKIS
quences
(mod p)
and M . W a r d
second paper
and c o n t a i n s
number
the periods
of a p e r i o d
one has
for w h i c h p
with
b ~ EO,t3
His
recur-
that for p r i m e
N.
D.D.WALL
j=I,2 ....
b ~t-1.
order
linearly
a.o.
a certain
of a recurrence,
concerning
a given
proved
of r e s i d u e s
for any r e c u r r e n c e
papers
that
algebraic
(mod p)
being
with
of the third
Engstrom's
period
un
Carmichael
v(p t) =pbv(p)
to r e c u r r e n c e s
general
(with
was considered.
respect
UD(modN) J.S.SHIUE to prime
[753, M . B . N A T H A N S O N
considered powers.)
only
Later
the case H.NIEDER-
in the case of primes.
condition
for
is of the forln
UD(mod m)
50
where
ml,m2,...,m s
by L.KUIPERS
[79].
REITER
[80]:
let
for i n t e g e r s
and
x i ~Yi(mod that
un
holds has
d)
satisfying
then
for a l l
N)
integers
A far-reaching f ( x l , . . . , x k) the
and
his
generalization
result
was
was
proved
extended
by H.NIEDER~
b e an i n t e g e r - v a l u e d
function
defined
following
if
and
condition,
diN
f(xl, .... xk) ~ f ( Y l ..... yk ) (mod d). A s s u m e
is a s e q u e n c e
a period
UD(mod
are d i s t i n c t
n >O.
with
the property
If for e v e r y
not divisible
for e v e r y
by
a prime
diN,
d, to
then N.
that
d #I the
Cf.
further
U n + k =f(un,Un+1,...,Un+k_1) the
sequence
sequence
G.J.RIEGER
Un(mod
v n =au n + n
d)
is
[79]°
Exercises
I. G i v e
a characterization
positive
integers
of o r d e r
two
2. F o r
a) . If then
such that
a given
lest period to M . W A R D
for w h i c h
of
sequence
U n ( m O d m) . P r o v e
m =ql...qt,
where
of the
set of a l l
sequence
{u n}
the
{Un}
by
v(m)
the
results,
which
denote
following
smal-
go b a c k
qi
are p a i r w i s e
co-prime
prime
powers,
v(m) = l . c . m . (v(ql) ..... v ( q t ) ) . there
c) . T h e n u m b e r
k
the p r o p e r t y ,
by the polynomial 3.
occuring
that
two.
p)
X v(p)
associated
of second
the discriminant
v n =un(mod
k z I
such that
n _>k.
(H.NIEDERREITER,
sequence
divides
exists
n : I , 2 ..... k
pn-kv(p)
than
X
a recurrent
[31a]:
v ( P n) = ] v(p)
rent
subsets
exists
M ( u n) = X .
recurrence
b) . For any prime
with
of t h o s e
there
does
not
in -I
with
J.SoSHIUE order,
(ii)
equals
the maximal
lies
in the
ideal
un
and
[77]).
then
it
is
satisfy
integer
ZEX]
s
generated
pS
Prove
of t h e a s s o c i a t e d
of
that
UD(mod
if
p)
{u n}
if a n d o n l y
polynomial,
a recurrential
is a r e c u r -
relation
and
the
if
p
sequence
of order
smaller
CHAPTER ADDITIVE
§ I. The
I. In t h i s f(n)
chapter
an additive
being
= f(m) + f ( n )
for all
such
a function
thus
one w o u l d
of the v a l u e s established concerning a special
we
mean
m,n
to h a v e
values
4.1.
i.e.
by
a criterion
for
the
aid
theorem,
function,
for all pairs
m,n
with g(p)
i.e.
and
UD(mod
a function
(m,n) = I .
for prime
p
g(2 k) = - I
deduce
M(g)
~ n~x
in terms was
~f E . W i r s i n g ,
shall
now
state
the c r i t e r i o n .
be a c o m p l e x - v a l u e d
satisfying
g(~)
that
:g(m)g(n) Ig(n) I ~I
lie in a convex polygon O. If either
contai-
the series
(4.1)
k >-I
one has
,
= lira ! x X÷ ~
shall
powers,
f
result We
g(p))p-]
then the mean value
for
then
P
or, for all
N)
=
Obviously
at p r i m e
functions.
g(n)
f(mn)
integers.
a criterion
Assume m o r e o v e r
ned in the unit circle and containing
diverges,
Such
of a d e e p
Let
[67]).
with
satisfying
prime
its v a l u e s
k=I,2,...).
[69] w i t h
f(1),f(2),..,
a function
of m u l t i p l i c a t i v e
multiplicative
(I - R e
sequences
determined
(E.WIRSING
and that all values
of D e l a n g e
of r e l a t i v e l y
(p - prime,
of W i r s i n g ' s
PROPOSITION
treat
function,
by H . D E L A N G E
case
shall
is c o m p l e t e l y like
FUNCTIONS
criterion
pairs
f(pk)
IV
M(g)
g(n)
of
g
defined by
52
exists
and equals
exists
and is non-zero.
For
the
origina l
very
paper
The proof consequence
N
If none of these
long and c o m p l i c a t e d
(E.WIRSING
of P r o p o s i t i o n
Let
= exp
has always either
holds,
p r oo f we refer
then
M(g)
the reader
to the
of D e l a n g e
be an additive,
f(n)
is based
on the f o l l o w i n g
4.1:
be a given positive
g(n)
conditions
[67]).
of the c r i t e r i o n
COROLLARY.
let
zero.
integer.
Then
integer-valued
function
and
the function
(2~if(n)/N)
a mean-value,
and this mean-value
vanishes
if and only if
the series
p-1
(4.2)
Nff(p)
diverges,
or for all
the resulting Proof.
ratio
Since
and b e c a u s e
series
(k ~ )
N
for
2. To state functions
integer
(Ad)
hy
N
and
are
when
if and only k ~I
the f u n c t i o n N - t h roots Observe
of unity,
the a s s u m p t i o n s
I -Re
g(p)
is an integer,
(4.2)
if and only
is m u l t i p l i c a t i v e
now that
f(n)/N if
g
diverges.
if w i t h
thus the
Finally
suitable
exceeds
note,
that
integral
= ~i + 2ziM k
2f(2k)/N = I + 2 M k
ditive
is divisible
2f(2 k)
one has
2~i f (2k)
thus
g
except,
divierges
holds
of
are satisfied.
constant,
g(2 k) = - I Mk
is additive,
the v a l u e s
(4.1)
the number
is odd.
f
of the p r o p o s i t i o n a positive
k ~I
is an odd
the n e c e s s a r y we introduce
integer.
and s u f f i c i e n t the following
d >I:
The
series
~ p-1 P dJf(p)
diverges,
D
condition
for
two c o n d i t i o n s
UD
of ad-
for every
53
and
(Bd)
For all
T H E O R E M 4.2. f(n)
is
either
(Ad) or
for every
the number
2f(2k)/d
if and only if for every divisor
(Bd)
(1.1)
r=I,2, .... N-1
the f u n c t i o n
rf(n)
d >I
of
N
holds.
By P r o p o s i t i o n
zero mean-value.
is an odd integer.
(H.DELANGE [69]). An i n t e g e r - v a l u e d additive function
UD(mod N)
Proof.
tion
k zl
f
will be
UD(mod N)
if and only if
gr (n) = e x p ( 2 ~ i r f ( n ) / N )
A p p l y i n g the C o r o l l a r y
to P r o p o s i t i o n
has a
4.1 to the func-
we obtain that this will take place if and only if either
the series
p-1
(4.3)
N~rf (p)
diverges,
or,
for all
k a l, the number
2rf(2k)/N
is an odd integer.
Now we shall t r a n s f o r m these conditions.
For
N =d(r,N),
C l e a r l y the conditions
and
r =r1(r,N)
d~f(p)
if the condition = 2rlf(2k)/d 2f(2k)/d integer.
with
(d,r I) =I.
are equivalent,
r e [1,N-1]
hence the series
(Ad) holds for
d =N/(r,N).
is an odd integer,
then
m u s t be an odd integer.
(4.3) diverges Further,
2rf(2 k)
if
(Bd)
if
N
odd and so
2f(2k)/N
is for
by
N~rf(p)
if and only
2rf{2k)/N =
m u s t be even, r I
Conversely,
It follows that the c o n d i t i o n
lent with the d i v i s i b i l i t y of
d
write
is an odd
d =N/(r,N)
equiva-
and the oddness of the
r e s u l t i n g ratio. The T h e o r e m results now from the o b s e r v a t i o n that if the integers
1,2,...,N-I,
d i s t i n c t from unity.
then
d =N/(r,N)
f
is
(Ap) holds for every prime divisor
(Ap) holds for all odd prime divisors f(2),f(22),..,
runs over N
D
C O R O L L A R Y I. An additive function if either
r
covers all divisors of
p
of
are all odd and in case
41N
UD(mod N)
if and only
p
of
N,
the numbers
N, or
the condition
N
is even.
(A 4) holds
as well. Proof.
Assume, of
thet
divisor
p
follows.
Hence
N. T h e n
4/2f(2)
the c o n d i t i o n
f
is
UD(mod N)
(Bp) ~ u s t hold,
(B 2) holds and so
f(2 k)
but
thus
(Ap) fails for a prime 2f(2)/p
is odd for all
(B 4) fails and we see that
is odd and
p =2
k-> I. Since
(A 4) must hold.
54
TO then sors may
get
(A D) d
the converse holds
on
invoke
N
except,
the
theorem.
2. If
COROLLARY
constant value Proof.
c,
If
implication
as w e l l ,
thus
f
N)
follows
UD(mod
N)
but
Since
(Ap)
2/f(2)
=c,
does
then
f
from
not c
that
d :2,
but
if
(Ad)
holds
imply
(Ad)
for all d i v i -
then
is an additive function, will be
then the
(N,c) # I
but
observe
assumptions
possibly,
(N,c) = I ,
UD(mod
our
(Ad)
holds
theorem.
and
hold,
UD(mod
let
the p r e v i o u s
is d i v i s i b l e
by
d i N , d #I
conversely
so w e
implies
and h e n c e
by
2,
(N,c) = I .
hence
that
prime divisor
corollary
(N,c)
diD,
attaining at primes a
for all
be any
and
if and only if
N)
Assume
p
(B 2) h o l d s
and
f of
i£ (N,c).
p =2
and
so w e
get a
contradiction.
COROLLARY indicated
3.
(S.S.PILLAI
a proof,
The functions divisors of
details
and
~(n)
[40].
of w h i c h
with their multiplicities,
3 to t h e
last
section theorem
only classical
tools.
we
are
UD(mod
N)
of Delange's
shall
without The
given
N :2
H.v.MANGOLDT
by E.LANDAU
[09],
the number of prime factors of
§ 2. A p p l i c a t i o n
I. In t h i s
were
functions,
however
in t h i s w a y
one cannot
obtain
show that
the use
same
of a d d i t i v e
[98] 8 167.)
which give the number of distinct prime
~(n)
n, repectively
In t h e c a s e
tauberian
one
works
the reader
may
theorem
can obtain
the Corollary
theorem,
also
utilizing
for a l a r g e r
convince
of Theorem
counted
N.
of W i r s i n g ' s
approach
a proof
for all
n,
4.2
himself
in its
full
class that gene-
rality. The main actually
a
tool
special
PROPOSITION
f(s)
=
here will
~ n:]
4.3.
an n-s
case
be
of it,
(H.DELANGE
t h e Ikehara-Delange which
[54]).
we now
Let
tauberian
states
a >0
and let
theorem,
55
be a Dirichlet
serie~ with non-negative
open half-plane functions with
and assume
s >a
#0
a real number
b
and complex constants
b. If for
that there exists an integer
not equal
to zero or
a
negative in-
with real parts smaller
al,...,a q
q ~1,
Re s ~ a
than
Re s >I
f(s)
gO(s) (s_a)b
then for
convergent in the
regular in the closed half-plane
go(S) .... ,gq(S)
go(a)
teger,
Re
coefficients,
x
q ~ j=l
+
-aj gj (s) (s-a)
tending to infinity
an =
(C + 0 ( 1 ) ) x a ( l o g
,
one has
x) b-1
,
n_<x
with
C =go(a)a-IF-1(b),
For
the p r o o f
special
case
of the
we r e f e r
or to c h a p t e r
III
~(n). of
distribution
Let
z,
Fz(S)
which prime
f
=
~ n=1
Fz(S)
either
m
F-function. theorem,
of w h i c h
to the o r i g i n a l
for
Re
s >I.
paper
is a
of D E L A N G E
[83a3.
4.3 ~ f o l l o w i n g
all
or
N,
of the
H.DELANGE
function
R, and c o n s i d e r
[56]) the
~(n)
for a fixed
and value
series
Since
Z
f (n)
is m u l t i p l i c a t i v e
(I + z p -s +
~ j=2
z f C P O ) p -is)
: g] ( s , z ) ~
p
(1 + ( ~ k=2
zf(pk) p-kS) (] + z p - S ) -I
(I + z p -s) p
where
= ~
and
we have
p
g] (S,Z)
this
zf (n)n -s
f(p) :I,
= ~
either
from Proposition
the D i r i c h l e t
converges p
the r e a d e r
the
tauberian
(mod N) , for
denote
Izl sl,
general
of W . N A R K I E W I C Z
2. N O W we d e d u c e uniform
denoting
F
)
.
,
for
56
LEMMA
hes at
only
s=l
Proof.
I( ~ k=2
with
The f u n c t i o n
4.4.
If
~
in the case,
denotes
when
the real
zf(pk)p-kS) (1+zP-s)-II
a certain
constant
is regular
g1(s,z)
B,
and
f ~
part
~ ( V k=2
for
and
of
Re
and vanis-
s >~
z =-1.
s, t h e n
P-k~) (l-P-°) -I = (P~-I)-2 ~ BP -2~
since
the
series
a >½
the
first
assertion
g1(1,z)
vanishes
p-2O P
converges
almost
To obtain
uniformly
for
the
second,
observe
that
nishes
one of the
factors
in the
product
factor
has
its a b s o l u t e
positive
for
and only
if
p ~2.
Now
value the
equal
factor
defining
to at
least
corresponding
results. only
it h o w e v e r
I - ( p - l ) -2 to
p =2
if v a the
p-th
which
is
vanishes
if
f (2 k ) z
I
k=2
and
2k
this
+ ~ z + I = 0
is c l e a r l y
z =-I.
Since
we have
for
3. T h e
LEMMA
possible
d(2 2) = 2 , z =-I
next
4.5.
indeed
lemma
For
(I + z p -s)
the
deals
Izl
only
if
z
function
f(2 k ) f =Q
z f(2k) = - I
with
the
and
Re
for
k=I,2,..,
is e x c l u d e d ,
for a l l
second
s > 7
=-I
k zl.
factor
in
and
for
and f =
D
(4.4).
one has
= g 2 ( s , z ) ( s - l ) -z
P
where
g2(s,z)
Proof.
For
is regular
Re
s > I
for
Re
s zl
and does
not vanish
P
s =~.
we have
oo
(l+zp -s) = e x p
at
~ l o g ( 1 + z p -s) = e x p ( ~ ( z p -s + ~ P P P k=2
(_I) k + s
z k p -ks) ) .
57
PZ p-S =log(s1__~) ÷ g ( s )
Since
with
g(s)
regular
for
Re
s al
and
k+1
(-1)
I ~ k=2
for
Re
zkp-kS I ~
I
k
p2a _ pa
s =o >½
we o b t a i n
that
for
Re s >I
(1 + z p -s) = ( s - 1 ) - Z e x p { z g ( s ) }g3(s,z) P
with
g3(s,z)
regular
For
COROLLARY.
Fz(S ) =
where For
Izl ~I,
the function
s >½
z ~-I
and n o n - v a n i s h i n g
and
Re
these
a positive
there.
D
one can write
s >I
integer that
N,
w e can
put
for
Re
Re
now return
z r =exp(2~ir/N)
Hz(1
s-> I.
to our m a l n for
#O.
D
task.
r=O,1,...,N-1,
Re s > I
N-1 I = N r~O Fzr ( s ) e x p { - 2 ~ i j r / N }
-s n
and
s >-I
is regular for
F z (S)
preliminaries
j, and o b s e r v e
E
is regular for
Hz(S)
the function
4. A f t e r
fix
Re
(s-1)-ZHz(S)
z =-I
Given
for
.
n f(n) -j (rood N)
Indeed, and we
this
follows
interchanging
the
by
substituting
summations.
here
Using
the
the
series
Corollary
for
Fzr(S)
to the
last
Lemma
obtain
-s
I N
n
HI(S) s-1
+
n
N-I H z (s) ~ __r exp{-2wijr/N} r=1 (s-l) zr
f(n)~j(rm:d N) Since for the
Re z r
function
In the n e x t problems
f
for
results
chapter
concerning
r=1,2,...,N-1
weak
the
from Proposition
we shall uniform
apply
the
assertion 4.3
since
about
HI (I) # O .
same P r o p o s i t i o n
distribution.
UD(mod
4.3 to
N)
58
§ 3. T h e
Let
s e t s M(f)
us n o w c o n s i d e r
for a d d i t i v e
THEOREM
the question
functiQns
If
4.6.
for a d d i t i v e
f.
X
The
X = { ~
to
it is g i v e n
the
f P
satisfying
sets
by the
is a non-empty set of positive
set of primes
p P:
of characterizing
answer
there exists an additive function if for a suitable
functions
M(f)
following
integers,
M(f) = X
(which may be empty)
then
if and only we have either
ap ->0} = A(P)
pEP or
X =
{2 a
Proof. follows
~ peP
p P: a
Observe
that
if
first
f
is
M,N
a r e even,
then
same
corollary
implies
UD(mod sults
2 m) now
for
functions
If
P
for
f(n)
P
for w h i c h
f(n) = 0
the C o r o l l a r y
if
and
UD(mod
UD(mod
MN).
is
UD(mod
f
The necessity
M(f)
if
n=O
[
if
21n
of a l l
set o f all
primes
primes
P'
in
and
X =A(P)
satisfying
will
or
I to T h e o r e m
N)
4.2
and n o t b o t h
Moreover 4),
note,
then
of t h e
it
numbers
that
the
it is a l s o
stated
condition
re-
primes,
the
If
form stated
it is e m p t y ,
holds
then
Let
for the
3 =Pl
+f(b)
A(P) P
nor
=B(P)
additive
in the T h e o r e m .
then
function
<''"
(j=1,2 .... ) .
be the
be the
a completely
for all
and
M(f) : A ( P ) defined
f(n) = n
its c o m p l e m e n t
ql < q 2 < ' ' "
then we define
"'" qJ
have
explicitly
by
.
is v o i d . let
now construct
2~n
that neither
f(ab) = f ( a )
= ql
shall
M(f) = B ( P )
= ] 0 I
we
set of p r i m e s .
and
So a s s u m e
f(PJ)
that
be a given
consists
If
is a l s o
sufficiency
our needs.
mes.
from
U D ( m o d M)
f
m=I,2, ....
its
f,
thus
holds
that
immediately.
To prove
Let
->0, 0 <_~ _
a,b)
P'
sequence
sequence additive
realizes in the of all
of a l l odd p r i function
by putting
f(2) = 2
f
(i.e. and
59
In c a s e w h e n formally
P'
qT+1
is f i n i t e
= qT+2
sense.
For
this
in
and
for n o n e
P
to Theorem If
4.2
=
X =B(P)
consists
"'" = 1 ,
function
so t h a t
P'.
Since
of,
say,
T
primes,
oar definition
the c o n d i t i o n
in
gives
and
(Ap)
(B 2)
holds
evidently
of
for
we define
f
makes
every
fails
prime
p
the Corollary
1
M(f) = A ( P ) .
(in w h i c h
case we may
assume
2 ~P')
we define
f
by
putting
f(2 n) = I
f(p~)
(with t h e dition (B 2) odd
(n=1,2 .... ) ,
= 2qlq2.°.qj
(n=1,2 .... ; j = 1 , 2 .... )
same convention
as b e f o r e
(Ap) holds
p
holds and
for
are divisible
4.2 to g e t
all p r i m e s
since
(A 4) by
fails
4) w e m a y
first
in H . D E L A N G E He c o u l d
results
[61],
assert
absolutely
=
by unity
(c + o ( I ) )
in the c a s e
~ .I " R e P P
P
and
(due to again
for n o n e
ql = 2
use
c =I
and
P') . T h e
in
con-
P' , h o w e v e r
all v a l u e s
the C o r o l l a r y
f(p)
a weaker of
M(f)
of T h e o r e m
version for
4.2
occur
of Wirsing's
f
already
Theorem.
multiplicative
and
4.2 w a s
b y H.
in t h e c a s e w h e n
x log x
the
series
f(p)
diverges.
2. T h e m a i n DELANGE
for
I to T h e o r e m
comments
in t h e d i r e c t i o n
proved
the vanishing
bounded
f(p)
who
p~x
and
in
of f i n i t e
M(f) = B ( P ) .
§ 4. N o t e s
I. T h e
in t h e c a s e
[69] w h o
result in t h e
of t h i s
chapter,
same paper
Theorem
considered
also
joint
proved
distribution
60
of the p a i r and
showed
then
for
if
NI,
g(n)mod
(N 1,N 2) = I
arbitrary
a,b
and
the p a i r
f,g
In a l a t e r = d #I with
and
is
paper
showed
respect
to
UD
that
and m o r e o v e r 0 < t I
cessarily
In H . D E L A N G E
sequences which with
of v a l u e s
is a s s u m e d a set
M
tive
f
pair
Recently that
if
which
x
is
the
condition f,g.
f
additive
an a n a l o g u e
N)
functions
UD(mod
tl,t 2
will
NI),
g
of c o p r i m e tlf +t2g
be
UD
is
~ntegers
is
UD(mod
for a r b i t r a r y ,
of T h e o r e m
function
result
4.2 w a s
restricted
d).
not ne-
is a l s o
obtained to a set
o f an a r i t h m e t i c a l
characteristic
may
for
is
(NI,N 2) =
true
contain
functions
all
of the
progression
function
integers, form
for A,
and with
thus one obtains
f(an+b)
with
addi-
a,b.
it w a s
UD(mod
U D ( m o d N 2) ,
functions.
to b e an i n t e r s e c t i o n
is an
is
the case
A corresponding
[72]
UD(mod
N I) , g
additive
N I , N 2.
is n e c e s s a r y
two
M
to
function
than
with multiplicative
for
UD(mod
of a d d i t i v e
for e v e r y
0 < t 2
o f an a d d i t i v e
and given
are both
he c o n s i d e r e d
if
I/p < ~ . In p a r t i c u l a r p~M criterias
f,g
_:b(mod N2) } - NIN21
respect [74]) f,g
functions
of m o r e
is
if a n d o n l y
this
additive,
for s y s t e m s
3.
that
with
a pair
NI,N 2
satisfying
also,
f
N I) , g(n)
(H.DELANGE
U D ( m o d N 2)
He n o t e d
where
one h a s
lira xl # { n _<x: f(n) - a ( m o d
i.e.
N2>,
shown
by H.DABOUSSI,
irrational N) , t h e n
real
the
H.DELANGE
number
sequence
and
f
[82]
(th.7
an a d d i t i v e
f([nx])
is a l s o
(iii)) function
UD(mod
N) .
Exercises
I. P r o v e then either
that f
if
or
2. C h a r a c t e r i z e P
(n) =
3. UD(mod UD
Z I pln pep
is
(H.DELANGE NI) , g
with
is odd, must
the
be
sets
P
N)
[69]).
Prove,
UD(mod to
NI
N 2) and
f,g
are
UD(mod
UD(mod
is
respect
N g
additive
of primes
for a l l
that and
N 2.
and
f+g
is
U D ( m o d N)
N) . for w h i c h
the
function
N.
if
f,g
are
(NI,N 2) = I ,
additive,
then
f
the p a i r
is f,g
is
61
4. if
Show
that
of the p r e v i o u s
exercise
may
be f a l s e
(N I ,N 2) ~ . 5. O b t a i n
additive
mine,
a criterion
for u n i f o r m
distribution
for a s y s t e m
of
functions.
6. L e t number
pect
the a s s e r t i o n
A,B
be
of d i s t i n c t when
the
to g i v e n
two prime
pair M,M
sets
~A(n), .
of p r i m e s ,
factors ~B(n)
of
n
and
let
lying
in
is u n i f o r m l y
~A' A
~B
denote
resp.
distributed
the
B . Deterwith
res-
CHAPTER
V
MULTIPLICATIVE
FUNCTIONS
§ I. D i r i c h l e t - W U D
I. In this chapter the
appropriate
be a i m i n g
notion
is that
at n e c e s s a r y
a possibly sider
we shall
notion
of u n i f o r m
sufficient
function
and then utilize
4.3)
to d e d u c e
condition
distribution
tauberian
a c~iterion
for
functions
theorem
WUD(mod
containing
for
functions.
for it to hold
N)
functions.
distribution
conditions
of m u l t i p l i c a t i v e
sary and
of m u l t i p l i c a t i v e
of weak u n i f o r m
and s u f f i c i e n t
large class
a weaker
study m u l t i p l i c a t i v e
and we shall
~D(mod We shall
than ~ D ,
Here
N~
for
first con-
obtain
a neces-
for a ~iven m u l t i p l i c a t i v e of D e l a n g e
(our P r o p o s i t i o n
for a r e a s o n a h l y
all those w h i c h
are
large class
"po!y~omial-
-like", i.e. satisfy f(pJ)
= Vj(p)
with certain for
polynomials
V~(x)
sufficiently
many
To i n t r o d u c e
the new notion,
function
all primes
indices
p
for all or at least
j.
w h i c h we shall
call
D-WUD(mod N), c o n s i d e r a m u l t i p l i c a t i v e ,
or shortly
positive
positive
and
f
and a given
integer
r
integer
N ~3.
with the p r o p e r t y
We assume that
Dirichlet-WUD(mod N), integer-valued
that there exists
a
the series
1/p P (f(pr),N)=1
diverges.
Let
m =m(f,N)
be the
smallest
integer w ~ t h
this property.
63
We only
shall
if for
say t h a t every
the f u n c t i o n
j
prime
to
is Dirichlet-~gD(mod
f
N
lira ( ~ n -s) : ( ~ n -s) rl s~l/m+O n f(n)-j (rood N) (f (n) ,N)=I
To
guarantee
abscissas are
that
this
of a b s o l u t e
at m o s t
definition
convergence
equal
to
I/m
by
e(n)
and
N)
if and
one has
=
I/~(N)
is c o r r e c t
of b o t h
.
we have to be sure t~at the
Dirichlet
it s u f f i c e s
(5.1)
series
to do this
for
occuring the
here
second
of
them. Denoting the
set of all
the
those
(multiplicative)
n's
for w h i c h
characteristic
(f(n),N)
=I
function
of
we can w r i t e
oo
Z n -s = E n n=1 (f(n) ,N)=I
at e v e r y with
s, w h e r e
A
the two
being
the
I ~j <m-1
one
has
and
denoting
A,
in
A
tely
B
resp.
B.
S(n)n-S
2. T o prime
=
N, by
j(mod
p-1 peAj
diverges.
the
the
second
N)
factor
the c r i t e r i o n set of all the
classes for w h i c h
being
the
integers
converge
absolutely,
with
a certain
set of r e m a i n i n g
composed first
of p r i m e s factor
primes lying
is a b s o l u -
+...)
converges
for
p
subgroup
of
(mod N ) , , p r i m e series
absolutely
D-WUD(mod
primes
the
e(n) n-s
for w h i c h
K I/p c o n v e r g e s , the peA Re s > 0 and in v i e w of
A =A(f,N)
of r e s i d u e
sidues
p
B
of all
7]- (1 + e ( p m ) p - m S peB
formulate
and d e n o t e group
that
to
~ neB
on the r i g h t
=I,
sets
a(n) n -s
primes
(f(pJ),N)
for
nEB
j
series
set of all
the
Z neA
Since
convergent
we c o n c l u d e
E(n)n -s =
N)
satisfying G(N) to
N)
for
denote f(pm)
Re s >l/m.
by
Aj,
=j(mod
for N)
(the m u l t i p l i c a t i v e generated
by t h o s e
re-
64
5.]. A multiplicative
THEOREM m(f,N)
is w e l l - d e f i n e d
racter
X ( m o d N),
is
trivial
integer~valued
D-WUD(mod
on
A
N]
function
f, for which
if and only if for every cha-
there exists a prime
p
such that with
m =re(f ,N)
X(f(pJ))p -j/m = 0 .
(5.2)
j=O
(Note,
that necessarily
] ~
p <_2m,
X ( f ( p j ) ) p - j / m 1 _> ] _
j=o
which
since
~
p-j/m = ]
I p I/m
j=1
is p o s i t i v e
Proof.
for
We shall
to look m o r e
I -
p >2 m.)
transform
carefully
the
formula
(5.1)
and to do this we h a v e
at the s e r i e s
n -$ n
f(n) -j (rood s)
occuring
I ~(N)
where
in it. N o t e
~ X(j) X
the
=
is a s e r i e s
~ n=1
to
,
(5.3)
all c h a r a c t e r s
X ( m o d N)
for
Since
absolutely
Euler's
For every character
Re s > I / m . products
X ( m o d N)
=
(s - 1 / m ) ~ ( X ) g ( s , X )
X(f(n))
is
to obtain:
one has in t~e half-plane
Re s > 1/m
F(s,X)
and
X ( f ( n ) ) n -s
we c a n u t i l i z e
5.2.
it e q u a l s
is t a k e n o v e r
convergent
multiplicative LEMMA
F(s,X)
summation
F(s,X)
first that
exp {~ X ( f ( p m ) ) p -ms} , P
65
where
is a non-negative
a(X)
if for some prime
the equality
which is positive
(5.2) is true,
is regular in the closed half-plane
g(s,X)
vanish at
Re
If we d e n o t e
s > I/m
F(s,X)
The
and the function and does not
s > I/m
=~
Tp(S)
first
=
factor
if and o n l y
thms
write
for
Re s a l / m
ties
listed
the
~ X ( f ( p k ) ) p -ks k:0
series
by
Tp(S),
then
we c a n w r i t e
~ Tp(S) p~2 m
p
s =I/m
Re
if and only
s =l/m.
Proof. for
p
integer,
is r e g u l a r
for
if for a c e r t a i n
it in the
form
Re s > O prime
and w i l l
vanish
at
we h a v e
(5.2).
We
p
(s -I/m) S ( X ) g 1 ( s , X )
and n o n - v a n i s h i n g
in the
~ Tp(s) p>2 m
lemma.
Since
at for
s =I/m
with and
p >2 m
and
p-k/m" = I -
(pl/m-I)
> O
Tp(S)
not v a n i s h
g1(s,X)
a(X)
has
Re s a l / m
can
regular the p r o p e r -
we h a v e
oo
ITp(S) I -> I -
hence
for t h e s e
From
~ k=1
p's
the p r o d u c t
to t h o s e
primes
p,
(f(pJ),N)
=I.
converges
and h e n c e
regular
for
By our
Re
the r e m a i n i n g
T
by
(s) = I +
(5.4),
hence
Tp(S) peA
with
a certain Putting
H T p ( S ) we p>2 TM for w h i c h t h e r e assumption the
s >O
the
separated
and d o e s
primes
P
does
p >2
TM
= exp
~ peA
function
everything
series
of
which
not v a n i s h
Re s > I / m
for now
is an i n d e x
part,
(whose
~ X ( f ( p k ) ) p -ks k=m
for
separate
(5.4)
at
Re
s al/m.
the p a r t
corresponding
I ~j ~m-1
inverses
of t h o s e
we d e n o t e s =I/m
set we d e n o t e
by
such
by
due
to
is
(5.4).
For
A) we h a v e
we can w r i t e
= exp{
~ X ( f ( p m ) ] p ~ms + g 3 ( s , X ) } peA
g3(s,X)
regular
together
we arrive
for at
Re s >_I/re.
p's
g2(s,X)
# O
log Tp(s)
that
66
F(s,X) = (s -I/m) ~(x) g1(s,X) g2(s, X) exp{ ~ X(f(pm))p peA
T M
+g3(s,X) ] ,
thus putting
g4(s,X)
= exp{- ~ X(f(pm))p -:ns} p~A
and g(s,X) =g1(s,X) g2(s,X)exp(g3(s,X) tion. D
+g4(s,X))
we obtain our asser-
Since for the principal character
Xo(mod N)
one has
F (S,Xo)
n-S
=
n
(f(n) ,N)=I we obtain that prime to
N
f
will be
D-WUD(mod N)
if and only if for every
j
one has
I
I
w(N) ~ X(j) F(s,X) lira X
_
~(N)
s-l+0
= ~ I
+
F(s,X o)
_
I
li4n (I + ~ ~(j) F(s,X) X#X o F(S'Xo)
~(N) s~+0
I Z X(j) lira F(s,X) ~(N) X#Xo s. ml+0 F(s,X o)
and using the lemma and the obvious equality is equivalent
a(X) = 0
we see that this
to
Z X(J----~g(I/m'X)s+l+0 lira (s- ~) I X~Xo
(X) exp I
k~ (X(k)-I)p~A~ ~ p-mS 1 =0
(5.5)
(k,N)=I holding for all
j
prime to
N. However,
since the matrix
(X(j))X#Xo (j,N)=I is of rank
o(N)-I
it follows that
(5.5) holds if and only if for all
67
X #X o
one has
lira
W
~
~÷~-+o [ (k,)=1 ~
(Re X(k)-1)
~
I
p-Sin +a(X) l o g ( s - I ) ~
~
=-
(5.6)
)
p~A k
A s s u m e now that for every n o n - p r i n c i p a l character which on
A
there exists a prime
then
a(X) al
and since
p
such that
Re X(k) ~ I
is not trivial on, then we may select such
r, Re X(r)
for
(5.2)
we obtain r
in
holds.
X
is such,
(5.6). If however
A
and since for r e m a i n i n g
If
is trivial
with
k's
X(r) #I.
Re X(k) 51
X
For we obtain
s >I/m
(Re X(k)-1)
E k (k,N)=I
Thus
f
is
~ p-Sin +a(X) log(s -~) ~ (Re X(r)-1) ~ p-Sin PeA k PEA k
D - W U D ( m o d N).
To obtain the c o n v e r s e i m p l i c a t i o n assume that but there exists a n o n - p r i n c i p a l c h a r a c t e r a(X) =0. k's
-~.
T h e n for
k
in
A
we have
f
is
D-~D(mod
X, trivmal on
X(k) =I
A
N)
for which
and for the r e m a i n i n g
the function
p-Sm P£A k
is regular at
s =I/m,
thus
(5.6) cannot hold,
contradicting
D-WUD(mod N) .
COROLLARY. if
q
If
A(f,N) =G(N),
is an odd prime and
A(f,q ) = G ( q
)
and thus
same assertion is valid, namely
then
f
is
A ( f , q 2) = G ( q 2) f
is
D - W U D ( m o d N). Moreover, then for all
k ~I
D - W U D ( m o d qk). In the case
we have
q =2
however under a slightly stronger assumption,
A(f,23) =G(23).
Proof.
The first part follows i m m e d i a t e l y from the theorem.
obtain the second note that in v i e w of residue classes
jl,...,jt (mo d q2)
Z P f(pm)~ji(mod q2)
I/p
the
A(f,q 2) = G ( q 2)
such that all series
(i=I ,2 ..... t)
To
we may find
68
(with
m = m ( f , q 2) = m ( f , q k)
is a p r i m i t i v e sider
those
There
must
the
root
residue exist
(k->I))
(rood q2). classes
at
least
For
r(mod
one
diverge given qk)
and
which
of them,
the p r o d u c t
k->2
say
fix satisfy
ri,
j=jl...jt
I -
with
and
r ~ Ji(mod the
conq2) .
property
that
series
1/p P f(pm)~ri(mod qk
diverges, but
so that
r ~ jl...jk
and
/hus
q =2 two
also
r ~ A(f,qk) . T h u s
~
(mod q2),
(mod qk),
is a n a l o g o u s , generators:
g2 z - 1 ( m o d
3.
It
WUD(mod
8),
then
is n o w
implying has
2 k)
lies
is a p r i m i t i v e
A ( f , q k) = G ( q k ) .
only
to r e m e m b e r
that
and
-1(mod
and
the pair
time
r =rl...r k r
g],g2
to s h o w h o w
2k),
generates
D-WUD(mod
also
The
has
in c a s e
for
gl ~ 5 ( m o d
G(2k) .
N)
{(f,qk)
(mod q2)
argument
G(2 k) if
in
root
k ->3
8),
D
is c o n n e c t e d
with
N) . We prove:
PROPOSITION
which is
The
is d u e
If
5.3.
WUD(mod
Proof. which
one
5(mod
hence
f
is a multiplicative,
then it is also
N)
assertion
essentially
results
D-WUD(mod
immediately
to R . D E D E K I N D
943,
from who
i n t e g e r - v a l u e d function N).
the
following
treated
Lemma,
the c a s e
a n =I.
LE~LMA 5.4.
series with have
a
Let
n=l
non-negative
an
resp.
f(s) = ~
b
,
and
bn
for their abscissas
and assume further, b
an n-s
that
over real values
> b.
[f(s) I
g s) = ~
r~l
arbitrary complex numbers, of convergence,
tends to infinity,
Put further
A(x)
= E an , n_<x
lim B ( x ) / A ( x )
=
I
X-~
then
a =b
lim s÷a+O
and
f(s)/g(s)
= 1 .
be two D i r i c h l e t
bnn-S
with
when
which
0 < a _
s
tends to
B(x) = Z b n . If n_<x
69 Proof. T h e first valid
for
s >a
1 ~ bnn-Sl n~x
assertion
and
=
I ~ B(n) (I n~x nS
_< C I ~ A(n)( n 1s n x which holds
=
=
from the f o l l o w i n g
constant
coincide
~ ann - s n_<x
C, and once we k n o w that the a b s c i s -
we use the e q u a l i t i e s
k A(n)( I .... I s ) = ~ A(n) n=1 nS (n+1) n=1
k s n= I
majoration
I )+B(x)[x]-Sl (n+l) ~
1(n+1) s ) + A ( x Y [ x ] -s 1 = C
with a certain
sas the c o n v e r g e n c e
f(s)
results
x z2
log (n+1) f e-StA(et)dt log n
s
log(n+1) -stdt S e = log n
= s 7 e-StA(et)dt O
(5.7)
and
g(s)
to get,
g(s)
and if
= s 7 e-st B(et )dt O with
R(x) =B(x)
= f(s)
-A(x) = o ( A ( x ) ) ,
+ s f e-stR(et)dt O
IR(x)/A(x) I <e
holds
I~ e - S t R ( e t ) d t l - < e s o
7 xo
for
x ZXo,
then w i t h
suitable
CI
log xo
implying
g(s) =f(s)
COROLLARY.
If
function for which it satisfies
e-stA(et)dt
+SOl
e - S t d t < esf(s) +0(I) o
:Log
+o(f(s)).
N ~3
and
m(f,N)
the conditions
f
is an integer-valued multiplicative
is defined and which is given in Theorem 4.1.
WUD(mod D
N),
then
70
4.
It s e e m s
that
which
is
D-WUD(mod
Hence
the
following
PROBL~4
no exal~ple N]
Are
III.
for
is k n o w n
a certain
t~e notions
of a m m l t i p l i c a t i v e
N, w i t h o u t
WUD(mod
being
and
N)
function
WUD(mod
D-~D(mod
N) .
equi-
N)
valent ? One
can
notions of all
however
coincide. those
find
The
large
first
multiplicative
E
classes
class,
of f u n c t i o n s
which
functions
we
shall
for w h i c h
for w h i c h
denote
the
5y
these FN,
consists
series
I/p
P (f(p) ,N)#I
converges.
For
namely
and we
is
N,
this
class
shall
our
two n o t i o n s
deduce
If
N ~3
this
is a given integer
PROPOSITION
5.5.
WUD(mod
if and only if it is
Proof.
N)
Observe
not n e c e s s a r i l y
first,
lying
that
in
FN,
if
f
and
~j(mod
N) }
-
D-WUD(mod
e(N)
for one m o d u l u s proposition
and
4.1.
then
f c FN,
f
N).
is a n y m u l t i p l i c a t i v e
(j,N) =I
I
#{n _<x: f(n)
coincide
from Wirsing's
function,
then
Z X'(mod N) X(j)
X(f(n))
,
thus
] - j (nod N) } - e(N)
dj = x÷~lim x -1 # {n _<x: f(n)
where The ence
M(g)
existence
of u n i t y
value
(mod N),
4.1,
M(Xo(f)),
and
with by
the
Xo(f(2k))
the m e a n
values since
of b o u n d e d
is n o n - z e r o
is v i o l a t e d tive.
as b e f o r e ,
of the m e a n
of P r o p o s i t i o n
or r o o t s mean
denotes,
occuring
Xo
of
Note
denoting
same
now
the
that
for
since d =
g.
is a c o n s e q u -
are
principal
that
function
formula
X(f(n))
Proposition,
It f o l l o w s
of the
in this
the v a l u e s
degrees.
~I.
value
~ N) X ( j ] M ( X ( f ) ) X(m0d
either
f e FN
zero
the
character
the c o n d i t i o n El dj (j ,N)=I
(4.1)
is p o s i -
71
Applying
Lemma
5.3 to the
fj (s) =
~ n -s n f(n) ~.j~nod N)
series
and g(s)
=
~ n -s n (f (n) ,N) =I
and o b s e r v i n g we o b t a i n
that
their
con/mon a b s c i s s a
the
limit
in
that
(5.1)
lira fj(s)/g(s) = l i m ( # { n ~ x : s+1+O x÷~
hence
D-WUD(mod
N)
however
that
deciding
whether
a given
since
one
can
of T h e o r e m
do this
5.1
this
N)
coincide
Proposition
function
applying
(see E x e r c i s e
much
larger
second and
in the n a t u r e ,
class
of
contains like
f
of d i v i s o r s ,
Ramanujan's
This
consist
f
which
class have
the
can write for
most
N ~3
Re
s >I
FN
is
D
useful
WUD(mod
N)
for
or not,
4.1 w i t h o u t
the
use
functions
which
we
shall
now consider
of the m u l t i p l i c a t i v e ~-function,
T-function
r=1,2,...,T
particularly
~roposition
the
functions
is
occuring
s u m of d i v i s o r s ,
the
number
etc.
of all m u l t i p l i c a t i v e
following
For every integer perty that for
from
case.
,
3).
functions
Euler's
unity,
(f(n) ,N) )=1} = d j / d
in this
is n o t
directly
§ 2. D e c e n t
I. T h e
equals
equals
f(x) ~j(mod N]}: #{n~x:
and W U D ( m o d
Note
of c o n v e r g e n c e
integer-valued
functions
property:
there exists an integer and for all integers
T
with the pro-
j, prime
to
N
one
72
p
-~
I
: a(r,j)log
s-~1 + gr,j
(s)
P
with
a(r,j)
and
~0
regular in the closed half-plane
gr,j (s)
Moreover at least one number Such
functions
of
T
(possibly
at
N.
In d e a l i n g
sufficient ver we
A broad
about
subclass
f
p
f(pJ)
If t h i s
over
= Pj(p)
condition
~-function
(with
Pj(x) = 1 + j ) , f
there
one has
f
is s a t i s f i e d
Let
Let
x I ..... x k
which
are prime
by
5.6.
say t h a t
E(f))
it w i l l
k, or
P
primes
multiplicative
over
p
Z
we
such
shall
exist
let
be all
solutions
to
if t h e r e
N,
polynomials
for the E u l e r ' s function
(with
order. T h e exact order
largest
integer
(or
~)
k.
and
of t h e exist
p Pr(p)~j(~d N)
it h a p p e n s
function
f
is decent and
~E(f).
N
~
as
as the
of order
is
that
speak
one has
for the d i v i s o r
Every polynomial-like N
be howe-
the polynomial-
by
if t h e r e
is of infinite
f
f
(5.8)
is d e f i n e d
r S E ( f ) , and
p-S=
p f(pr)sj(modN)
for a l l
Pj(x) = x J - 1 (x-l))
the order of its decency at
Proof.
N
precisely,
k,
value
observation.
a polynomial
for all
is p o l y n o m i a l - l i k e
PROPOSITION
of
integer-valued
of o r d e r
that
this
is f o r m e d
f(p) = P ( p ) . M o r e
such
value
(j=1,2 ..... k) .
then we
(denoted
for w h i c h
exists
possible
of decency of
for t h i s o n e n u m b e r ,
to a p p l y
as t h o s e
function Z
only
functions
we define
a polynomial-like
the order
for a p a r t i c u l a r conditions
of d e c e n t
for w h i c h
primes
and t h e m a x i m a l
be called
an o p p o r t u n i t y
which
PI ( x ) , . . . , P k ( x )
of
WUD these
not have
-like f u n c t i o n s , functions
with
will
Re s ~I.
j ~ I) should be positive.
(r ~ T ,
decent,
be called
infinite)
to c o n s i d e r
shall
for all
will
a(r,j)
p-S=
j , with congruence
any.
Pr(X)
b e given.
~j (mod N)
Then
k
j=1
(j,N) :I
p-S = P p--xj(mad N)
k
log
+ g(s)
7S
with
g(s)
rem.
We m a y
regular
that
congruence,
hence
for
put
Re
s al
a(r,j)
then
the
by the D i r i c h l e t ' s
=k/o(N).
If there
are
Prime
Number
no s o l u t i o n s
Theoof
series
p-S P
Pr (p)-j (mod N) reduces
to a f i n i t e
2. T H E O R E M
sum and
If
5.7.
f
so we m a y
take
of
vanish and
f
Proof. sible,
f
at
is
for
a(r,j) .
r
assume
then it is also
N)
that
the
i=I,2, .... r-1
index
and
all
Q
function,
not exceeding
such that not all numbers
D-WUD(mod
We m a y
thus
N
for
is a decent m u l t i p l i c a t i v e
a given integer and there exists an index of decency
O
a(r,j) WUD(mod
((j,N)=I) N).
r
is c h o s e n
as s m a l l
j
prime
N
to
N a 3
the order
the
as p o s -
series
p-S P
f(pi)---J (mod N) represents
a function
regular
for
Re
s a I, and h e n c e
the
same
applies
to E p-S P (f(pi) ,N)=I
Note
also
that
E
the
series
I/p
P (f (pr) ,N)=I
certainly
diverges,
since
otherwise
the
series
p-S P (f (pr) ,N)=I
would
represent
r. T h i s also
shows
that
for
a function
that
for
f
any c h a r a c t e r
regular
at
the n u m b e r X(mod
N)
s =I,
contrary
m(f,N) and
to the c h o i c e
coincides
Re s > I / r
with
of
r. O b s e r v e
74 X(f(pr))p-rS =
Z X(j) (j ,N] =I
p
X(j) (a(r,j)log E (j ,N) =I =
holds
+gr,j (rs)) =
with
hj (s)
regular
for
+hi (s)
Re s -> 1/r
n-S -~(N)I I ~ X(j) E n [X#X o f (n) -=j (rood N)
Also
~ X(j)a(r,j) (j ,N)=I hy Lemma 5.2 we get
F(S)
=
and
B (X) =
Since
~ n -s = n (f~n) ,N)=I
and thus by Lemma
5.2
hj (s,x)~(X___~+ h(s'X°) ~ (s-I/r) (s-I/r) B (X°) ]
hj (s,X)
regular
for
Re s a I/r,
h ( s , X O) (s-I/r)B (Xo)
by a s s u m p t i o n
l'm s÷~+0
F~ (s)/F(s)
and also = B(X O)
Re
: 1
8 (X) _< 6(X o)
it follows
equality
g o (s)
go(S) ..... gq(S)
f(n)-j(mod
+
regular Proposition
N) } =
possible Fj (s)
only
in case
B (x)
in the form
I -aj
q
(s-I/r) 8 (Xo)
go(l/r) = h ( I / r , X o) A p p l y i n g finally
#{n_<xz
with
that we can write
I
Fj (s) - ~(N)
with
I
I(j ,N)~ =I X(j)a(r,j) }log ~ i
Fj (s)
with
~
~ p-rS p f(pr) __-j(mod N)
~ gj (s) (s - r ) j=1 for
Re s -> I/r,
4.3 to
F(s)
go (I/r> (r (p(N)F(B(Xo))
Re ~j < ~ (Xo) and
+o(1))
Fj (s)
xlog
and
we arrive
B (Xo) -I
x
at
75
and
#{n-<x:
(f(n),N)=1}
rg o(I/r) B (Xo)-1 (~(N) F(S(Xo)) +o(I)) x l o g x
:
which due to the arbitrariness
of
We may thus use T h e o r e m
j
proves that
5.1 for checking
f
is
WUD(mod N).
WUD(mod N)
D
for decent
functions. 3. In the case of polynomial-like a criterion P1,P2,...
for
COROLLARY. order
WUD(mod N)
occuring Let
the m u l t i p l i c a t i v e sisting
of those
RI,R2,...,R E are empty). Then
such
(in case
m =m(f,N)
Let
the function
f
Let
E =~
~D(mod
be
X(mod N)
N)
prime
to
N,
con-
be the sub-
that not all
sets
that not all sets
Rm
index with if and only
trivial
G(N),
of
the congruence
Aj :Aj (f,N) further
we assume
be the smallest
will
character
G(N).
for which
and assume
of exact
the subset
(mod N)
classes
in
Rj
by
and p o l y n o m i a l - l i k e
r ~G(N)
classes
is given by the following
Rj :R~(f,N)
by
of residue
generated are empty
non-principal p
be m u l t i p l i ~ a t i v e
has a solution
G(N)
of
f
group
Such a criterion
denote
it is useful to have
in terms of the polynomials
(5.8).
residue
Pj(x) ~r(mod N) group
expressed
in
j:],2, .... E
E. For
fnnctions
on
Am
is
~D(mod
non-empty.
if for
there
every
exists
a prime
that
X(f(pj))p-j/m
=
0
•
j=O In particular, is an odd
Am(f,N) =G(N)
if
prime
such
Am(f,pk) = G ( p k)
and
holds
however
for
Proof.
with
p =2,
is
WUD(mod
one
has
Note that the group
A m . Indeed,
the series
p-I P
f
f (pro)_--j(rood N)
then
f
Am(f,p 2) = G ( p 2)
that
p )
(with for
to assume
A
occuring
N)
and if
m =m(f,p2))
k ~I.
A similar
then result
Am(f,23) :G(23) in Theorem
Ri
5.1 coincides
76
diverges
if and o n l y
x
to
p r im e
having
N.
It s u f f i c e s
in mind,
k=I,2, . . . .
if the c o n g r u e n c e
cases
one can
Am(f,p2)
cases
PROPOSITION
corollary,
if
Prmof.
has a s o l u t i o n
5.1 and
m[f,p)
its C o r o l l a r y ,
= m ( f , p k)
for
in the last part of this C o r o l l a r y
= G ( p 2)
is a prime P m (x)
follows
by the w e a k e r
larger than
Let
Am(f, P) = G ( p ) .
and notation 2n, where
then from
"
for
condition
replace
in the n e x t P r o p o s i t i o n .
of the preceding
n
denotes
the de-
the equality
A m (f,p) = G ( p )
k=1,2 .....
In v i e w of the p r e v i o u s k =2.
a •G(p2).
Corollary
it s u f f i c e s
By a s s u m p t i o n
to c o n s i d e r
one can find
rl,...,r s •
satisfying
Pm(rl) ...Pm(rs)
Observe all t h o s e
now,
- a ( m o d p).
t h a t if the d e r i v a t i v e
x • G(N)
X P m ( X ) P 'm (x)
for w h i c h
p, and we get it v a n i s h e s
identically
1+n+n-1 = 2 n a p (mod p)
Since
find
ro EG(N)
Pm(ro)P-1
(with
the d e r i v a t i v e p)
(mod p),
p),
then
then
its d e g r e e
by
p
for
we have
p. If
equals
our a s s u m p t i o n .
P~(x) (mod p) V(x)
by
then the p o l y n o m i a l
points divisible
contradicting
polynomial
x = r o cG(p)
of the l e f t - h a n d there
such that
~I (~od p),
Pm(rl)...Pm(rs)Pm(ro)P-2pm(x)
x ~ro(mod
IO(mod
is d i v i s i b l e
at least
If h o w e v e r
vanishes
identi-
Pm(X) = v ( x P ) ,
thus
a contradiction.
H e n c e we m a y
is s o l v a b l e
P'm (x)
at i n t e g e r
identically,
thus w i t h a c e r t a i n
p, again
Pm(x)
has all its v a l u e s
it d o e s not v a n i s h
c G(N).
one has
Under the assumptions
5.8.
p
Am(f, p ) = G ( p k)
n
p
is p r e s e n t e d
gree of the polynomial
cally,
n o w to r e c a l l T h e o r e m
that for p r i m e
the a s s u m p t i o n One of t h Q s e
G(N)
(x) z j(mod N)
D
In c e r t a i n
the case
P
eG(N)
and
P~(r o)
the c o n g r u e n c e
-a
and side
exists a solution
Pm(rl) . . . P m ( r s ) P ~ ( r o ) P - 2 p m ( y )
Pm(ro)
- O ( m o d p)
Pm(x)
~P
(r o) @ O ( m o d
d o e s not v a n i s h of
- a ( m o d p2)
p))
(mod p)
and at
since
77
with
y-ro
/O(/~od p).
Pro(Y) - P m ( r o ) (rood p~ proving
Since implies
Pro(Y) E A re(p21)
I. N O W we g i v e
of d i v i s o r s
some e x a m p l e s .
for a l a r g e c l a s s
a e A re(p2) ,
for c u r v e s
we p r e f e r
to u t i l i z e
only simpler
first example
l e a r n the r e a s o n s
PROPOSITION
Proof. j
converse
however
the n u m b e r
N
d(pJ) = l e j
d(n)
d(n)
of p o s i t i v e
complicated.
root
holds
it s u f f i c e s
we assume
that
~D(mod
Later
divisors we shall
thus
p
X mod N
is
q
N-I ~ X(k) k=1
Aq_ I ~ G ( N )
thus N-I X(i)z i = 0 .
~ zJ = z - I ( I j ~2 j-k (mod N)
denotes
(5.2)
N-I ~ X(i)z i. i=I
q the
c a n n o t be
p. To o b t a i n write
the
if
N). To p r o v e
then
z =p'1(q-~)
-zN) -I
the d i v i -
the first
and we see that %~D(mod
and ~ p r i m e
h o l d s and p u t t i n g
j zl
q-l, w h e r e
Aq-1 = g P { q } d(n)
and
Rj = {1+j} n G(N),
equals
then
(mod N).
0 = k X ( d ( p J ) ) z 0 = ~ X ( 1 + j ) z J = 1 + z -I ~ X(j)zJ = ~=0 ~=0 j=2
i=I
stage
if and only if
N)
root
for p r i m e s
Because
to show that if
with a character (5.2)
N,
(mod N),
is
is a primitive
it is n o n - e m p t y
satisfied
= I + z -I
to
at this
tools.
The function
not d i v i d i n g
is a p r i m i t i v e
leads to an
on a C o r o l l a r y
by A . W E I L ,
looks r a t h e r
is p o l y n o m i a l - l i k e .
for w h i c h
least prime
M*(f~,
proved
an a l g o r i t h m
f
for that.
5.9.
Since
set
concerns M*(f)
the least prime not dividing
sor f u n c t i o n
L a t e r we shall p r e s e n t
based
conjecture
n. Here the set
~-function
of the
determination
Our
and E u l e r
functions
effective
index
we o b t a i n
of p o l y n o m i a l - l i k e
Riemann's
of
(rs) eArn(p21), and
Am(p2) = G ( p 2) .
§ 3. T h e n u m b e r
which
Pm(ro~,Pm(r11) ..... P
this
78
This tainly
shows
not
the
that
z
case,
and
must
b e an a l g e b r a i c
this
contradiction
integer, shows
but
that
this
(5.2)
is c e r -
cannot
hold.
One
sees
dividing list:
easily,
N
that
all n u m b e r s
is a p r i m i t i v e
N =4,
N =2"3 a
root
(a k l ) ,
is a p r i m i t i v e
root
(mod pa))
is a p r i m i t i v e
root
(mod pa)).
2. O u r set
second
M*(f)
PROPOSITION
to
example
looks much
5.10.
N,
for w h i c h
(mod N)
N =pa and
are
(p - an o d d N =2p a
concerns
the
least
included
prime
in the
prime
a ~I,
(p - an o d d
prime,
the E u l e r
function
~(n).
not
following and
2
a ~I, 3
Here
the
simpler:
The set
consists
M*(9)
of all numbers
prime
6. Proof.
values and
If
for
N
2 cG(N)
{r:
are all
the
set
I -
Now we rule necessarily
out
p =2,
O = I +
=
is even,
n k3
~ j=1
then even.
RI
(r(r+l) ,N) : I }
the
cannot thus
contains
I,
be N
thus
WUD(mod be odd.
N)
since
Since
is n o n - e m p t y
P1(x) and
its =x-1
equals
.
possibility
of
(5.2).
Indeed,
if it h o l d s
then
thus
X(~(2Jll2-J
(3-X(2))/(2-X(2))
a contradiction.
~ Let
Thus
=
I +
~ j:1
X ( 2 ) J - 1 2 -j
1
I + ~w ( I
=
X(2) --T-)
@ 0 ,
N cM*(f)
if a n d o n l y
if the
set
R1
generates
G (N) . If to u n i t y 3~N. and
If
N
is d i v i s i b l e (mod 3),
by 3,
then every
RI
does
aI at N = P l "''P , then
G(N)
so w e m a y
thus
represent
[Yl ..... Yr ]
every
not
is t h e
element
ai) (Yi ~ G ( P i ) "
element
generate
of
of
product G(N)
R1
G(N).
is c o n g r u e n t
Assume
thus
of the g r o u p s
in t h e
form
that ai) G(Pi
79 In particular
the set
RI
equals
a.
{[Yl ..... Yr ]: Yi £G(Pi ~)' PiXI+Yi } •
To prove that it generates
G(N)
solve,
for given
[yl,...,yr] , the
congruences
2Wi
(i=1,2,...,t) ,
- Yi (rood pi I)
define
y±
if
Vi = [ 2
Yi ~-I (mod pi)
otherwise
and
[ I
if
zi = ] w i
and observe product
Yi ~-I (rood pi )
otherwise
that
equals
[V1,...,V
;~*(a)
consists
[zl,...,z
for
involved reasoning
o(n) = d ~ n d.
of all integers
sult later
and
I. In the previous One can however
G(N). X
N
we have
~733)
that
~2(n) = d~N d 2
P1(x) =I e x 2,
M*(o)
group
(5.2)
%~D(mod N)
A=(f,N)
exactly
coincided with
of m u l t i p l i c a t i v e
W~D(med N}
with
A=
One needs only to satisfy the equality
of the quotient
for the function -like, with
and their
We shall deduce this re-
of the sum
easily produce examples
which w i t h a suitable
(J.~LIWA
by 6.
two examples we get
when the appropriate
subgroup of
RI
6.9).
§ 4. The vanishing
characters
lie in
leads to the d e t e r m i n a t i o n
It turns out
not divisible
(see Proposition
those cases,
]
[Yl ..... Y ]"
A similar but more of
]
group and
G(N)/Am(f,N). N =40.
In fact
P2(x] =I + x 2 + x 4, thus
in G(N),
functions
for
being a proper (5.2)
for all
This can be done ~2
is polynom±al-
~(~2,40) = 2
(Pl(X)
80
is always even for odd tion).
The set
R2 Xo
and
R2(o2,40)
112 ~ 1(mod 40)
and
X(mod
equals
{3,11}
is the principal
which
(mod 8)
hence
~2(2 k) ~ 5 ( m o d
G(40). A2
(mod 40)
8),
so
x =I A2
and
X8
34 ~ 1(mod 40), generated by
X =XoX8,
where
is the only non-prinn =3.
22k+2 -I ~7(mod
X8(o2(2k))
for solu-
The only non-trivial
equals
which equals unity for
= I + 2 2 + 2 4 +... + 2 2 k = (22k+2 -I)/3,
has
the group
is trivial on
character
40)
and in view of
40)
thus is of index two in
40)
cipal character
P2(x) ~3(mod
3,32,33 ~ 1 1 ( m o d
has 8 elements,
character
x
Since
8)
q'2(2k) =
for
k al,
=-I.
Moreover = ~ 0
k
odd
[ I
k
even ,
X°(a2(2k))
thus finally
/
I + ~ X(o2(2k))2 -k/2 k=1
and
~D(mod
40)
follows
2. The index of
A
I-
w 2 -k/2 k>O 21k L
from the Corollary
(see exercise
f(pk).
This Proposition
dispose quickly of the possibility
,
to T h e o r e m 5.7.
large without
I), however
that this is not the case under
on the values
0
=
can be arbitrarily
weak uniform d i s t r i b u t i o n implies
=
spoiling the
the next P r o p o s i t i o n
certain additional
assumptions
allows also in many cases to
of having
I,~D(mod N)
without
A(f,N) =G(N) . PROPOSITION
5.11.
(mod N)
of order
propert y
that
is an integer
d
Let
N a3,
and
al,a2,..,
the sequence T
such
- X(aj)(rood N)
If for
k=1,2,..,,d-1
1 +
Xk(aj)p -j/M = 0
~
be integers, a sequence
x(aj) (mod N)
X
a character
of integers
is purely
with
periodic,
the
i.e.
that
X(aj+ T)
j=1
M ~ I
(j=1,2 .... ) .
there
exists
a prime
p =p(k)
such
that
there
81
then
p(k)
=2,
X(aj)
the character
-I
cf
M]j
o
if
Mfj
X
i.e.
d =2
and m o r e o v e r
=
Proof.
Let
T
be a p e r i o d
of
restricting
the
generality
characters
Xk
(k=I,2 .... ,d-l)
with
is real,
that
X(aj) (mod N) T
exceeds
and
M.
and d e n o t e
assume
Let
by
×
dk
without
be one
of the
its order.
We have,
p =p(k),
T
O = 1 +
~
x(aj)p -j/M = I +
j=1
,~
X(ar)
Z
j:1
p-j/M
j_>1 j-r (rood T)
= I +pT/M(pT/M
T
_1)-I
X(ar)P -r/M r=]
Putting
Y
T
for
shortness
T-I ~ r=1
+
x(a r) yT-r
Since
the v a l u e s
have
x(a T) = 0
or
which
it is not.
sible
by
and
to
D
the
is a unit, must we
×(a i]
]
since
Thus
D
dk
Now
K
If
be
also
y
would
is d i v i s i b l e
in the
would
a unit,
of u n i t y
denotes
cannot unit
integer,
of u n i t y
divi-
×(a T)
we
the field g e n e r a t e d
N(x)
a prime
which
we
be an a l g e b r a i c
algebraic
and by
is n o t
and
form
the n o r m power
then
is a n o n s e n s e .
since
N(y)
d
let
is a p o w e r
by
from
K
X(aT)-] Thus of
p
Then
X
t
= pt[Dt]dt
by
be a p r i m e
q,
and
We
show
p(k), divisor
applying
that
D
_ s D --po
say
q
now
if
-I).
power,
or r o o t s
y
of the r o o t
N(y) IN(×(aT)
is of o r d e r shall
order
of u n i t y
a certain
let
zero
is a n o n - z e r o
the
equality
(5.9)
case
d k. M o r e o v e r ,
p(k) I N ( y ) i N ( x ( a T) -I)
hence
either
root
hence
this
.
d-th
be a p r i m e
get w i t h
are
= 0
in that
×(a T) -1 by
divides primitive
Q, t h e n
we can w r i t e
+ × (aT) _i
of
y. D e n o t i n g
see that y
Y =pl/M
q =2
the and
of
last
and
observation
to do
this
X =xd/q. we
get
we a s s u m e ,
q =p(d/q). a contrario,
D
82
that
q =p =p(d/q)
vial Y
powers
of
is an odd
X
are
( = p(d/q) I/M)
y
for
T
T-I ~ r=1
+
also
prime.
Note
of o r d e r
q
also
that
thus w i t h
since the
all
same
non-tri-
value
for
we have
X k(a r)y
T-r
Xk
+
(a T ) - I = O
k = I , 2 , .... q-1. Adding
these
(q-1)yT
equalities
(q-l)
+
~
we
get
yr
_
yr
I ~r<-T -1 X
- q = 0
(5.1o)
1 <_r_
(aT_r) =1
x (aT_r) ~0,1
because
[ q-1 q-1 ~. k=1
[
Now root
observe
of u n i t y
unit
if
x (c) = I
0
if
X (c) = 0
-I
if
X (c) ~O, ]
I ×k(c) = ~
c
that ~q
we can w r i t e
and
a suitable
×k(a T) = ~qs
with
] ss sq-1.
Since
a primitive with
q-th
an a p p r o p r i a t e
we h a v e
q-1 q = ~j=1
(I -~J) q
= ~-I (I _~q) q-1
thus
e-q =
Denote obssrve
( ] - xk (aT)) q-1 =
by
that
j
the m i n i m a l
it is i n d e p e n d e n t
T-I
eY M
(yT +
= eq =
(yT +
X
k
T-] ~ X k (aT_r)yr) q-1 r: ]
index of
(aT_r)y
a ~ k.
r
+
such
that
xk(aT_j)
~0
and
Now
xk
" q-1 (aT_j)y3)
=
yj (q-l)
-A
r=1+j
where
A
is an i n t e g e r
of the
field
K
not d i v i s i b l e
by any
prime
83 i d e a l of
K
which divides
M = j(q-1)
y, t h u s
-> 2j > j
results. On the o t h e r
O -= yT +
hand from
~ yr s I-
The r i g h t - h a n d by
yj+1,
hence
divisor dk
of
th a t
d,
are p o w e r s
so of
we get,
yT + y j (I +
using
results, q
d
must
M =q,
is d i v i s i b l e
by
yJ
but not
a contradiction.
c a n n o t be odd,
thus
be a p o w e r of
2 and since
y
Z yr-j) (mod yM) ~
side of t h i s e x p r e s s i o n
j aM
We see thus
(5.10)
q =2
is the only p r i m e
2. It f o l l o w s
p(k) Id k
we o b t a i n
X =xd/2
which
that all numbers
p(k) = 2
for
k = I , 2 ..... d-1. Consider Since ger,
from
n o w the c h a r a c t e r (5.9)
follows
ylx(aT]-1
and
it m u s t be e v e n h e n c e we o b t a i n
T y
is of o r d e r
X(aT]-I
x(a T) =-I
2, h e n c e real.
is a r a t i o n a l
inte-
, thus
yT-1 +
(a I)
S i nce
y
+... + (×(aT_M)-I)yM +... +X(aT_1)y
is not a u n i t we m u s t
X(aT_I ) = X ( a T _ 2 ) =
= O
have
... : X ( a T _ M + I) : O
thus
T-M y
which
yT-M-I + x(a I)
gives
+... +
X ( a T _ M) -I
= 0
X ( a T _ M) = -I, and the r e p e t i t i o n
of our a r g u m e n t
X ( a T _ M) =-I and X(aT~M_I) = . . . = X ( a T _ 2 M + I ] =O. C o n t i n u i n g in this w a y we a r r i v e f i n a l l y at
x(aj ) = ~ -I O
if
j is of the
otherwise
form
T-rM
(r ~ Z)
leads to
84
and since the c o e f f i c i e n t sibility
of
x(aj)
T
by
of
Y
T -M
is n o n - z e r o
we o b t a i n
also the d i v i -
M, thus
= I -] 0
if MIj otherwise
results. To c o n c l u d e l i t i e s we get
the p r o o f X(aj) = 0
it r e m a i n s for
M~j,
d =2,
to p rove
N o w f r o m last
equa-
thus
0 = I + ~ X ( a j ) 2 - j / M = I + ~ X ( a j ) 2 -j/M = I + Z X ( a k M )2-k j=1 j->1 k:1
MIj and this e a s i l y
implies
d #2,
w o u l d be a n o n - p r i n c i p a l
then
X2
X(aj) = - I
for
j
divisible
character,
by
M.
If it w e r e
and so we w o u l d
have
0 : 1 + ~
X2(a.)2 -j/M : I + ~
j=1
J
2 -j/M
j>_1
=
2
MIj a clear contradiction.
COROLLARY~
Let
N ~3
g e r - v a l u e d function, prim~
p <2 m
(k=I,2,...)
on
~m(f,N) = G ( N )
m(f,N) = m
then or
one has
X(f(2k))
= I -I
If
f
if
character
X ( m o d N),
and
f(pm) =V(p)
can happen only if
first assertion
last p r o p o s i t i o n ,
is a subgroup
N)
X(f(p¢))(mod p) if and only
of index
2 in
which is trivial
m k
is p o l y n o m i a l - l i k e
The
D-WUD(mod
inte-
If for every
the sequence
will be
Am(f,N)
a multiplicative
otherwise.
the second p o s s i b i l i t y
Proof.
f
is defined.
X ( m o d N)
f
and for only n o n - p r i n c i p a l Am(f,N)
and we are ready.
be an integer and
and every character
O
then
d =2
for which
is periodic,
if either G(N)
Hence
will
o n c e we e s t a b l i s h
follow
with a p o l y n o m i a l N
from Theorem
t h a t the
V(x),
is even.
index of
5.1
and the
A m (f,N)
in
85
G(N)
is in c a s e
that not
of
~D(mod
the c o r r e s p o n d i n g equal
say
C2,
Xl,X 2
satisfy If Am(f,N)
then we have
and
XlX2,
X(f(2m)) f(pm)
=V(p), then
c Am(f,N)
thus
all
=-I,
=G(N),
-I = X ( f ( 2 ) )
N)
factor
is o d d
I. In t h e (5.2)
to e l i m i n a t e and
only
this
that
section
and we
this
reduced The
Let
we
proved
we
shall
have
for a l a r g e
to the case, condition,
m
prime powers
class
the
when
and
N
WUD(mod
on
as w e l l
m
If
is their product,
P(2)
e
we have
for the
permit
D-WUD(mod
one has
at this
equality
in m a n y
N)
cases
holds,
Am(N) = G ( N ) .
condition
functions, others,
of the condition
is t h e
integer.
Am(f,N)
which
as c e r t a i n
is a p r i m e
without i.e.
= G(N).
cases,
look
should
D
of m u l t i p l i c a t i v e
it,
characters, all
N)
=I,
the c o n d i t i o n s
of
shows
if it d o e s
possible.
(2P(2),N)
Am(N)
In s u c h
study N
ensuring
be a positive
not
results
value
C 2, b u t
non-principal
trivial
analyzed
a closer
functions
T-function
X
eertain
possibility.
polynomial-like
Ramanujan's
thus
equality
if for a n a p p r o p r i a t e
section
prove the
last
to h o l d ,
last proposition
of
Am(f,N) , so t h e y
a contradiction.
§ 5. T h e
The
and we have =I,
for a n y c h a r a c t e r =I,
three
on
is o b v i o u s l y
(f(2m),N)
= X(P(2))
2.
is a p o w e r
at l e a s t trivial
which N
at m o s t
group
and
if In
shall
which
includes
including
Am(N) = G ( N )
can be
are pairwise
coprime
power.
following: a aS ql I .... 'qs
then the series
i/p
(5.11)
P f (pro)~r (roodN )
diverges
for a certain
Z P
I/p a.
f(p~)zr(mod qjJ) diverge.
r
if and only if all series (j=1,2 ..... s)
(5.12)
86
It can be also formulated in the following,
equivalent, way:
aj If
R(f,q~
classes then
)
(j=1,2 ..... s),
(mod ~qjj),a resp.
R(f,N)
denote
for which
aj
if We regard
G(N)
Observe,
as the direct product of the groups
that this condition,
least equal to
m. R a m a n u j a n ' s
m =1,2
G(qj
which we shall d e n o t e by
satisfied by all p o l y n o m i a l - l i k e m u l t i p l i c a t i v e
tion for
the set of all residue
(5.12) resp. (5.11) diverges, is the direct product of the sets R(f,qia i ) (i:1,2,...,S),
R(f,N)
(mod N)
).
(Cm), is
functions of order at
T - f u n c t i o n also satisfies this condi-
however this fact lies m u c h deeper.
2. We prove now
aj
s
PROPOSITION
5.12.
Let
N =
~
qj
a3
be an integer and
f
a multi-
j=1 plicative
i n t e g e r - v a l u e d function,
Assume further G(N)
that
f
satisfies
will be g e n e r a t e d by
aj
characters fying
Xj(mod qj )
Xj(n) =cj
their product
for
for which
R(f,N),
aj
cic2...c s
Proof.
The c o n d i t i o n
R(f,N)
=
equal
)
is defined.
(Cm). Then
except in the case,
(j=1,2 ..... s), n ER(f,qj
m(f,N) = m
the condition
the group
when there exist
not all principal,
and satis-
with suitable constants
cj
having
to I.
(C m) gives
S
~ R(f,q~ J) j=1
so we may appeal to the f o l l o w i n g e l e m e n t a r y observation:
L E M M A 5.13. Let direct product. and let
R cA
generate
A
i=1,2,...,n,
Ri
not all trivial, say and
and
A
their
be a non-empty subset of Ri's.
if and only if there exist characters
ci,
Proof. XI...X
Let for
be finite abelian groups
be the direct product of the
(i=1,2,...,n), there to
AI,...,A n
Then
R
X±
which are constant on
Ai
does not
of
Ai
Ri, being equal
Cl...c n =I.
If such c h a r a c t e r s
Xl, .... X n
exists, then their p r o d u c t
is n o n - t r i v i a l and equals unity on
R, hence
R
lies in the
n
kernel of R
X1...Xn, which
cannot generate
is a proper
A. Conversely,
if
subgroup of R
A. Thus in this case
does not generate
A, then
87
there on
is a c h a r a c t e r
R. D e f i n e
the
X i = X I A i. T h e n r i £A i
X
of
A, w h i c h
charcters
clearly
Xi
of
X = X i . . . X n.
is n o n - t r i v i a l Ai
for
If n o w
and
equals
i=1,2,...,n
for
by
i=1,2,...,n
unity
putting we h a v e
then
n
(5.13)
Xi(r i) = I i=1
holds
and
if
X~(sj)
I -<j -
and
~ X i ( r i] = 1~i~n
s. e R , t h e n J J
I
i#j implying equals
Xj (rj) = X j ( s j ) .
to
cj
To a p p l y prime
general
number
LE~4A subset which ter on
is
case
If
constant
on
q2)
will
to know,
Here
to d e a l
R~
whether
character
for w h i c h
if
be
of
be a prime,
there R, q
we
shall
with
and
we
for a g i v e n which
shall
is p o s s i b l e only
primes,
if it
[]
(mod qa)
chapter this
It s u f f i c e s there
exists
(mod q2)
a ~3
there
q2),
on the
and
exists
also
(mod 8)
resp.
point their
give
a
in the
out
that
squares
R
and
of
that
every
always
such
a power
(mod 8).
be a n o n - e m p t y
character
X(mod
a non-principal
if
(mod 8).
reduction
to note
resp.
let
a non-principal
odd resp.
R(mod
trivial
exists
then
is
Proof.
character
powers
functions.
q
R,
and
needs
on
cic2...c n =I.
R(f,qa) . In the next
prime
it s u f f i c e s
Let
G(qa).
the r e d u c t i o n ×
be c o n s t a n t
is a n o n - p r i n c i p a l
set
all
one
must implies
8:
5.14.
of
×(mod
then
on
find
Xj
(5.12)
proposition there
on the
to
Thus
then
of p o l y n o m i a l - l i k e
in the the
qa
unity
procedure case
this
power
equals
there,
q =2 If
X
which
qa)
charac-
is c o n s t a n t
is t r i v i a l
on
R,
R.
power
of
which
X
must
be c o n s t a n t
is a n o n - p r i n c i p a l
88
§ 6. R a m a n u j a n ' s
z-function.
I. Before d e v o t i n g our a t t e n t i o n e n t i r e l v to p o l y n o m i a l - l i k e tions we want to c o n s i d e r R a m a n u j a n ' s tance in the theory of m o d u l a r
function
func-
T(n) , which is of impor-
forms. We recall,
that it is defined by
the e x p a n s i o n
~] T(n)x n = x ~i- (I -xJ) 24 n=1 j=l
is m u l t i p l i c a t i v e and for
T(p~+1)
and prime
p
satisfies
= T(pn)T (p) _ p11T (pn--1)
(See e . g . T . A P O S T O L of
n~I
(I~I
(5.13)
[76]). The values of
n, used below,
T(n)
for certain small v a l u e s
are taken from the table given by J . P . S E R R E
[68], but
with a certain pain can also be d i r e c t l y e x t r a c t e d from the above expansion. We shall need also certain deeper p r o p e r t i e s of based on P . D E L I G N E
L E M M A 5.15. Let
satisfying exists
the conditions
p-S = c(A)
most of t h e m
Re s a~
log ~
1
c(A)
p
~(p) ~ J 2 ( m o d M2) . Then
p ~ J l ( m O d MI) ,
constant
satisfying
be the set of all primes
A =A(MI,M2,Jl,J2)
a non-negative
in the half-plane
be given integers,
MI,M2,Jl,j 2
and let
(JI,MI) =I
T(n)
[69] which we state in the f o l l o w i n g three lemmas:
and a function
such that for
Re s >I
g(s;A)
there
regular
one has
+ g(s;A)
peA
The constant qk
dividing
c(A)
(MI,M 2) such
p ~ J 1 ( m o d qk) we have
belongs
vanishes
to the set
that for every prime
T(p) l J 2 ( m o d qk).
p ~q
satisfying
This can happen
only if
{2,3,5,7,23,691}.
(This lemma is T h ~ o r ~ m e
~J of J . P . S E R R E
that there it is assumed that c o n c l u d e d that
if and onl~ if there is a prime power
M2
[72] with the d i f f e r e n c e
is prime to
2-3-5-7-23.691
c(A) >O. The proof is in both cases the same.)
and
89 LEMMA
5.16.
(i) (ii)
T(p) T(p)
(iii)
T(p)
(iv)
~(p)
(v)
and
p
(p)
- -I ( m o d
(vi)
(p #7) 72).
~(p) =-O(mod 23),
then
is of the form
x 2 + 2 3 y2 , then
but
p
23) .
691).
(All those
congruences,
save the second
in J . P . S E R R E
[68],
by H . P . F . S W I N N E R T O N - D Y E R
shows
that
we shall there
7, 23 and 691. not need
in
(v) are classical,
and H . P . F . S W I N N E R T O N - D Y E R
of t h e m are d e s c r i b e d .
Finally
x 2 + 2 3 y2 , then
is not of the form
=- I + p 1 1 ( m o d
listed
The
second
[73],
congruence
in
where (v) was
and the found
[73].)
need
a result
of H . P . F . S W I N N E R T O N - D Y E R
are no c o n g r u e n c e s
(There are similar
of this
results
type
[77] w h i c h
for h i g h e r
for other
powers
primes,
of
but we shall
them here.)
LEMMA tant
(p 35)
<(p)
origins
hold:
232 )
I +p11(mod
(p/23) =+I,
and if
52 )
~(p) =-p + p 1 0 ( m o d
then
(p/23) =-I,
If
~
the f o l l o w i n g congruences
_- p + p 4 (rood 7)
(p/23) =+I • (p)
p
- p +p10(mod
(p/7) =-I
and if
if
For primes
- I + p ( m o d 8) (p #2) _ p2 + p 3 ( m o d 32 ) (p #3)
5.17.
c(A)
(i) If
for
(ii) If
(a/7) =I
A =A(72,72,a,b)
(a/23) =-I
A =A(232,232,a,b),
and
and
b :a + a 4 ( m o d
7),
then the cons-
is positive.
b-O(mod
23),
then
and the same holds in the case
c(A)
>0
for
(a/23) =+I
provided
b ~--I (rood 23) . (iii) tant
If
c(A) 2. U s i n g THEOREM
b-I
+a11(mod
691),
then for
A ( 6 9 1 2 ,6912,a,5)
the cons-
is positive.
these 5.]8
lemmas
(J.P.SERRE
if and only if ei'ther and not divisible
we p r o v e
N
[75]).
now The function
T(n]
is odd and net divisible by
neither by
3 nor by 23.
is 7 or
~UD(.mod N) N
is even
90
Proof. fies of
Lemma
5.15
the c o n d i t i o n N
odd and
N
=
shows
(C I)
that
from the
•
is d e c e n t
of order
last
section.
Consider
z l and first
satis-
the case
let
s
a.
~
qi =
i:I
be
its c a n o n i c a l To a p p l y
5.14,
factorization
Theorem
to k n o w
that
with
qi~T(p) , but
take
p =2
good
and
lues the
of
residue of
T
equality
character we would
will
for ~5.2)
in v i e w there
and
for
the
be
WUD(mod
(mod N)
would
hold with
equal
are
for
qi # 3
Theorem
only
prime
group
m =~,
to u n i t y
one prime
the prime
Hence
if a n d
the
last part of Lemma
to
p :2
on
RN,
~%N' a n d
thus
N
is a p p l i Using
set
it of
and contain
va-
Indeed,
for
is
RN
and every
thus
we may
5.1.
if t h e
G(N).
p #q
p =7
5.7
given by Theorem
N)
which
generates
since
qi = 3
3).
criterion
of t h e
is at l e a s t
to e s t a b l i s h
p~N
X ( m o d N)
prime-powers.
~ - 1 6 744 ~ O ( m o d
apply
classes
z(p)
is e a s y
~(2) = - 2 4 )
T(7)
that
into
it s u f f i c e s , i=1,2,...,s
this
so w e m a y
we deduce those
for
(because
in v i e w
cable
5.7
otherwise
non-principal
such a character
have
X(~ (2 j))
however
= -I
~(2)
(j=1,2 .... )
mod N
lies
in
X(~(2)]
=I,
a contradic-
tion. We d i s p o s e in t h i s RN
case
cannot
shall
that
of the c a s e
element
generate
To prove we
now quickly
every
of
G(N)
for
N
RN
and we odd
71N.
Lenm]a 5 . 1 5
is a q u a d r a t i c
(iv)
residue
shows
that
(mod 7) t h u s
are r e a d y .
and
not divisible
by
7 we
get
WUD(mod
N)
s h o w t h a t for k=1,2,...,t the set Rk of r e s i d u e ak classes (mod qk ) p r i m e to qk containing the value of T(p) for ak P #qk generate G ( q k ) . In v i e w o f t h e C o r o l l a r y t o T h e o r e m 5.1 w e
may
first
assume
assertion
that
qk s2.
is c e r t a i n l y
5.16
(ii),
root
for a l l p o w e r s
congruent case 5.15 G(232)
resp.
to
qk = 2 3 (v),
5.77
which
Lemma
satisfied.
(iii),
2(mod
taking
of 3 a n d 32),
(ii)
and
the
congruent
shows For
that
the
same
(mod 52)
fact
that
the
we
qk #3,5,23,691 get
Because
applies
2 ER k
way,
to e v e r y
using
set of t h o s e
t o 2 or to
our
b y Len]ma
2 is a p r i m i t i v e
our a s s e r t i o n
in an a n a l o g o u s
either
for
qk = 3 , 5
p ~ 1(mod q~).
5 and
resp.
can be treated
are
5.15
integer
results.
The
the Lemmas residues
-1 (mod 23)
in
generates
91
G(232).
Finally
ment
of
thus
find
G(6912)
Rk
number
not
generates
show that
that
power.
such
that
congruent
Using ~(p)
11X~(691)
thus
Lemma
(iii)
5.17
is c o n g r u e n t
to u n i t y
(mod 691)
every
ele-
we can
(mod 6912 )
and
this
implies
G(6912).
the p r o o f
there
we n o t e
11-th
p ~691
To c o n c l u d e
not
qk = 6 9 1
is an
primes
to a g i v e n that
for
cannot
all p r i n c i p a l ,
exist
which
X k ( R k) = c k
it s u f f i c e s ,
in v i e w
of P r o p o s i t i o n 5.12, to ak X k ( m ° d qk ) (k=l,2,...,t),
characters
satisfy
(k=1,2 ..... t)
and
cic2.,.c % =
Assume
that
qk # 3 , 5 , 2 3
I .
this and
(5.J4)
holds.
691
Again
the
we may,
In c a s e
the
of all r e s i d u e s
c = Xk(2)
we
get
Xk
Xk(2 ) =±I.
is e i t h e r In c a s e
Since
classes 2
For
qk = 2 3 ,
u {x ~ - 1 ( m o d Xk ver
c =-I,
Xk(5)
=±i,
possible,
then thus
since
Finally, elements (mod 691) principal
of
if
one
that
for
For
a k s2. of L e m m a
5.16
(ii)
from
5)
since from
4 divides qk = 6 9 1 ,
of
(iii)
2(mod
9)
this
i.e.
we
shows
that
~(23]
as w e h a v e
G(691)
which
character
=Xk(25) by
~(232] o
seen
above,
constant
c =1,
G(232).
nor
are not
Rk
then
If h o w e -
=-I,
4, w h i c h
hence
is im-
contains
congruent
on all
=c =Xk(1)
23)}
If
and
Xk(52)
consists
be p r i n c i p a l .
~2(mod
be d i v i s i b l e
Rk
Xk(2)
must
c =±I.
G(23) get
that
thus
Xk
R k ={x
=I
must
5),
52 )
that
23)
Xk
then
only
5.16
(mod
seen
neither
resp.
(mod
-I g e n e r a t e
25 e 2 ( m o d
order
so the
and
root
c 2 =Xk((-1)2) 2 and
root
= (m/3}.
of t h e m
all
to u n i t y is the
character.
We see t h a t when
easily
be p r i n c i p a l .
assu~e
(in v i e w
from Lemma
already
thus
G(6912) and
Xk(m)
£1(mod
we h a v e
the
must
2 is a p r i m i t i v e or
is a p r i m i t i v e
23)},
is p r i n c i p a l ,
hence
one d e d u c e
of all r e s i d u e since
character
consists
sees
= Xk(2)2
= I
and
one
to Ler~na 5,14 Rk
3),
principal, qk = 5
due
set
z 2(mod
= Xk(8)
Len~na 5.~5
corresponding
qk • { 3 , 5 , 2 3 , 6 9 3 } qk = 3
using
of the
the
only non-principal
primes
qN,
say
ql'
character equals
Xk
occurs
3, in w h i c h
case
in case 2
ci=(])=-I,
=
92
but
then The
Now
the p r o d u c t obtained
assume
that
Cl...c %
equals
contradiction
proves
N
is even.
= T(p) 2 _ p 1 1 ,
thus
the v a l u e
and
N
t (p) m o d
order
22
and
Since
contradicting theorem
for p r i m e
T(p 2) m o d N
and we d e d u c e satisfies
-1, our
the
from
p
case
we h a v e
5.15
that
(C2) . As
22N.
T(p 2) =
is d e t e r m i n e d
Lemma
condition
(5.14).
in the
by
~
p(mod
is d e c e n t
in the p r e v i o u s
N) of
case
~,
write
s
N =
and
ai qi
~ i=I
observe
that
T(5) = 4 8 3 0 , apply
Theorem
First
for
hence 5.7
and
we d i s p o s e
elements
of
3 is not
a primitive
cannot for
RN
m =2
but
since
21T(2 n) equals
equal
(mod
or
3)
231N.
n 21,
thus
Lemma
because
to r u l e
RNO
and
due
(iv)
to
shows
3(mod
311 e 1 ( m o d
X(mod
the
since
that
23),
N)
(5.2)
are n o n -
for
p =2
this
eO(mod
2)
we have
left-hand
since RN
of
which
all
and
23),
the p o s s i b i l i t y
However
21N
and
thus w e m a y
5.1.
of
t (2) = - 2 4
to
2-23),
5.16
out
characters
on
(5.13)
=-26-23,
~O(mod
be c o n g r u e n t
23)
for
to u n i t y of
q ~(22)
~-511
of t h e o r e m
case
It r e m a i n s
p =2
all
case
in that root
in v i e w
for
we h a v e -511
criterion
of the
G(N).
(thus
-principal,
the
must
generate
sible,
qi ~ 2 , 2 3
x(52) = 4 8 3 0 2
side
is i m p o s -
of
(5.2)
unity.
Similarly, w e get
if
~(3 n+1)
left-hand
1 -
side
~
p =3,
~(3n-I) of
(5.2)
I
then (mod
to
(5.14)
and
I-5
~ =
3-k
and
T(3 2k+I)
is in a b s o l u t e
I
21k k-VTY: 3
due 2)
value
T(3) = 2 5 2 ~O(mod
at
2)
least
e O(mod
2)
and t h u s
equal
the
to
I > O
k=O
:2
"
k>O
A
similar
argument
shows
also
WUD(mod
N) . Indeed,
from
Lemma
follows
immediately
that
all
(rood 3), using
T(2)
by 3 for gruent =
thus
RN,
3)
n >-J
to u n i t y
on
cannot
-O(mod
all
(m/3)X o(m)
unity
RN
(mod
(with in c a s e
3) Xo of
5.16
~(3)
~(2 n)
and
of
RN
G(N).
-=O(mod
the
WUD(mod
n.
we c a n n o t
congruent
that
T(3 n)
3)
Since
have
T(p2) = ~ ( p ) 2 _p11 are
from
principal N)
31N
Moreover
is d i v i s i b l e
for e v e n being
in case
(ii)
elements
generate
and
and
that
by 3 for
(5.~3]
character satisfy
we
infer,
is d i v i s i b l e
odd
the c h a r a c t e r
it s h o u l d
it
to u n i t y
n
and c o n X(~n] --
(rood N]) (5.2]
equals
with
93
p =2
or
3 and
m =2,
T(3 n)
show
that
ruling
thus
out
The
N.
residue
classes
nerates
G(q~)
of the
It s u f f i c e s
is c o n s t a n t
the
~O
argument
(mod q~)
on
it.
Lemma
5.15
follows
the
for
for
holds
resp.
further,
shows
same
~(2 n) for
lines
i=1,2,...,s
qk ~2)
and
=O
and
all
n ~I,
(5.2).
that
(if
G(8)
congruences
X(~(3n))
of
to s h o w
resp.
obtain
and
the p o s s i b i l i t y
remainder
of odd
however
X(T(2n))
(mod
that
that
as
the 8)
in the c a s e
set
(if
Rk
qk =2)
to n o n - p r i n c i p a l
it is e n o u g h
of ge-
character
to do this
for
qk e { 2 , 5 , 7 , 6 9 1 } . In case In the hence
Rk
Since
qk = 2
same
contains
-principal (n/5),
Rk
In c a s e
qk = 7
(mod 72 ) that
all
However
~(32)
elements
in
if
provided
does
must
qk = 6 9 1 ,
not
Lemma
Rk to
to 7)
for
and
6
42 < 2 - 2 2
Since
only
5).
non-
character R k.
a(mod
elements,
72 )
residue
7) lie
at l e a s t
and in
this
R k.
3-7 +I = 2 2 the
only
one.
implies
that
is c o n g r u e n t
11~(6912)
3(mod
52 )
generates
every
(mod 7)
so we h a v e
(iii)
which
any
(a 4 +a) 2 - a 1 1 ( m o d
and
5.17
in
contain
be the p r i n c i p a l
(mod 6912 )
I or Rk
so the
3 lie
that will
3,5
has
to
the q u a d r a t i c
1 and
implies
Rk
to
every
contains
I + a 11 +a22(mod 691),
residue
(mod 6912)
thus
6912 ) : x -~1+y+y2(mod
Rk
contains
character
occur.
be
R k =G(8).
~I + p 2 + p 1 1 ( m o d
that
10 e l e m e n t s ,
must
set
G(72)
Rk
(1 +I +I 2 691 ) = -1 ,
this
(i)
~2(mod
if a n o n - p r i n c i p a l
quadratic
shows
this
and b o t h
congruent
since
on
power,
see t h a t that
and
congruent
Rk
the
to get
~(p2)
(mod 52 )
is c o n g r u e n t
643
class
R k = {x(mod
shows
which
691~a(1+a11+a22).
ll-th
and we
5.17
(5.14)
that
25) least
on
(mod 7)
=-113
Rk
residue
is an
Lemma
residues
constant
Finally, every
at
(i) and
qk = 5
(mod
trivial
a £1,2,4
character
5.16
(I/5) =1 ~ - I = (3/5)
satisfying
shows
root
contains
character
but
class
Lemma in case
all r e s i d u e s
3 is a p r i m i t i v e
G(52) . M o r e o v e r
the
we u s e
way we obtain
at
least
character
(n/691),
691(691-3)/2 is c o n s t a n t
however
'I +4__+42.) 691
This
691) , 6 9 1 Z y ( l + y + y 2 ) }
establishes
elements. on
in v i e w
of
Theorem
for
Rk,
This
it m u s t
= +I
the
N
even.
D
be
94
§ 7. N o t e s
I. The n o t i o n J.~LIWA
[76],
Theorem proved
of D i r i c h l e t - W U D ( m o d
where
in W . N A R K I E W I C Z
5.1 can be
(mod N)
KIEWICZ
[66],
was o b t a i n e d the general
of
images
In m a n y
cases
It w o u l d
fill
is due
they
m a n y values
of these
to
happens
p(n)
and
jectured
c(n))
(A.O.L.ATKIN,
that for every
N
was
settled.
D.W.MCLEAN
[80]
functions
for all
m,n)
satisfying
are
31.
are known.
of the m o d u l a r unadapted
values
~67]).
of
of
13
M.NEWMAN
(mod N) p(n),
[59] proved In T . K L ~ V E
solved
[77].
in-
to ful-
N
there class
(both for [60] con-
contains
and proved
it for
N =2
and
F70] the case
by A . O . L . A T K I N
in-
this
N =121
[68].
Cf.
results. (mod N)
[77],
who g a v e
was c o n s i d e r e d necessary
and
for m u l t i p l i c a t i v e sufficient
of c o m p l e t e l y m u l t i p l i c a t i v e
for all primes
the e x i s t e n c e UD(mod N]
coefficients
in e v e r y r e s i d u e
class
strongly multiplicative
f(pk) =f(p)
them he d e d u c e d tions w h i c h
and
and
The
for the p a r t i t i o n
for all powers
function
was also
for n u m e r i c a l
in the case
for certain
O.KOLBERG
N =7
distribution
by H . D E L A N G E
for that
13.
N =7,17,19,29
The case
3. U n i f o r m
tions
N =5
for
WUD c(n)
J.N.O'BRIEN
and
for
and
every r e s i d u e
conjecture
only odd
and m u l t i p l i c a t i v e
representations
functions
of the p a r t i t i o n
[68]
and in
no new problems.
seem c o m p l e t e l y
e.g.
finitely many values
T.KL~VE
case
5.11
by H . P . F . S W I N N E R T O N - D Y E R
UD
that
(mod N)
N. This
[82]
to the F o u r i e r
coefficients
are i n f i n i t e l y prime
A special
who c o n s i d e r e d
also
Z-adic
the known m e t h o d s only
proved in W . N A R -
Proposition
F.RAYNER
are integral
to study
It is known
[75],
applicable
and the Fourier
task.
[45].
case p r e s e n t s
image was d e t e r m i n e d
be i n t e r e s t i n g
5.7 was
theorem)
5.10 appear.
f
class
[83b].
of the c o r r e s p o n d i n g
p(n)
to T h e o r e m tauberian
5.9 and
to J . P . S E R R E
provided
[76]. Delange
function
in a fixed r e s i d u e
in L . G . S A T H E
the general
forms,
this
form in H . D E L ~ q G E
lies
5.5 appears
from it and
integer-valued
in W . N A R K I E W I C Z ,
in W . N A R K I E W I C Z
5.18
j, however
this
case
is in p r i n c i p l e
and the
function
already
in W . N A R K I E W I C Z ,
Proposition
resulting
The C o r o l l a r y
also P r o p o s i t i o n
N, however
same a p p r o a c h
f(n)
first
proved.
also via D e l a n g e ' s
in a special
of other m o d u l a r
variant
for w h i c h
5.9 occurs
case
5.1 was
the c r i t e r i o n
a density.
where
2. T h e o r e m values
n's
appears
in an e q u i v a l e n t
(although
of P r o p o s i t i o n
N)
for a m u l t i p l i c a t i v e
has always way
however
found
that
the set of those
in a n o t h e r
also T h e o r e m
[77],
further
and c o m m e n t s
p
(i.e. m u l t i p l i c a t i v e and
k ~I]
N.
and
functions.
of i n f i n i t e l y m a n y m u l t i p l i c a t i v e
for all
condi-
(if(mn) = f ( m ] f ( n )
Using
func-
95
Exercises
I. S h o w t h a t cative Am(f,N)
f
f
M(f,q)
D-~D(mod
be
=I.
qk)
assumption 4.
is
5. P r o v e set of a l l if a n d o n l y
WUD(mod
N)
mean-value 8. N's
that
those if
for
=I
if
is
such
D-WUD(mod
function
f
is
and
N
and a multipli-
that
the
index of
N). q
D-WUD(mod
excercise
by the mere Let
f
satisfying
n's
N.
Prove
[76]).
an odd
q2)
then
functions
of t h e M o e b i u s [803,
from
f(p) a 2 that
squares,
prime
such
it is a l s o
replace
the
M(f,q). completely
for a l l p r i m e s
p
multi-
and which
f(n) = n . integer-valued has
function
a positive
f
the
density
FN.
from Proposition
4.1
a criterion
for
F N-
function
5.7
to d e d u c e
of p r i m e
r2(n) , c o u n t i n g WUD(mod
the v a n i s h i n g
of the
~(n) .
in t h e c a s e
is
of
(f(n),N) = I
to T h e o r e m
function
cannot
integer-valued
to the class
Deduce
one
existence
be an
for w h i c h
belongs
(O.M.FOMENKO
as a s u m of t w o
f
integer
m(f,N)
for a m u l t i p l i c a t i v e
f
the
find an
k al.
the Corollary
for w h i c h
and
that
for a l l
(H.DELANGE
7. U s e
T
[77]).
function, N)
can
in the p r e c e d i n g
M(f,q)
UD(mod
6.
than
Prove
(H.DELANGE
plicative
one
well-defined
a multiplicative
for a l l
3. S h o w t h a t
T
with
is l a r g e r
2. L e t that
for a n y
function
N) .
N).
Determine
all
the r e p r e s e n t a t i o n s
those of
n
CHAPTER
VI
POLYNOMIAL-LIKE
§ I. G e n e r a t i n g
I. T h e uniform
x
to h a v e
P(x)
with
shall
[48]),
stated
p
P(x)
with
Zet
shows
that
for c h e c k i n g
set of values
(xP(x),N)
=1
in o r d e r
which
P(x) the
is b a s e d
conjecture
it is
for a g i v e n
by
generate
to c h e c k w e a k
function
whether,
attained
does
such a procedure,
(For a p r o o f
X
If
polyno-
at integers
group
G(N).
on a c o r o l l a r y
for a l g e b r a i c
and
let
see e . g . W . S C ~ 4 I D T P(x)
P
[76],
be a p o l y n o m i a l
be a n o n - p r i n c i p a l
the p o l y n o m i a l
curves
We to
(A.WEIL
character
does
not
Ch. II,
over
Z
(mod p)
satisfy
the
th.2C) . of degree
and denote
by
K. d
congruence
- c W d(x) (mod p)
a certain
1
5.7
of a p o l y n o m i a l
below:
6.1.
order.
s e t of v a l u e s
for a p o l y n o m i a l - l i k e
o n the R i e m a n n
be a p r i m e
Further, its
property
theorem
LE~4A Let
Z, t h e
now present
A.Weil's
(mod N)
a procedure
over
the
by the
to T h e o r e m
distribution
important mial
Corollary
G(N)
FUNCTIONS
~
constant
c
and a polynomial
W(x),
then
one
has
X(P(x)) I -< (K - I ) P ½ •
x(mod p)
First ERDOS,
l e t us d e d u c e
posed
He a s k e d , function
on one
whether will
from this
lemma
an answer
of the n u m b e r - t h e o r e t i c a l
a "well-behaved"
be necessarily
(in a c e r t a i n
WUD(mod
p)
to a q u e s t i o n
meetings
for all
sense)
o f P.
at O b e r w o l f a c h . multiplicative
sufficiently
large
p.
97
This
cannot hold
way,
since
this
property.
the answer
However
Let
degree
Let
6.2.
d ~1
is a large
question
f
all sufficiently
large primes p .
We need
If
6.3.
P(x)
reducible factors not divide
Proof.
of
P
into
implies
P
W(x)
of
D
in
a certain P
and by
prime
ideal of
P(x)
splits
is
~D(mod
of degree
with a constant
p)
for
and a poly-
p
P(x)
factors
that
is a prime which does
over
with
i.e.
now that constant
its h i g h e s t
Our
p
cannot as above.
coeffi-
factorization assumption
is a p r i m e w h i c h
c, k a 2
~cWk(x) (mod p) . Let
P(x)
c,k,W(x)
be the
the rationals.
Assume
Then
P(x).
cWk(x)
is monic,
bmt with a c e r t a i n P(x)
c, k ~2
which
and assume
P(x) =V~I (x)...vSn(x)
GF(p),
d a I
of the product of all ir-
P, W
and a p o l y -
be the images
thus
in
ZK
non-zero ZK
constant
the ring
containing
p
K, the p o l y n o m i a l
write
n
P(x) =
~
(x-ai)
ai
i=1
with
of
= cw(x) k
holds with field
func-
P(x)
the d i s c r i m i n a n t
that
Let
we have W
polynomial-like
for a certain constant f
is a p o l y n o m i a l
(al,a2,...,a n) =I.
resp.
~(x)
Then
to a polynomial
irreducible
does not d i v i d e
of
D
P(x)
first
unity.
that
nomial
of
Assume
not have
for w h i c h
with a p o l y n o m i a l cWk(x)
k ~2.
P(x) = c w k ( x )
(mod p)
equals
in a natural does
of f u n c t i o n s
D, nor the leading c o e f f i c i e n t of
be congruent
cient
and
e Z[x]
Denote by
W(x).
d(n)
a lemma.
is not of the form nomial
p
which is not of the form W(x) ~Z[x]
LEM~
class
be a multiplicative,
c, a polynomial
Proof.
occuring
function
is positive.
for all primes
f(p) =P(p)
functions
5.9 the divisor
there
to Erd~s's
PROPOSITION
tion.
for all m u l t i p l i c a t i v e
by P r o p o s i t i o n
a l , . . . , a r ~ZK,
distinct,
and
c. Denote
of integers and let P(x)
by
K
of it. Let
the s p l i t t i n g P
be any
K = Z K / P ~ GF(pf) . must
split
in
K,
Since
thus we m a y
98
n
[(x)
~
=
(x-Ti) al
=
c
W
(x)
k
i=1
with
~i
all
~i's
being
the
were
it e x i s t s
a pair
divisible
by
trary If
~i =~j and
ai
under
then
k
with
i #j.
since
=Ax d +...
the
would
canonical
divide
But
map
Z K + K.
( a l , . . . , a n) =I
then
it is a r a t i o n a l
is not
~i,~2,...,~n
suitable
integer
of
~i - ~ j ~ P
integer
so
w e get
If
hence D
pID,
is con-
assumption.
P(x)
distrinct with
P
to our
image
distinct,
algebraic
q, w h o s e
all
then
monic
for
n ai ~ (x -~i) with i=I we m a y w r i t e ~i =Si/q
P(x)
=A
i=1,2,...,n
integers
prime
and
81,...,8 n
factors
divide
and A
a rational
hence
pXA.
positive If we n o w
put
n
F(X)
then
:
~ i=I
(x - 8i) ai
F(x) = q d A - I P ( x / q ) If n o w
P(x)
for
certain
F(x)
tain
c,
F(x)
k a2
(Z[x].
and
W(x)
we h a v e
eZ[x]
- c W k(x) (rood p)
and we d e f i n e
and
and
A' , q
by
cc'
- qdA'cW(xq')k(mod
since
F(x)
constant
is m o n i c a,
k >_2
P (x) = A q - d F (qx)
contradicting
our
_--AA' - I (mod p) , then
p)
we m u s t
and
have
V(x)
either
eZ[x]
F(x)
= a V k(x)
for a c e r -
hence
= a A q - d v k (qd)
assumptions,
or
the d i s c r i m i n a n t
of the p o l y n o m i a l
n
H (x-Si) i=I P(x) must
is d i v i s i b l e be
TO p r o v e
R =
also
by
divisible
the p r o p o s i t i o n
{P(x) : (xP(x) ,p) = I }
p, but by
in t h a t
p, w h i c h
it s u f f i c e s
case
the d i s c r i m i n a n t
is i m p o s s i b l e .
to
show
that
the
set
of
99
generates
G(p)
all p rime factors
for s u f f i c i e n t l y
factors
of
of
satisfying
and a s s u m e P(x) . If
a non-principal x
of the d i s c r i m i n a n t
P(x)
coefficient
large
R
character pfxP(x)
further
L e t thus
p
p), e q u a l
we h a v e
p
of the p r o d u c t that
does
d o e s not g e n e r a t e X(mod
p-1 ( d - 1 ) / p >_ I Z X(P(x) I = x:O
and thus we h a v e o n l y
p.
be l a r g e r
of all i r r e d u c i b l e
not d i v i d e
G(p),
the
then t h e r e
to u n i t y on
X(P(x)) = I ,
than
R. Thus
and we o b t a i n
leading exists
for all
from Lemma
6.1
l#{x(mod p) : p ~ x P ( x ) } + X ( P ( O ) ) ] - > p - d - 2
finitely many possibilities
for
P(x)
Z
p.
2. N o w we p r o v e
THEOREM
6.4.
not
of the form
and
let
pa
Let
cWk(x)
be a prime
X(mod
pa),
vides
the d i s c r i m i n a n t
or the
constant
leading
max{d 2 +2d,
power.
on the set D
aO
term
3d +2}.
be ~ p o l y n o m i a l (with a constant
exists
X
unity
in
d ~1
,
W(x) cZ[x]) character
then either
of i r r e d u c i b l e
P, or finally equals
of degree and
a non-principal
R ={P(x) : p~xP(x)}
of the product of
If
If there
over c, k ~2
factors
p
does
not exceed
R
then
either
p
di-
of
pIDao
P,
or
p ~ d 2 +2d.
Proof.
Lemma
5.14
We may
our a s s u m p t i o n s Cp
that
Since
imply
cyclic
x(mod
p2) ÷ < x ( m o d p ) , ~ >
x
of
~
elements)
element
of P r o p o s i t i o n
p. If 6.2.
is odd and
of
moreover
at least
n o w that
G(p 2) =G(p)
the i s o m o r p h i s m
G ( p 2)
the c h a r a c t e r
eCp,
of elements, (with
g i v e n by
satisfying X
in v i e w
pa-1(p-l-d)
x z 1(mod p)
and
acts by
= ~ (x)× (x)
is a c h a r a c t e r
order d i v i d i n g
p
contains
Observe
p2) . M o r e o v e r
X ( < x mod p,~>) where
p
is the u n i q u e
~p-1 ~ x P - 1 ( m o d
R
R #~.
being
where
have
assume
a s 2 .
~
(mod p) , and
×
is n o n - p r i n c i p a l
Indeed,
since
X p =]
is a c h a r a c t e r we p r o c e e d and
~P
(mod p2)
of
as in the proof
is n o n - p r i n c i p a l ,
we
100
xP(R)
with
= YP(R)
a constant
Lemma
6.3)
: c
c,
thus
we o b t a i n ,
with
P (d-l) /p _> ] ~ X x=O
thus
p < (d+1) 2. Assume
for
all
~
with
1 ~r ~p-] X(X)
= X(X)
By a s s u m p t i o n ,
utilize
the
of
reduction
in v i e w
R(mod
of
p) ,
_> p - d - 2
we
are r e a d y
y
is the p r i n c i p a l generated
by
character.
1+p(mod
p2)
Since
we can write
satisfying
n =exp{2~i/p}, = X (1+p) t(x)
for e v e r y
x
Q 5t(x] then
for
all
If x
X(1+p)
=n r
with
we h a v e
= n rt(x)
satisfying
C],
(in s e q u e l
p~xP(x) C2,C3,...
we h a v e all w i l l
X(P(x)) denote
= C I, cons-
thus
q
and
(which w e m a y
(mod p2)
a certain c o n s t a n t
tants)
and
in the g r o u p
t(x)
and
being
a =I
a =2
lies
a unique
with
In case
(I +p) t ( ~
R'
6.1
(P(x)) ] _> # R ' - I
now that
x, ~
by L e m m a
rt(P(x>) = ~1
so
rt(P(x))
holds
- C2Cmod
implying
p]
in turn
(1+p) rt (P(x))
_ C3(mod
p2)
i.e.
P(x)r
and t h i s assumes
_= C 3 ( m o d
p2)
shows,
that
at m o s t
p-3
for
x
values
subject
to
(mod p2),
p~xP(x)
the polynomial
all d i s t i n c t
P (x] r
tmod p). D e n o t e
101
these values x ~O(mod
by
p)
of
c I, .... c r
and
let
N(c)
be the number
of s o l u t i o n s
the congruence
P(x) r - c ( m o d
p2)
On one h a n d w e h a v e
r
N(cj)
= ~ R > p(p-I
-d)
~=I
a n d on t h e o t h e r
N(c)
_< # { x
+ p #{x
hand
mod
p: p ~ x , ( P ( x ) r] ' ~ O ( m o d
rood p:
p) ,pr(x)
pXx, (Pr(X)) ' -O(rNod p),
Pr(x]
-c(mod
~c(mod
p) } ÷
p) } ,
thus with
S = ~ { x rood p;
p~x,P(x)
/O(mod
p]
(Pr{x))'
~O(mod
p) }
we get
r
p (p-1 -d]
N(C
_<
) < S +p(p
_ c _J]
j=1
Hence
p a S >_p(S-d]
(pr(x))' x(mod
vanishes
for
at
S-d <_ J least
and
p _<~+d. We
~R'-1-d->p-2d-2
see t h u s r t h a t residue
classes
p) eR.
Now implies
(pr(x])' = r P r-1 (x) P' (x) , t h u s
(Pr(x]] ' - O ( m o d
P' (x) - O ( m o d
is of d e g r e e
congruence
has
p _<3(d+I) , or = PI (xP] thus
thus
p'
holds
X
solutions,
vanishes
identically
a certain
leading
polynomial
to
for
d-1
the
last
p - 2 d - 2 _
and
(mod p) , in w h i c h PI,
but
this
case
implies
x £R
P(x) = p <~d,
p <_3d+2.
equals
P(x) = a x ÷ h ,
P' (x]
d-1
with
in a n y c a s e If
p) . S i n c e
at m o s t
p]
unity
(pr(x)
on
R
and
' = r a ( a x + h } r-1
p-1
if
plb
[p-2
if
pXb
S = <
3d+2 >d2+2d, t
thus
then
d =~
hence
102
and
from
blishes
S _
we
get
p _<4, i.e.
I. N o w w e c a n d e d u c e polynomial
integers
N
not
P(x)
the
= {P(x)mod
generate
THEOREM
If
6.5.
p
exists
The
the
conditions
for a
of T h e o r e m
6.4,
all
the conditi~on~ of theorem
6.4,
then
=7}
integers
runs set
an u p p e r
with the property,
K I , K 2 .... ,K s
if and only if
G(N)
N
is divisible
as f o l l o w s : A
of a l l
character
X(mod
a <2
p
a certain
gives
for d e t e r m i n i n g ,
K'is.
algorithm
determine
with
6.4
the
(xP(x),N)
determine
a non-principal
R ( P , p a) rem
N:
~atisfies
by at least one of the Proof.
procedure
set
does not generate
R(P,N)
First
esta-
G(N).
one can effectively that
This
algorithm
an e f f e c t i v e satisfying
for w h i c h
R = R(P,N)
does
3 _
theorem.
§ 2~ A n
given
p =2,
bound
for
primes
p
pa),
which
odd
resp.
for t h e p r i m e s
for
which
there
is c o n s t a n t
a_<3
for
on
p =2°
Theo-
p
t o be i n v e s t i g a t e d . aD In the n e x t s t e p c o n s i d e r all p r o d u c t s H p with ap _<2 for p6A a p ~2 and a 2 _<3 and f i n d out, for w h i c h of t h e m the set R(P, ~ p P) peA does not generate G( H pap). L e t { K 1 , K 2 .... ,Ks} = B b e t h e set of in-PEA t e g e r s o b t a i n e d in t h i s way. W e s h a l l p r o v e t h a t t h e s e i n t e g e r s s a t i s f y our a s s e r t i o n . In o n e d i r e c t i o n N -O(mod
unity will
on
Ki),
then
R ( P , K i) , h e n c e
be n o n - p r i n c i p a l
generate
To
this
there
is easy: exists that
if for
character
the
X ~I
lifted
and
trivial
on
R(P,N),
converse
we need
a lemma:
G (N) •
prove
a certain
a character
i of
we have G ( K i)
equal
to a c h a r a c t e r thus
R(P,N)
of
could
to
G(N) not
103
L E M M A 6.6. bp = m i n ( a p , 2 ) such
if
R(P,N)
that
G(N I )
nerate
and
then some power
Write
and for
case
and
or
s =3
and
~p
A
in
A
of
N/M
(mod
G(N/M).
Since
form
2 tp
with
ap ~2.
Let
s =maX{tp:
p
of
N
and
N.
0 ~tp ~ a p - 2
with
prime
Denote
and h e n c e
XM
by
t
piN} =s.
2, M
the p r o d u c t A
if
in and let
If
P then
in
that
R(P,N)
R(P,N),
of
s ~2
N =N I
so let of all
s ~4
N
If
(mod pap)
suitable
with
constants
and
trivial
we are ready.
is a c h a r a c t e r
5.12 we h a v e
papNN
and
Xp
R(P,N/M)
D
G(N).
is not d i v i s i b l e
as a cha-
on
In v i e w of the
neither
by
(mod N)
24
nor
trivial
X =
H X where Xp is PIN p equals Cp on R ( P , p ap)
H Cp = I . If C is the set pIN of all p r i m e s pln for w h i c h the c h a r a c t e r X is n o n - p r i n c i p a l , then P ^ C cA and m o r e o v e r H X e q u a l s U n i t y on R(P,N) with N = K pap peC p p~C p r o v i n g that R(P,N) does not generate G(N). The t h e o r e m f o l l o w s n o w f r o m the o b s e r v a t i o n
Primes
p EA
that
Cp
N
X #I
then by p r o p o s i t i o n
a character
can be r e g a r d e d
d o e s not g e n e r a t e
that
b y a c u b e of an o d d prime.
XM
is n o n - p r i n c i p a l ,
is a p r o p e r d i v i s o r
Now assume
with
of
of G(D ap) for piN. t dpp p with dplp-1 and
and of all p r i m e s
pap-1 )
last l e m m a we m a y a s s u m e
on
N2
R(P,N),
on
(mod N 2)
character
divisor
form
of the s i n g l e
s =3
trivial
X' = X M = H X M. S i n c e for peA the order of P d p p S / ( d p p S , M ) = d ~ p s-1 (with d'Ip-1)p thus ~4p is i n d u c e d
by a c h a r a c t e r
and
of the
is not the case.
if
assertion:
the c h a r a c t e r
equals
racter
does not ge-
is a c h a r a c t e r of the
in c a s e
consists
that this
odd p r i m e s consider
p =2
tp ~I
G(N)
of
proper
set of all p r i m e d i v i s o r s
us a s s u m e
R(P,N I)
then
be a n o n - p r i n c i p a l
for a certain
X = H X where Xp P P Xp is for p #2
be the
let
the f o l l o w i n g
is a character
of it ~ill
0 ~ t p ~ap-1
A
G(N),
to e s t a b l i s h e d
The o r der of
ap a3
bp N I =H p where P P(x) is a p o l y n o m i a l
and
b 2 =rain(a2,3) . I f
not g e n e r a t e
X ~I
R ( P , N 2)
on
be an i n t e g e r
either.
N #N I
trivial
does
It s u f f i c e s
Proof.
If
pap N =~ P p #2 and
Let
satisfying
N15]
w i l l be c a l l e d
and
N eB.
exceptional
for the p o l y n o m i a l
P.
104
2. The f o l l o w i n g question arises
immediately:
P R O B L E M IV. Characterize finite sets
{KI,K2,...,K n}
the K.'s divisible by 2 4 or a cube of an odd prime)
(with none of
for which
there
generates
G(N)
l
exists a polynomial if and only if
N
P(x) cZ[x3
such that
R(P,N)
is not divisible by any of the
Ki's
It is clear that some conditions have to be imposed on the since e.g. one cannot have primes,
n =I,
as the following a r g u m e n t
nomial w i t h the n e e d e d property, N
"bad" if
number
R(P,N)
P~P2
K I = (plP2)2 shows:
m u s t be "bad".
X i(R(P,p2i)) = c i
and
d
say.
Since
c's
is a polyan integer
G(N). We shall show that the
Xi(mod p~)
5.12 (i=I,2)
such that
must be roots of unity of the same order
dl ( ~ ( p ~ , ~ ( p ~ ) ~
cannot be d i v i s i b l e by
ced by a character
Ki's,
being odd
(i=1,2)
ClC 2 =I. Now the
d ~I,
P(x)
for shortness,
In fact, P r o p o s i t i o n
implies the e x i s t e n c e of c h a r a c t e r s
Pl
assume that
and call,
does not generate
with
= (pl (p1-1),p2(P2-1)~
P2' thus
(mod p2 )
dlP2-1,
and
showing that
and this implies that
P~P2
(See also exercise 5 which shows that a l r e a d y in case
n =I
Pl
is indu-
is "bad". the solution
of the p r o b l e m looks rather complicated.)
§ 3. A p p l i c a t i o n s to the study of M*(f)
I. Our first a p p l i c a t i o n is an immediate c o n s e q u e n c e of T h e o r e m 6.5 and strengthens
PROPOSITION
P r o p o s i t i o n 6.2:
6.7. If
such that for p r i ~ e ~ not of the form W(x))
p
cWk(x)
is an i n t e g e r - v a l u e d m u l t i p l i c a t i v e
one ~as (with
f(p) =P(p) k ~2,
that if
N
function
with a p o l y n o m i a l
a constant
c
P(x)
and a polynomial
then there exists an effectively d e t e r m i n a b l e finite set
primes with the property, f
f
has no prime divisors
in
E
of
E, then
is WUD(mod N) .
Proof. The a s s e r t i o n results from T h e o r e m 6.5 to T h e o r e m 5.7. D
and the C o r o l l a r y
105
From T h e o r e m 6.5 and the C o r o l l a r y to T h e o r e m d i a t e l y an a l g o r i t h m for d e t e r m i n i n g t i p l i c a t i v e function
f, p r o v i d e d
M*(f)
5.7 one d e d u c e s imme-
for a p o l y n o m i a l - l i k e mul-
it satisfies the following three con-
ditions:
(A)
The value of
is for all
N
(B) For with
, as defined in the
m(f,N)
b o u n d e d by a constant
k =I,2,...,T
Pk(X) £Z[x]
and
one has
Pk(X)
(C) If for an integer
R(Pm,N) {with
N
= {Pm(x)mod N:
m =m(f,N))
f(pk) =pk(p)
not equal
d-th power of a p o l y n o m i a l with
for all primes
to a constant m u l t i p l e
p, of a
the set
(XPm(X) ,N) =I} G(N),
then
The last c o n d i t i o n says that the e q u a l i t y
use of P r o p o s i t i o n
to Theorem 5.7,
d >_2.
does not generate
the d e t e r m i n a t i o n of
Corollary
T.
M*(f) . In practice
f
(5.2)
cannot be k73D(modN).
is i r r e l e v a n t for
it is m o s t l y checked w i t h the
5.11, the case of a q u a d r a t i c character being dispo-
sed by a separate argument.
For further r e f e r e n c e we state now a simple
o b s e r v a t i o n w h i c h w i l l f a l i c i t a t e this task:
LEMle~ 6.8. If a p o l y n o m i a l - l i k e
m u l t i p l i c a t i v e function
for a certain
N = H pap the condition P bp for the integer N I =H p with P [ min(2,ap)
if
p is odd
bp = ~ min(3,ap)
if
p =2.
Proof.
O b s e r v e first that
then Lemma 6.6 shows that and since since Thus
NI[N
f
If
R(Pm,N 1)
is assumed to be
(C) fails for the number
R(Pm,N)
W U D ( m o d N) , it is also
N I.
N
N
and
N1
does not g e n e r a t e
does not g e n e r a t e
and every prime d i v i d i n g
violates
(C) then it does the same already
m(f,N) =m(f,N1) , since
composed of the same prime factors.
f
G(N I) ~D(mod
are G(N)
either NI) ,
is also a prime factor of
N I.
0
2. TO illustrate our a l g o r i t h m we consider now the sum of d i v i s o r s (n)
and prove'.-
106 PROPOSITION
has
M*(a)
7.9
={N:
Proof.
Since
m(o,N)
= ~
I
if
2~N
[
2
if
21N.
Assume set
cannot
generate
:0
equality get
X(a(3n))
check
I
aO.
now
5)
this
that
shows If
a s3,
N b s2
cases
and
way
odd
F.RAYNER
with
(5.2)
is s a t i s f i e d .
one
E d aln
N
or
3.
aO,
p =2,3
a(3 n) ~ 1(rood 3)
their
obliges
products. n(mod
exceptional ~D(mod
the we
us to
Since
52 )
prime
must
denotes
~(2 n) :
for
is e x c l u d e d .
all
(5.2)
Xo
Since
thus
algorithm
we h a v e
R(P2,N)
(where
our
and all
a short
hence
N)
contains
all e l e m e n t s
with is
p =3
and
N).
to c h e c k
computation
2a,5 b, shows
7b
that
with in all
G(N).
can
find
[82])
the d e t e r m i n a t i o n
final
=
the c o n d i t i o n
since
t h e n we have
products
one
then
the o n l y
3~N,
generates
arise
the
every
p =2
~D(mod
case
(mod 3)
X(o(2n))
is odd,
that
in t h i s
N)
similarly,
R(PI,52)
and
their
with
3,32,5,52
and
for
In a s i m i l a r
only
61N
N
we O b t a i n
R(P2,N)
(W.NARKIEWICZ,
for
powers
is even,
}~D(mod
we h a v e and
If
that
X(n) = (n/3)Xo(n) ,
hold,
Thus
={2,5,8]
n /O,1(mod
o(n)
we have
to u n i t y
of
(mod N))
(mod 3),
6/N.
the p r i m e
R(PI,32)
In c a s e
character
cannot
:x2+x+1
and o b s e r v e
are c o n g r u e n t
character or
(5.2)
Assume
61N
G(N).
for the
the p r i n c i p a l = 2 n+1 -I
that
R(P2,N)
satisfied
For the function
[73]).
PI (x) = x + 1 , P2(x)
first
in the
be
(J.~liwa
6~N}.
the
set
although of t h o s e
We do n o t
enter
M * ( o 2)
for
here
certain
N's
for w h i c h
into
o2(n)
additional
=
Z d2 aln problems
the c o n d i t i o n
the d e t a i l s
here,
and q u o t e
result:
The function
o2(n)
is
~D(mod
N)
for all integers
N
except
when (i)
8IN,
40~N,
or
(ii)
and there is a prime divisor
401N
and the order of
4(mod
p)
p
of
N
such that
p ~7
is odd,
or finally (iii)
N
i~ divisible by one of the n~mbers
12,
15,
28,
42 or 66.
107
3. The above result shows that the set
M*(f)
may have a c o m p l i c a -
ted s t r u c t u r e even for r e l a t i v e l y simple p o l y n o m i a l - l i k e however of
f
fulfills the c o n d i t i o n s
M*(f)
T H E O R E M 6.10. Assume
satisfying with
DI,D2,...,D T
gers such,
M*(f)
= {N:
Proof.
if
Let
Di
Aj
multiplicative
(C). Then there exist integers
and finite sets
X1, .... X T
of inte-
j=I,2 ..... T
f
(N,D k) =I, N • S ( X k ) }
can be effectively determined.
Xi
f(pJ) :P~(p)
the function
the set of all
for all primes
will be
p
~.~D(mod N)
with
N's
for which
P~(x) eZ[x].
if and only if
G(N) . Due to T h e o r e m 6.5 this happens if and only
where
Xj
is the set of integers c o r r e s p o n d i n g to
P~
6.5.
P~(x)
N •A~ but
R(P~,N)
if and only if
R(P~,N) = [ R ( P , p ap] P and the set R(P~,p)
and since
p
R(Pi,N)
are e m p t y
is n o n - e m p t y and note that to a given an intecer
(D~,N) =I.
D~
Indeed,
with the p r o p e r t y
N = H pap then P is e q u i v a l e n t to R ( P ~ , p ) # O
R(P~,p ap) # O
if
can be empty only for finitely m a n y primes
6.5), hence
over all primes
if and only if the sets
there corresponds
R(P~,N) # ~
(by T h e o r e m
is a polynomial-like
j -
and sets
be for
i=I,2,...,j-I
polynomial
for
(N,D~) Jl
Observe now that
that
If
denotes the set of all positive integers with-
S(X)
generates
N e S(Xj)
f
(B) and
T = m ~ x m(f,N)
and let
N •Aj
by T h e o r e m
for
functions.
(C) then the structure
X, then
All integers
m(f,N) =j,
that
(A),
that if
out a factor in
R(P$,N)
(B),
can be d e s c r i b e d a priori:
function,
For
(A),
it suffices to put
for which
R(P~,p) =@.
D~ = H p,
p
the p r o d u c t taken
D
§ 4. The functions o k for k ~3
I. Our last section is devoted to the functions which we dealt already in the cases
k=O,~,2
the Fourier c o e f f i c i e n t s of E i s e n s t e i n
series.
ok(n) =
Z d k, with dln and w h i c h for odd k are We shall show that the
a l g o r i t h m of § 2 is a p p l i c a b l e in this case and using it we d e t e r m i n e
108
the
set of all
when
k
odd
is an odd
PROPOSITION (B),
We
ck~n)
start
k ~3,
Proof.
Pj(x)
in this
thus
that
m(ak,N)
~.
dividing
(B)
if
then
the
pll +I
+I 2
p =7
in the case~
ok
satisfies
the conditions
(A),
hence there is an effective procedure
+ x ~k =
is o b v i o u s l y
~
is the
Indeed, N
N)
case
= I + x k + x 2k +...
j al,
WUD(mod
M*(ak).
We h a v e
blish,
is
with
of the preceding section,
(C)
thus
for w h i c h
If
6.11.
for determining
for
N's
prime.
if
satisfied.
smallest
R(~_I,N)
set
R(P
+...
+1 (7-I)
and m o r e o v e r
(x k(j+1)
must
To p r o v e
number
is empty,
_i, p)
for
prime
- 1 ) / ( x k -1)
(A) we
satisfying
then
be empty,
for
shall
a certain
and this
esta-
~-1~k,
then
prime
p
implies
=
a primitive
root
g(mod
p)
we w o u l d
have
pl (g kp - 1 ) / ( g k -1)
thus
gkp ~I (mod p)
divide
k, b u t
in f a c t
to s h o w
6.12.
k ~3,
Since
for
ok(q
hence,
by our
p-11kp, choice
hence of
7-I = p - 1
7.
(One can
must
s h o w that
7-1). the c o n d i t i o n 5.11
and
(C) holds.
to be a b l e
Here
to do
we
shall
use
this we p r o v e
result:
If
N
is not divisible by a fourth power of a prime q
~k(q3) (mod N)
the sequence
is periodic. It s u f f i c e s j,T al
to c o n s i d e r
) - ok(q j) = q
p #q
N =pa
with
p
prime
and
a ~3.
we have
j+T.
if
implies
then for every prime
(j=1,2 .... )
Proof.
that
to P r o p o s i t i o n
an a u x i l i a r y
LEMMA
and
equals
N
the C o r o l l a r y
this
is e x c l u d e d
m a x m(ok,N)
It r e m a i n s
first
but
this
and
b
k(j+]) (qkT
is d e f i n e d
- 1 ) / ~ q k -I)
by
pbllqk-1
,
then
the o r d e r
of
109
qk(mod pa+b) view of
is a period of
a ~3
k(j+1)
~k(qj) (mod pa).
If however
p =q,
then in
we get
pa[~k(qj+T ) - ok(q j) thus our sequence
is constant
Since Lem~a 6.8 permits not divisible
neither by
lemma and the corollary WUD(mod N) G(N)
character
Moreover
R(Pm,N)
Since
N's
which are
(with
5.11
imply now that if
m =m(Ok,N))
ok
is
does not generate
a subgroup of index two and for the only non-
X(mod N)
~
which is trivial on
~ -I
)=
in that case
only those
nor by a cube of an odd prime the above
2 k(j+1) -I
x(ok(2J)) =x(
D
as to consider
24
to Proposition
but the set
then it generates
-principal
(mod pa).
N
k a3, we have
[ o
if
mlj
if
m~j .
must be even and
X
Ok(2J)
for
~ 1(mod 8)
R(Pm,N)
one has
real. j a I, thus if we write
X = H Xp, where Xp is a character (mod pap) (with pap[IN) then piN X2(Ok(2J)) =I holds for j al. If p is a prime dividing N, but not dividing j
2 k -I
divisible
o=
and
r
P
denotes
the order of
2k(mod pap),
by
If
rp
pin p#2 pX2k-1
we get
Ok(2J)
I (mod pap) , thus
It follows that if
x' =
L I xp piN p#2 p]2k-1
then for
j ~O(mod D)
X' (~k(aJ))
we have
= <~
-I
if
mlj
[
0
if
adj.
Xp(Ok(2J))
=I
holds.
then for
110
Since ter
Xp
Ok(23)
X
was
real,
is i n d u c e d ~I + j ( m o d
= Xp(1+j),
is a l s o
such,
by a character
p)
thus
X'
thus
for
hence
(mod p).
such primes
X' (Ok(2J)) = X ' (1+j)
But
p
and
for
p #2
for
p12 k -I
we have
this
the
characwe have
Xp(ok(2J))
leads
=
to a contradictory
equality
I = X' (I + D m ) X ' (I +Dm)
2. U t i l i z i n g for all o d d shown
result
F.RAYNER
For very
small
in W. N A R K I E W I C Z
details
of t h e
BROWOLSKI ging
this
k sI07.
to
proved
= X' (I + D m ( 2
computation,
M*(Ok)
and p r o v e
exceeding
in the c a s e
for a n y
k
the
(k+1) 2. T h i s
THEOREM
which
k
is composed,
ver
divisible
by
K
[ q
Proof. tion
6.11
primes
if
k-
shows
p=3 assume which
p
of
set of a l l Thus we with
X
r-th
a certain
Without
ok
will
p
then
is
belon-
a prime
[80] number
6.4.
the
M ~ ( o k)
~D(mod
If
N).
If howe-
of all integers
powers that all
fixed
restricting
provided
value
is a p r o p e r of
G ( p a)
G ( p a)
not
elements
elements
with
c
the generality
of
of
G ( p a)
H
X
of
G ( p a)
and
assume
a character If
H
to
denotes
R(P1,pa)
H,
and
coincides
with
R(Pl,pa)
suitable r
since
with
a certain
a r e of the
that
if
is a p r i m e .
G(Pa) ,
respect
of
we may
and
R(P1,pa).
and
set Proposi-
exceptional
last number
subgroup
the =I.
if and o n l y
a ~2 on
N
m(Ok,N)
N)
the
is c y c l i c
of e l e m e n t s
for o d d
thus
s h o w the o n l y
is an odd p r i m e
in a c o s e t thus
since
~D(mod
shall
a constant H
and
be
we
p = q = 2k+I
that
prime,
see
that
assumes
be c o n t a i n e d
is an o d d
of E . D O -
O.M.FOMENKO
of all odd integers.
it is n o n - e m p t y ,
G(N) . F i r s t
and
of
the k e r n e l
result
from Theorem
ok
into the
odd n u m b e r s
contain
consists
M ~ ( e k)
P1(x) = x k +I
2 = I + I k,
now
To do this
must
consists
M * ( o k) as
I (mod 4) .
case
generates
G ( p a)
cannot
for which
then
k - 3(rood 4)
In our
are
N
M 2 ( o k)
if
contains
R(PI,N)
the
k a 3. E a r l i e r
M * ( o k)
not go
a partial
describes
sets
by hand,
where
K = < [ 3q
R(PI,N)
then
is a prime,
q = 2 k +I
We s h a l l
instead
the
this
be an odd prime and denote bw
set of all those odd integers 2k +I
D
computed
can do
k:3.
of p r i m e set
: -I .
has
one
c a n be a l s o d e d u c e d
Let
6.12.
[83] k
[83b] in the c a s e
(oral c o m m u n i c a t i o n ) ,
that
+Dm))
form
the
r. cy r
y eG(pa). is a p r i m e .
111
N o w let
g
then
is d i v i s i b l e
k
since
k
be a p r i m i t i v e
is a prime.
R(P1,32)
= {2,5,8}
the v a l u e
-I
is a prime.
for
on t h o s e two sets,
or
p,
and
X3(n) = (n/3)
in the s e c o n d
by
(p-I)/2 =k,
R(P1,3) = { 2 }
character
and
is d i v i s i b l e
p =3,
case
assumes
p =2k+I =q
- 1 +
(p)
and thus e v e r y c h a r a c t e r of
(mod p)
I + x (p-I)/2
is c o n s t a n t
is not d i v i s i b l e
on
by
p
the c o n g r u e n c e s
I + x (p-1)/2
have
1+g k
In thi s c a s e
S i n c e the d e r i v a t i v e
x #0
If
thus e i t h e r
In the f i r s t case we find
R(PI, p) = { 2 }
R(PI,p) .
(mod pa).
(p-I)/2,
thus the q u a d r a t i c
I + x k = I + x (p-1}/2
hence
root
by
solutions
- 2 +jp(mod
for
j=O,1, .... p-l}
p2)
j=O,1 .... ,p-1
thus
and e v e r y c h a r a c t e r
R ( P I , p 2)
(mod p)
equals
lifted
to
{2 + j p : G(p 2)
is c o n s -
tant on it. This
s h o w s that
ly e x c e p t i o n a l . the case, p.
Then with
suitable
showing
and
that
and this take
c=I G(pa).
Since
if
k - t h power (k,~(p))
G(pa) . T h u s
to
that
r-th power
in
it r e m a i n s
l+gk
we have
are c e r t a i n to consider
is d i v i s i b l e
1 + gk ~ c y ~ ( m o d
by
pa),
c
+gk)
and
(mod pa)
indeed,
in this c a s e
which
rlk and so r=k. k I _ g k ~1 + (_g) k - c y 3
leads to
k-th power
of
I (mod p)
but this m e a n s
hence
we have
_ c2(ylY3)k(mod
consists
=I,
p)
pa)
is i t s e l f a
for all
pa)
G(pa),
y3,y 4 ~O(mod
R(Pl,pa) O
p)
_ cgky~(mod
and thus e v e r y e l e m e n t
not c o n g r u e n t a
is a
_ (I _ g k ) ( I
implies
g ( m o d pa)
y],y 2 ~ O(mod
with a suitable
- I _g2k
it is a prime,
hence
_-- gk(1 + g - k )
gk
in case
are no o t h e r s
root
I _ g 2 k _ I + (_g2)k - Y 4 - c y ~ ( m o d
cy~
in
pa)
- I +gk
Moreover
q =2k+I, that there
w h e n for no p r i m i t i v e
I +g-k =cy~(mod
cy[
3 and
To p r o v e
R(PI,pa)
pa)
in
must
G(pa), be a
of all e l e m e n t s we o b t a i n
j ~O,1(mod that there
H
easily p)
so we m a y
k-th power
] +xk(mod that
which
pa)
j(mod p)
is p o s s i b l e
is not a p r o p e r
are no e x c e p t i o n a l
is only
s u b g r o u p of primes.
112
N o w we c h e c k If
2k+I
R(PI,N)
WUD(mod
N)
root
it is clear
If
k ~3(mod
(mod q), hence
for no m u l t i p l e
of
q
integers
we have
root
(mod q)
hence
R(PI,N)
we have
R(PI,N)
is non-cyclic. and only
= {2}
for them.
N's
except
[83])
that
and
this
=
{N:
with
do not
a solution
k
for
N =3
theorem
and
3q
2 cannot be a G(q), but
thus
for
is a p r i m i t i v e
G(N)
is g e n e r a t e d
32 . However
for
N =3q
G(N}
this
group
since
we do not get
approach
does not
WUD(mod
6.10.
seem to work
The results
any p a t t e r n However
M*(o k) =M*(s)
N) ,
in the
structure
in r e m a r k a b l y
={N:
6~N}.
for
of c o m p u t i n g
many
It would
of
M*(Ok),
cases
be i n t e r e s t i n g
to the f o l l o w i n g
those odd ~ntegers
6.4 was proved
covering
k, for which
M*(o k) =
with
2. P r o p o s i t i o n
that g i v e n here,
[82])
7.9 was o r i g i n a l l y
but
to P r o p o s i t i o n
proved 5.7.
of T h e o r e m
6.5
M,N
by J . ~ L I W A
His p r o o f was
of
of integers
[73]
~D.
on the than
of A.Weil,
M*(a 2)
here,
is
longer
on
(W.NARKIEWICZ,
by a computer.
is i n a p p l i c a b l e
In W.
given by T h e o r e m
the d e e p r e s u l t
role was p l a y e d
general
polynomials.
all p a i r s
In the d e t e r m i n a t i o n
an i m p o r t a n t
in a more
the a l g o r i t h m
the pair
it did not u t i l i z e
6.4 rests.
the a l g o r i t h m
to w h i c h
[82]
of several
to obtain
of this a l g o r i t h m
respect
of the c o r o l l a r y
F.RAYNER
in W . N A R K I E W I C Z
it was u t i l i z e d
As an a p p l i c a t i o n
which Theorem
and c o mm e n t s
also the case of systems
[81]
were d e t e r m i n e d
since
2
two
6~N}.
NARKIEWICZ
basis
4), then
generate
of
so
we have
distribution
the group
composed.
indicate
Characterize
I. T h e o r e m
6.5.
(2/q) =+1,
q2
it cannot
by it. H e n c e
is prime,
does not g e n e r a t e
k ~3(mod
§ 5. N o t e s
form,
then
the form of
is g e n e r a t e d
q =2k+I
and
elementary
~k
it obeys
P R O B L E M V.
If
N =q
for m u l t i p l e s
(19 out of 54) we have to have
4),
G(N)
If
and their powers.
but from
D
or for
(F.RAYNER
that N.
primes
32
R(PI, q)
same holds
Hence
Unfortunately even
~D.
for
and the
and
we can get weak u n i f o r m
other
by
3
for all odd
to consider.
primitive
of e x c e p t i o n a l
they are
in that case
we get cases
the p r o d u c t s
is composed,
In fact,
a computer
113
experiment form
of
was made,
integers
Theorem 6.11
was
6.10
f
N) then
found
conditions
that p
occurs
to f o r m u l a t e
which
was
a conjecture
subsequently
in W . N A R K I E W I C Z
is a p o l y n o m i a l - l i k e and
tors,
namely,
allowed
M * ( o 2)
[83b],
about
the
proved.
where
also
Proposition
obtained.
3. If ~D(mod
which
in
N1
obviously
that
f
under
if
multiplicative
is a d i v i s o r is a l s o which
m =m(f,N)
pm(p ) =f(pm)
of
the
N
WUD(mod
for
the
which
same
the
holds.
polynomial
is n o n - c o n s t a n t
and
is
prime
E.J.SCOURFIELD
implication
,
primes)
function
has
NI).
converse
is d e f i n e d
holds
which
fac-
[74] She
proved
Pm(X)
(such
for
a prime
we d e f i n e
8p = inf{caO:
and
put
is
WUD(mod
for a c e r t a i n
N2 =
H p2Bp+1 pIN N2).
x
, then
with
f
is
p~xPm(x)
WUD(mod
one has
N)
pelIP~(x)}
if and o n l y
if it
Exercises
I. D e t e r m i n e =
n
2
H
the
same
representation 3. P r o v e
5. P r o v e generates
of
n
following
(b) K I
function
J2(n)
=
is odd,
6 ar,
po,pl,...,pr
is odd
satisfies
has
or
exists
and there
P(x) if
positive
the use
with
KI~N
the
form
to an odd
and has
there
and
of t h r e e
counts
the n u m b e r
of
factors.
of T h e o r e m
6.4.
the p r o p e r t y
that
exists,
if and
only
R(P,N)
if o n e
of
holds.
and
to u n i t y
stisfying
d3(n) , w h i c h
7.9 w i t h o u t
conditions
and
(c) K I
as a p r o d u c t
a polynomial
is e v e n
equal
function
if and o n l y
pl,...,p r
which
for J o r d a n , s
M*(~4).
that
G(N)
(a) K I
for the
Proposition
4. D e t e r m i n e
either
M * ( J 2)
(I _ p - 2 ) .
pin 2. Do the
the
set
has
the
form
a common
the
exists
pol61po(Po-1)
form
K I =2eq
K I =pl...pr divisor
6
1 ~
with of
2 K I =poPl...pr
a common and
with
~3
and
q
prime,
divisor
6 zl+r.
with 6
distinct
primes
P1-1,P2-1,...,Pr-1
of
distinct
primes
Pl-l,P2-1,...,Pr_l
REFERENCES APOSTOL, T.M. E763 Modular functions and D i r i c h l e t series in number theory, Springer 1976. MR 54 ~ 10149 ATKIN, A.O.L. [68] M u l t i p l i c a t i v e c o n g r u e n c e p r o p e r t i e s and d e n s i t y problems for p(n), P r o c . L o n d o n Math. Soc., 18, 1968, 563-576. MR 37 # 2690 ATKIN, A.O.L., O'BRIEN, J.N. [67] Some properties of p(n) and c(n) m o d u l o powers of 13, Trans. Amer.Math. Soc., 126, 1967, 442-459. MR 35 ~ 5390 BETTI, E. [513 Sopra la r i s o l u b i l i t ~ per radicali delle equazioni algebriche irredutibili di grado primo, A n n a l i di Sci.Mat.e Fis., 2, 1851, 5-19. E52] 49-115. [55]
Sulla r i s o l u z i o n i delle equazioni algebriche, Sopra la teorica delle sostituzioni,
ibidem,
ibidem 3, 1852, 6, 1855,
5-34.
BRAWLEY, J.V. [76] P o l y n o m i a l s over a ring that permute the m a t r i c e s over that ring, J.of Algebra, 38, 1976, 93-99. M R 52 # 13755. BRAdlEY, J.V., CARLITZ, L., LEVINE, J. E75] Scalar p o l y n o m i a l functions on the n×n m a t r i c e s over a finite field, Linear A l g e b r a and App]., 10, 1975, 199-211. M R 51 # 12800 BRUCKNER, G. [70] F i b o n a c c i sequence m o d u l o a prime p ~ 3 ( m o d 4), F i b o n a c c i Quart., 8, 1970, 217-220. MR 41 # 3384 BUMBY, R.T. [753 A d i s t r i b u t i o n p r o p e r t y for linear r e c u r r e n c e of the second order, Proc.Amer.Math. Soc., 50, 1975, 101-106. MR 51 ~ 5475 BUNDSCHUH, P. [743 On the d i s t r i b u t i o n of F i b o n a c c i numbers, T a n k a n g J.Math., 1974, 75-79. MR 50 # 12933
5,
BUNDSCHUH, P., SHIUE, J.S. [73] S o l u t i o n of a p r o b l e m on the u n i f o r m d i s t r i b u t i o n of integers, Atti A c c a d . L i n c e i , 55, 1973, 172-177. M R 51 # 3072 [74] 135-144.
A g e n e r a l i z a t i o n of a paper by D.D.WalI, M R 51 ~ 5476
ibidem,
56, 1974,
BURKE, J.R., KUIPERS, L. [76] A s y m p t o t i c d i s t r i b u t i o n and i n d e p e n d e n c e of sequences of G a u s s i a n integers, Simon Stevin 50, 1976~7, 3-21. MR 54 ~ 5131 CARLITZ, L. [53] Permutations 538. M R 15 p.3
in a finite field, Proc.Amer.Math. Soc.,
4, 1953,
115
68,
[62a] 1962,
Some theorems on p e r m u t a t i o n p o l y n o m i a l s , 120-122. M R 25 # 5052
Bull.Amer.Math. Soc.,
[62b] A note on p e r m u t a t i o n functions over a finite field, Duke Math. J., 29, 1962, 325-332. M R 25 ~ 1151 [63] P e r m u t a t i o n s in finite fields, A c t a Sci.Math. (Szeged), 1963, 196-203. MR 28 # 81.
24,
CARLITZ, L., LUTZ, J.A. [78] A c h a r a c t e r i z a t i o n of p e r m u t a t i o n p o l y n o m i a l s over a finite field, A m e r i c a n M a t h . M o n t h l y 85, 1978, 746-748. MR 80a # 12022 CARMICHAEL, R.D. [20] On sequences of integers defined by r e c u r r e n c e relations, Quart.J.Math. (Oxford), 48, 1920, 343-372. CAVIOR, S.R. [63] A note on octic p e r m u t a t i o n polynomials, 1963, 450-452. M R 27 # 3628
Math. Comput.,
17,
CHAUVINEAU, J. [65] C o m p l ~ m e n t au th~or~me m ~ t r i q u e de Koksma dans R et dans Qp, C.R.Acad. Sci. Paris, 260, 1965, 6252-6255. MR 31 # 3400 [68] 225-313.
Sur la r ~ D a r t i t i o n dans R et dans Qp, A c t a Arith., MR 39 # 3865
14, 1968,
COHEN. S.D. [70] The d i s t r i b u t i o n of p o l y n o m i a l s over a field, A c t a Arith., 17, 1970, 255-271. MR 43 # 3234 DABOUSSI, H., DELANGE, H., [82] On m u l t i p l i c a t i v e a r i t h m e t i c a l functions whose m o d u l u s does not exceed one, J . L o n d o n Math. Soc., 26, 1982, 245-264. DAVENPORT, H., LEWIS, D.J. [63] N o t e s on congruences, 51-60. MR 26 ~ 3657.
I, Quart. J.Math.
DEDEKIND, R. [97] II S u p p l e m e n t to D i r i c h l e t ' s B r a u n s c h w e i g 1897.
(Oxford),
14, 1963,
" V o r l e s u n g e n ~ber Zahlentheorie",
DELANGE, H.. [543 G ~ n ~ r a l i s a t i o n du t h ~ o r ~ m e de Ikehara, Ann. Scient.Ec.Norm. Sup., 71, 1954, 213-242. MR 16 p.921 [56] Sur la d i s t r i b u t i o n des entiers ayant c e r t a i n e s propri~t~s, ibidem, 73, 1956, 15-74. M R 18 p.720 [61] Un t h ~ o r ~ m e sur les fonctions m u l t i p l i c a t i v e s tions, ibidem, 78, 1961, 1-29. M R 30 ~ 71 [69] 419-430.
On i n t e g r a l - v a l u e d additive functions, M R 40 ~ 1359.
et ses ~ p p l i c a -
J . N u m b e r Th.,
I, 1969,
[72] Sur la d i s t r i b u t i o n des valeurs des fonctions additives, C.R. Acad. Sci. Paris, 275, 1972, A 1139-A 1142. M R 46 ~ 7187 [74] On i n t e g r a l - v a l u e d a d d i t i v e functions, 1974, 161-170. M R 49 ~ 7227
II, J . N u m b e r Th., 6,
[76] Sur les fonctions m u l t i p l i c a t i v e s ~ v a l e u r s entiers, C.R.Acad. Sci. Paris, 283, 1976, A IO65-A 1067. MR 55 # 300 [77]
--.,
i b i d e m 284,
1977, A 1325-A 1327.
M R 56 ~ 290
DELIGNE, P. [69] Formes m o d u l a i r e s et r e p r e s e n t a t i o n s l-adiques, Bourbaki, 1969, nr.355.
S~minaire
116
DESHOUILLERS, J.M. [73] Sur la r ~ p a r t i t i o n arithm ~ t i q u e s , C.R.Acad. Sci.
des nombres Paris, 227,
In c] dans les p r o g r e s s i o n s 1973, A 647-A 650. MR 49 # 2 6 0 3 .
DICKSON, L.E. [97] The a n a l y t i c r e p r e s e n t a t i o n of s u b s t i t u t i o n s on a power of a prime number of letters with a d i s c u s s i o n of the linear group, P h . D . T h e s i s , C h i c a g o 1897 = A n n a l s of Math., 11, 1896-97, 65-120 and 161-183. [01]
Linear
Groups,
Leipzig
1901.
DIJKSMA, A., ~ I J E R , H.G. [69] Note on u n i f o r m l y d i s t r i b u t e d sequences Arch.Wisk., 17, 1969, 210-213. M R 41 # 1676.
of integers,
DOWIDAR, A.F. [72] S u m m a b i l i t y m e t h o d s and d i s t r i b u t i o n of sequences J.Nat. Sci.Math., 12, 1972, 337-341. MR 50 # 2098. ENGSTROM, H.T. [313 On sequences d e f i n e d by linear Amer. Math. Soc., 33, 1931, 210-218.
recurrence
Nieuw
of integers,
relations,
Trans.
FOMENKO, O.M. [80] The d i s t r i b u t i o n of v a l u e s of m u l t i p l i c a t i v e functions with r e s p e c t to a prime modulus, Zapiski Nau~n. Sem. LOMI, 93, 1980, 218-224 (in Russian). MR 81k ~ 10068 FRIED, M. [70] On a c o n j e c t u r e MR 41 # 6188.
6,
of Schur,
FRYER, K.D. [55] Note on p e r m u t a t i o n s 1955, I-2. M R 16 p.678
Michigan
in a finite
Math. J.,
field,
17,
1970,
P r o c . A m e r . M a t h . Soc.,
HALL, M. [38a] An i s o m o r p h i s m b e t w e e n linear r e c u r r i n g sequences braic rings, T r a n s . A m e r . M a t h . Soc., 44, 1938; 196-218. [38b] Equidistribution 1938, 691-695.
of r e s i d u e s
HAYES, D.R., [67] A geometric approach field, Duke Math. J., 34, 1967,
in sequences,
and alge-
Duke Math.J.,
to p e r m u t a t i o n p o l y n o m i a l s 293-305. M R 35 # 168.
HE~4ITE, E. [63] Sur les f o n c t i o n s de sept lettres, 1863, 750-757 = Oeuvres, II, 280-288.
41-55.
over
C.R.Acad. Sci.
4,
a finite
Paris,
57,
HULE, H. MULLER, W.B. [73] Cyclic groups of p e r m u t a t i o n s induced by p o l y n o m i a l s over Galois fields. An. A c a d . B r a s i l . , 45, 1973, 63-67. M R 48 # 8453 (In Spanish). KL~VE, T. [68] R e c u r r e n c e formulae c o n g r u e n c e s for the p a r t i t i o n (j(T)
-1728) ½ and
for the c o e f f i c i e n t s of m o d u l a r forms and function and for the c o e f f i c i e n t s of j(T) ,
j(T) ~/3 , Math. Scand.,
[70] Density problems -508. MR 42 # 219.
for p(n),
KNIGHT, M.J., WEBB~ W.A. [80] Uniform distribution Arith., 36, 1980, 17-20.
23,
1968,
J.London
of t h i r d - o r d e r
133-159.
Math. Soc.,
linear
5~ 40 ~ 5545. 2,
1970,
sequences,
504-
Acta
117
KOKSMA, J. [35] Ein m e n g e n t h e o r e t i s c h e r Satz 0ber die G l e i c h v e r t e i l u n g m o d u l o Eins, C o m p o s . M a t h . , 2, 1935, 250-258.
KOLBERG, O. [59] Note on the parity of the p a r t i t i o n function, Math. Scand., 1959, 377-378. MR 22 # 7995
7,
KRONECKER~ L. [81] Zur Theorie der E l i m i n a t i o n einer V a r i a b e l n aus zwei algeb r a i s c h e n Gleichungen, Monatsber. Kgl. Prenss. Akad. Wiss. Berlin 1881, 535-600 = Werke, II, 113-192. KUIPERS, L. [793 Einige B e m e r k u n g e n zu einer A r b e i t von G.J.Rieger, 34, 1979, 32-34. M R 8Oh # 10063
Elem. Math.,
KUIPERS, L., N I E D E R R E I T E R , H. [74a] A s y m p t o t i c d i s t r i b u t i o n (mod m) and i n d e p e n d e n c e of sequence of integers, I, II, Proc. Japan Acad. Sci., 50, 1974, 256-260 and 261-265. M R 51 # 404 [74b] U n i f o r m d i s t r i b u t i o n of sequences, W i l e y - I n t e r s c i e n c e M R 54 # 7415
1974.
KUIPERS, L., N I E D E R R E I T E R , H., SHIUE, J.S. [753 Uniform d i s t r i b u t i o n of sequences in the ring of G a u s s i a n integers, Bull. Inst. Math. Sin., 3, 1975, 311-325. M R 54 # 2612. KUIPERS, L., SHIUE, J.S. [71] On the d i s t r i b u t i o n m o d u l o M of g e n e r a l i z e d F i b o n a c c i numbers, T a m k a n g J. Math., 2, 1971, 181-186. M R 46 ~ 5231 [72a] A d i s t r i b u t i o n p r o p e r t y of the sequence of F i b o n a c c i numbers, F i b o n a c c i Quart., 10, 1972, 375-376, 392. M R 47 # 3302 [72b] A d i s t r i b u t i o n p r o p e r t y of the sequence of Lucas numbers, Elem. Math., 27, 1972, 10-11. M R 46 # 144 [72c] Asymptotic distribution modulo m of sequences of integers and the n o t i o n of independence, Atti Accad. Lincei Mem. CI. Sci. Fis. Mat. Nat., 11, 1972-3, 63-90. M R 51 # 3106 [72d] A d i s t r i b u t i o n p r o p e r t y of a linear r e c u r r e n c e of the second order, Atti Accad. Lincei, 52, 1972, 6-10. MR 48 ~ 3862. [80] On a c r i t e r i o n for u n i f o r m d i s t r i b u t i o n of a sequence in the ring of G a u s s i a n integers, Rev. Roum. Math. Pures Appl., 25, 1980, 1059-1063. MR 81m # 10070 KUIPERS, L., UCHIYAM~, S. [68] N o t e on the u n i f o r m d i s t r i b u t i o n of sequences of integers, Proc. Japan Acad., 44, 1968, 608-613. MR 39 # 148. KURBATOV, V.A. [49] On the m o n o d r o m y group of an algebraic 25, 1949, 51-94. MR 11 p.85. (in Russian).
function, Mat. Sbornik,
KURBATOV, V.A., STARKOV, N.G. [65] On the a n a l y t i c r e p r e s e n t a t i o n of permutations, U~. Zap. Sverdlovsk. Gos. Ped. Inst., 31, 1965, 151-158. MR 35 ~ 6652. (in Russian). LANDAU, E. [09] Handbuch der Lehre ~ber die V e r t e i l u n g der Primzahlen, 1909; r e p r i n t e d by C h e l s e a 1953. MR 16 p.904
Teubner
LAUSCH, H., MULLER, W.B., NOBAUER, W. [73] Uber die Struktur einer durch D i c k s o n p o l y n o m e d a r g e s t e l l t e n P e r m u t a t i o n s g r u p p e des R e s t k l a s s e n r i n g e s modulo n, J.reine angew. Math., 261, 1973, 88-99. MR 48 ~ 2231
118
LAUSCH, H., NOBAUER, W. [73] A l g e b r a of polynomials,
Amsterdam
1973.
MR 50 ~ 2037.
LIDL. R. [71] Uber P e r m u t a t i o n s p o l y n o m e in m e h r e r e n U n b e s t i m m t e n , Math., 75, 1971, 432-440. M R 46 # 5290.
Monatsh.
[72] ~ber die D a r s t e l l u n g der P e r m u t a t i o n e n durch Polynome, Abhandl. Math. Sem. Hamburg, 37, 1972, 108-111. M R 46 ~ 9012. [73] T c h e b y s c h e f f p o l y n o m e und die d a d u r c h d a r g e s t e l l t e n Gruppen, Monatsh. Math., 77, 1973, 132-147. M R 47 ~ 6655 LIDL, R., MULLER, W.B. [763 Uber die P e r m u t a t i o n s g r u p p e n die dutch T s c h e b y s c h e f f - P o l y n o m e erzeugt werden, A c t a Arith., 30, 1976, 19-25. MR 54 # 5196. LIDL, R., N I E D E R R E I T E R , H. [72] On o r t h o g o n a l systems and p e r m u t a t i o n polynomials variables, A c t a Arith., 22, 1972-3, 257-265. MR 47 # 6661 LIDL, R., WELLS, C. [72] C h e b y s h e v p o l y n o m i a l s in several variables, Math., 255, 1972, 104-111. MR 46 # 5291. LUCAS, E. [78] T h ~ o r i e des fonctions n u m ~ r i q u e s can J. Math., I, 1878, 184-240.
in several
J. reine angew.
simplement p~riodiques,
~eri-
MACCLUER, C.R. [663 On a c o n j e c t u r e of D a v e n p o r t and Lewis c o n c e r n i n g e x c e p t i o n a l polynomials, Acta Arith., 12, 1967, 289-299. MR 34 # 7453. McLEAN, D.W. [80] Residue classes of the p a r t i t i o n function, Math. 1980, 313-317. MAMANGAKIS, S.E. [61] Remarks on the F i b o n a c c i 1961, 648-649. M R 24 # A73
sequence, A m e r i c a n Math.
Comp.,
34,
Monthly,
68,
~.~NGOLDT, H. v. [983 Norm.
D ~ m o n s t r a t i o n de l'~quation k=| ~ --~---~(k) = 0 , Ann.
Sup.,
(3),
15, 1898,
Scient. Ec.
431-454.
MATHIEU, E. [61] M ~ m o i r e sur l'etude des fonctions de plusieurs quantit~s, sur la m a n i ~ r e de les former et sur les s u b s t i t u t i o n s qui les laissent invariables, J. de Math., 6, 1861, 241-323. ~ I J E R , H.G. [70~ On u n i f o r m d i s t r i b u t i o n of integers and u n i f o r m d i s t r i b u t i o n (mod I)., N i e u w Arch. Wisk., 18, 1970, 271-278. MR 44 # 2712 MEIJER, H.G., SATTLER, R. [723 On u n i f o r m d i s t r i b u t i o n of integers and u n i f o r m d i s t r i b u t i o n Mod. I, ibidem 20, 1972, 146-151. MR 47 # 151 NARKIEWICZ, W. [663 On d i s t r i b u t i o n of values of m u l t i p l i c a t i v e functions sidue classes, Acta Arith., 12, 1966/7, 269-279. MR 35 # 156. [74] E l e m e n t a r y and a n a l y t i c 1974. M R 50 # 268.
theory of a l g e b r a i c numbers,
[77] Values of i n t e g e r - v a l u e d m u l t i p l i c a t i v e functions classes, Acta Arith., 32, 1977, 179-182. M R 55 # 7956 [81] Euler's function and the sum of divisors, Math., 323, 1981, 200-212. M R 82g # 10077
in reWarszawa
in residue
J. reine angew.
119
[82] On a kind of u n i f o r m d i s t r i b u t i o n for systems of m u l t i p l i c a t i v e functions, Litovsk. Mat. Sb., 22, 1982, 127-137. [83a3
N u m b e r Theory,
Singapore
1983.
[83b] D i s t r i b u t i o n of c o e f f i c i e n t s of E i s e n s t e i n classes, Acta Arith., 43, 1983, 83-92. NARKIEWICZ, W., RAYNER, F. [82] D i s t r i b u t i o n of values of o2(n) Math., 94, 1982, 133-141.
series in residue
in r e s i d u e classes, Monatsh.
NARKIEWICZ, W., ~LIWA, J. [763 On a kind of u n i f o r m d i s t r i b u t i o n of values of m u l t i p l i c a t i v e functions in residue classes, Acta Arith., 31, 1976, 291-294. M R 58 # 559. NATHANSON, M.B. [75] Linear r e c u r r e n c e s and u n i f o r m d i s t r i b u t i o n , Soc., 48, 1975, 289-291. MR 51 # 379
Proc. Amer. Math.
[77] A s y m p t o t i c d i s t r i b u t i o n and a s y m p t o t i c independence of sequences of integers, Acta Math. Acad. Sci. Hung., 29, 1977, 207-218. MR 56 # 11943 N I E D E R R E I T E R , H. [70] P e r m u t a t i o n polynomials in several v a r i a b l e s over finite fields, Proc. J a p a n Acad., 46, 1970, 1OO1-1OO5. MR 44 # 5298 [72a] D i s t r i b u t i o n of F i b o n a c c i numbers mod 5 k, F i b o n a c c i Quart., 10, 1972, 373-374. MR 47 # 3303 [72b] P e r m u t a t i o n p o l y n o m i a l s in several variables, (Szeged) , 33, 1972, 53-58. M R 46 # 8998 [75] 243-261.
R e a r r a n g e m e n t theorems M R 52 # 8068
Acta Sci. Math.
for sequences, A s t ~ r i s q u e
24-25,
1975,
[80] V e r t e i l u n g von Resten r e k u r s i v e r Folgen, A r c h l y f. Math., 1980, 526-533. M R 82c # 94011
34,
N I E D E R R E I T E R , H., LO, S.K. [793 P e r m u t a t i o n p o l y n o m i a l s over rings of a l g e b r a i c integers, Abh. Math. Sem. Hamburg, 49, 1979. 126-139. M R 80k # 12002 N I E D E R R E I T E R , H., ROBINSON, K.H. [82] C o m p l e t e m a p p i n g s of finite fields, J. Austral. Math. 33, 1982, 197-212. NIEDERREITER, H., SHIUE, J.S. [77]~ E q u i d i s t r i b u t i o n of linear r e c u r r i n g sequences fields, Indag. Math., 39, 1977, 397-405. MR 57 # 3085.
Soc.,
in finite
[80] E q u i d i s t r i b u t i o n of linear r e c u r r i n g sequences in finite fields, II, A c t a Arith., 38, 1980, 197-207. M R 82f ~ ]0048 NIVEN, I., [61] U n i f o r m d i s t r i b u t i o n of sequences of integers, Trans. Amer. Math. Soc., 98, 1961, 52-61. M R 22 # 10971 NOBAUER, W. [64] Zur Theorie der P o l y n o m t r a n s f o r m a t i o n e n und P e r m u t a t i o n s p o l y home, Math. Ann., 157, 1964, 332-342. [65] Uber P e r m u t a t i o n s p o l y n o m e und P e r m u t a t i o n s f u n k t i o n e n f~r P r i m z a h l p o t e n z e n , Monatsh. Math., 69, 1965, 230-238. MR 31 # 4754 [66] Polynome, w e l c h e f~r g e g e b e n e Zahlen P e r m u t a t i o n s p o ! y n o m e sind, A c t a Arith., 11, 1966, 437-442. M R 34 # 2562
120
PILLAI, S.S. [40] G e n e r a l i z a t i o n of a theorem of Mangoldt, Soc., sect.A, !I, 1940, 13-20. MR 1 p.293 RAYNER, F. [83] Weak u n i f o r m d i s t r i b u t i o n for divisor
Proc.
Indian Math.
functions,
to appear
RIEGER, G.Jo [77] B e m e r k u n g e n ~ber gewisse n i c h t l i n e a r e Kongruenzen, 32, 1977, 113-115. MR 57 # 3054 [79] Uber L i p s c h i t z - F o l g e n , MR 82a ~ 10055
Math.
Scand.,
45,
1979,
Elem. Math.,
168-176.
ROBINSON, D.W. [66] A note on linear r e c u r r e n t sequences m o d u l o m, A m e r i c a n Math. Monthly, 73, 1966, 619-621. M R 34 ~ 1260 ROGERS, GoL. [91] M e s s e n g e r of Math.,
21,
1891-1892,
44-47.
SATHE, L.G. [45] On a c o n g r u e n c e p r o p e r t y of the divisor function, Amer. J. Math., 67, 1945, 397-406. MR 7 p.49 SCHMIDT, W. [763 E q u a t i o n s over finite fields: an e l e m e n t a r y approach, Notes in Math. 536, 1976. MR 55 ~ 2744
Lecture
SCHUR, I. [233 ~ber den Z u s a m m e n h a n g ewischen einem P r o b l e m der Z a h l e n t h e o r i e und einem Satz ~ber a l g e b r a i s c h e Funktionen, Sitz. Ber. Preuss. Akad. Wiss. Berlin 1923, 123-134 = Ges. Abh., II, 428-439, Springer 1973. [333 Zur T h e o r i e der einfach t r a n s i t i v e P e r m u t a t i o n s g r u p p e n , 1933, 598-623 = Ges. Abh. III, 266-291, Springer 1973. [73] A r i t h m e t i s c h e s ~ber die T s c h e b y s c h e f f s c h e n Polynome, III, 422-453, Springer 1973.
ibidem
Ges. Abh.
SCOURFIELD, E.J. [74] On p o l y n o m i a l - l i k e functions w e a k l y u n i f o r m l y d i s t r i b u t e d (mod N), J. London Math. Soc., 9, 1974, 245-260. M R 50 # 7OO1 SERRE, J.P. [683 Une i n t e r p r e t a t i o n des congruences r e l a t i v e s ~ la fonction T de Ramanujan, S~minaire D e l a n g e - P i s o t - P o i t o u 9, 1967/68, nr.14, 1-17. MR 39 ~ 5464 [72] C o n g r u e n c e s et formes m o d u l a i r e s , nr.416. MR 57 ~ 5904
S~m. Bourbaki,
[75] D i v i s i b i l i t ~ de certaines fonctions arithm~tiques, -Pisot-Poitou: 1974/75, Th~orie des Nombres, Exp.20, 28pp. SHAH, A.P. [683 F i b o n a c c i sequence modulo m, F i b o n a c c i Quart., -141. MR 40 # 86
6,
24,
1971/72,
S~m. Delange-
1968,
~LIWA, J. [733 On d i s t r i b u t i o n of values of o(n) in residue classes, Math. 27, 1973, 283-291, corr.332. MR 48 # 6044
139-
Colloq.
SWINNERTON-DYER, H.P.F. [73] On l-adic r e p r e s e n t a t i o n s and c o n g r u e n c e s for c o e f f i c i e n t s of modular forms, In "Modular Functions" III, Lecture Notes in Math., 350, 1973, 3-55. MR 53 # 10717 [77] On l-adic r e p r e s e n t a t i o n s and c o n g r u e n c e s for c o e f f i c i e n t s of m o d u l a r forms, II, In "Modular F u n c t i o n s V", L e c t u r e Notes in Math., 601, 1977, 63-90. MR 58 # 16520
121
UCHIYAFu~, M., UCHIYAMA, S. [62] A - c h a r a c t e r i z a t i o n of u n i f o r m l y d i s t r i b u t e d sequences of integers, J. Fac. Sci. Hokkaido, 16, 1962, 238-248. MR 27 # 1433 UCHIYAMA, S. [61] On the u n i f o r m d i s t r i b u t i o n of sequences of integers, Japan Acad., 37, 1961, 605-609. MR 25 ~ 1145
Proc.
[681 A note on the u n i f o r m d i s t r i b u t i o n of sequences of integers, J. Fac. Sci. Shinshu Univ., 3, 1968, 163-169. M R 40 # 115 VAN DEN EYNDEN, C.L. [62] The u n i f o r m d i s t r i b u t i o n of sequences, 1962.
Diss.,
Univ. of Oregon
VEECH, W.A. [71] Well d i s t r i b u t e d sequences of integers, Trans. Amer. Math. Soc., 161, 1971, 63-70. MR 44 # 2715. VINCE, A. [81] Period of a linear recurrence,
Acta Arith.,
39,
1981,
WALL, D.D. [60] F i b o n a c c i series m o d u l o m, A m e r i c a n Math. Monthly, 525-532. MR 22 # 10945
67,
303-311. 1960,
WARD, M. [31a] The c h a r a c t e r i s t i c number of a sequence of integers, satisfying a linear r e c u r s i o n relation, Trans. Amer. Math. Soc., 33, 1931, 153-165. [31b] The d i s t r i b u t i o n of residues in sequences s a t i s f y i n g a linear r e c u r r e n c e relation, ibidem, 33, 1931, 166-190. [33] The arithmetic 1933, 600-628,
theory of linear r e c u r r i n g series,
ibidem,
35,
WEBB, W.A., LONG, C.T. [751 D i s t r i b u t i o n m o d u l o p of the general linear second order recurrence, Atti Accad. Lincei, 58, 1975, 92-100. M R 54 ~ 7396 ~ G N E R , U. [28] Uber die g a n z z a h l i g e n Polynome, die fur unendlich viele Primzahlmoduln P e r m u t a t i o n e n liefern, Diss. Berlin 1928. WEIL, A., [48] Sur les courbes a l g ~ b r i q u e s et les v a r i ~ % ~ s qui s'en deduisent, Actual. Sci. Industr. v. 1048, 1948. M R 10 p.262 WELLS, C. [67] Groups of p e r m u t a t i o n polynomials, Monatsh. 148-262. MR 35 ~ 5421.
Math.,
71, 1967,
[68] Generators for groups of p e r m u t a t i o n p o l y n o m i a l s over finite fields, Acta Sci. Math. (Szeged), 29, 1968, 167-176. MR 38 # 5903. WILLETT, M. [763 On a theorem of Kronecker, MR 53 # 264 WILLIAMS, K.S. [683 On e x c e p t i o n a l polynomials, -282. M R 38 # 140
F i b o n a c c i Quart.,
Can.
Math.
14,
Bull.,
1976,
27-29.
11, 1968, 279-
WIRSING, E., [67] Das a s y m p t o t i s c h e V e r h a l t e n von Summen 0ber m u l t i p l i k a t i v e n Funktionen, II, A c t a Math. Acad. Sci. Hung., 18, 1 9 6 7 , 4 1 1 - 4 6 7 . M R 3 6 #6366. ZAME, A. [723 On a problem of N a r k i e w i c z c o n c e r n i n g u n i f o r m d i s t r i b u t i o n of sequences of integers, Colloq. Math., 24, 1972,271-273. MR 46 # 5272.
INDEX
absolutely
irreducible
additive
function
additive
functions,
bad
integers
~eby{ev
decent Dickson
(C~)
72 21
mod N
, criterion
62
64,
84
function
function,
~D
mod N
77
WUD mod N
78
of a field
31
62
Euler's
function,
Euler's
function
exact order
72
exceptional
polynomial
exceptional
primes
103
sequence
28,
Fibonacci ,
UD mod N
independent
linear
26
86
number
D-WUD mod N
53
18
polynomial
function
distribution divisor
UD mod N
polynomials
Dirichlet-WUD --
51
104
complete mapping condition
polynomial
32,
sequences
recurrent
25
47,
10
sequences
, least period
49
48
48,
49
28
25
123
, order - - -
28
, polynomial , second
order,
, third
mean
value
M(f) - -
UD mod
30,
32,
33,
N
37
33,
37,
37,
38,
38
49
32
M(f)
51
, characterization for
additive
for
polynomials
4
functions
4
17,
22
function
78
9 for
Euler's
for
d(n)
77
for
o(n)
106
for
o2(n)
106
for
ok(n)
110
for
polynomial-like
, algorithm
modular
94
function
51
, polynomial-like , ,
- -
of
,
UD mod
,
%~D
62, N
mod
decency
N
75,
linear
polynomial-like
recurrent
function
sequence
several
12,
permutation
vector
polynomial,
absolutely
25
25 irreducible
with
, exceptional
a linear
12
function for
62,
WUD nod
72 N
25
recurrent
25
, permutational
, criterion
18
15
~ariables
, associated
28 72
94
polynomial
polynomial-like
104
function
, characterization in
96,
72
of
permutation
72
94
of
partition
104
functions
105
invariant
multiplicative
order
N
28
4
M* (f)
- -
N
sequence
with
UD mod
order,
UD m o d
,
Lucas
associated
75
sequence
28
124
Ramanujan's , ~D
Schur's
T-function mod
conjecture
second
order , UD
sequence
21
linear
mod
recurrent
weakly
distributed
uniformly
of
divisors,
W~D
sum
of
powers
divisors,
of
of
Hermite
mod
Ikehara-Delange
of
Wirsing
of
Zame
distribution
N
, systems
weak
uniform
- -
, for
polynomials
WUD
, for
systems
N
mod
N
106,
I
1 of
functions
distribution
8
mod
of
N
8
106
54
mod
, criterion
N
mod
I
4
- -
mod
N
51
- -
WUD
UD
15
of
uniform
mod
distributed
sum
theorem
sequence,
38
N
uniformly
sequence
88
89
N
9,
60
N
8
mod 23
functions
10,
11
107
p
33,
37
ADDENDA
i. p°2.
I~ v~as recently sho~,~
bj l.Ruzsa (On the uniform and
almost uniform distribution of (anX) mod i, S@m. Th@orie des Eombres,1932-3,exp.20,Universit@ e~cists a sequence N v hereas
an
de Bordeaux 1933.) that there
of intec;nrs
vhioh is UD(mod N) for all
(anx) is D~D(mod i) for no real x.
2. p.~. a study of uniform distribution of sequences of alcebraic in~ej;ers ras carried out in H.i~IEDr~£~<TITER,S.K. LO, Uniform di~tribution of sequences of a!sebraio in'~egers, [~th. J. Okayama Univ. ,13,1975,19-29. 3° p.10. Cf. also H°NiEDE~]IYDi:{,
On a class of sequences of
la~tice points, J.Nu.u~ber Theorj ~,1972,m77-502. %. p.26, line +6° H.NiEDE~EiT?Ld
[70~ should be also quoted he-
re. 5. p.32.
i~.Hal]'s result v~as gre~eralized by H. NIEDE~L~EITER
(On vhe cycle structure of linear recurrin~ sequences,
-~,iath.
Scand. ,~8,1976,p~-77). 6. Frob!em I, stated on P.9 has just been solved ±ndeFendentij by E.Rosocho~icz
a~d l.luzsa.