Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1253 JL~rgen Fischer
An Approach to the Selberg Trace Formula via the Selberg Zeta-Function
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo
Author
J~rgen Fischer Mathematisches Institut Einsteinstral3e 62, 4 4 0 0 ML~nster, Federal Republic of G e r m a n y
Mathematics Subject Classification (1980): 10 D 12, 10 D 40, 10 H 10, 58 G 25 ISBN 3 - 5 4 0 - 1 5 2 0 8 - 3 Springer-Verlag Berlin Heidelberg N e w York ISBN 0-387-15208-3 Springer-Verlag N e w York Berlin Heidelberg
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TABLE
OF C O N T E N T S page
Introduction The m a t h e m a t i c a l
The c o n t e n t s
background
of this v o l u m e
I.
Basic
1.1
Notations
14
facts
14 groups,
fundamental
15
1.2
Cofinite
1.3
The
1.4
The e i g e n v a l u e p r o b l e m of a u t o m o r p h i c forms, the r e s o l v e n t of the d i f f e r e n t i a l o p e r a t o r -Ak
22
1.5
Eisenstein
28
1.6
Spectral decomposition e x p a n s i o n in D k
linear o p e r a t o r s
domains
. I [S,k],
multiplier
series of
-~k ' o r t h o g o n a l
The trace of the
2.1
Representation
2.2
The c o n t r i b u t i o n
of the h y p e r b o l i c
2.3
The c o n t r i b u t i o n
of the e l l i p t i c
2.4
The c o n t r i b u t i o n
of the p a r a b o l i c
2.5
The r e s o l v e n t
3.
The e n t i r e f u n c t i o n ~ Selberg zeta-function
trace
resolvent
40
kernel
3.3
The d i s t r i b u t i o n the W e y l - S e l b e r g
-~k
57
elements
69
elements
associated
41 47
elements
IO5 w i t h the 113
equation
of
~ , 113 117
of of the e i g e n v a l u e s asymptotic formula
of
-~k
l
W e i e r s t r a B f a c t o r i z a t i o n of the e n t i r e f u n c t i o n an a n a l o g u e of the E u l e r - M a s c h e r o n i c o n s t a n t The g e n e r a l
of
formula
D e f i n i t i o n and f u n c t i o n a l the S e l b e r g z e t a - f u n c t i o n The g r o w t h
4.
iterated
of the trace by the e i g e n v a l u e s
3.2
3.4
series 35
2.
3.1
17
systems
Selberg
trace
formula
127 Z , 145 162
Index
176
Index of n o t a t i o n s
177
References
180
INTRODUCTION
THE
In
1949
ence
H. M a a 6
between
forms.
For
tions
that
respect
wave
equation
some
forms
and
ticle
[I] by H.
the MaaB
its a p p l i c a t i o n s
wave
holomorphic (see
forms
automorphic
[GGPS],
Selberg
[Se
lebrated values
problem I],
operator,
the H i l b e r t
of
planation
space
Selberg A
=
up
wave
of a u t o m o r p h i c
half-plane
forms
development
of
theory
[He 2],
,
IH, and
func-
automorphic satisfy
metric
in the
turned
the
on
way
out
theory
(see e.g.
[He 3],
in a n a t u r a l
forms
automorphic
the
IH
xf
MaaB
I],
and
correspond-
to be o f k e y
of modular
the
survey
ar-
[Ve
I]).
In a d d i t i o n ,
jointly
with
the
classical
of
SL(2,]IR)
representation
theory
[La]).
eigenvalue
linear
come
a new class
~ < PSL(2,]IR)
to n u m b e r [He
equation
for t h e h y p e r b o l i c
f
subsequent
Riemann-Hecke
functional
subgroup
+
Iwaniec,
classical
o n the u p p e r
I . These
for t h e
BACKGROUND
introduced
for t h e L a p l a c i a n
parameter
importance
the
with
real-analytic
-y \ ~ x 2
with
he
to a c e r t a i n
(I)
A.
series
purpose
are
with
The
[Ma I ] e x t e n d e d
Dirichlet
which
MATHEMATICAL
and
of t h i s
(I) w a s
considered
[Se 2] as an e i g e n v a l u e
that
is,
the L a p l a c i a n
L2(T \ ~). Trace
These
Formula
some data background
which
Roelcke
problem
defined
introduce
T . For some
of a s e l f - a d j o i n t
finally
is a r e l a t i o n by
[Ro 3] a n d
on a suitable
researches
determined we
by W.
led to the
between a more
notations.
domain
the
in ce-
eigen-
detailed
ex-
A
fundamental
bits
of
~
domain
in
F
of
[
is a s e t o f
~ , measurable
measure
~ .
The
foM = f
(M 6 T)
with
e-measurable
representatives
respect
functions
to t h e f:
of
hyperbolic
~
, C
such
the orarea
that
flfi2d~ < ~ constitute a Hilbert space F H ~ L2(~ ~ ~), e q u i p p e d w i t h the s c a l a r p r o d u c t (f,g) = f f g d ~ F Since af is i n v a r i a n t u n d e r ~ whenever f has this property, A
: D
~ H
Roelcke
has
The
defines shown
key problem
This
is c a l l e d
this
problem
of
the
groups
i.e.
bolic
area.
by Selberg
~
I],
[Se
are
with
for w h i c h
(It is n o t
an explicit
determination
eigenvalue
0
first
is r e a l l y
PSL(2,Z) , and eigenvalues
the o t h e r
different conjecture
that
O the
for
it is n o t
exists
even
a t all.
generic
domain
was
Recent
cofinite
completely
[Ro
Of
2],
example
For
for t h e
in t e r m s
whether research
probproblems
of a cofinite
should
rational
mean
only
by
the
modular
in d e t e r m i n i n g
([He 2],
has
3]
c a n be e x p l i c i t l y one
spent
co-
determined
example,
"explicitly
group
For
A , and these
doubt what
was
[Ro
first
hyper-
Hence, t h e m a i n
a single
known
These
the
finite
groups.
of e i g e n v a l u e s
is
groups
[Pa 3]).
of
cofinite
series.
on a computer
PSL(2,~)
of Fuchsian
of
I],
effort
U p to n o w
groups
eigenvalues.)
much
A .
Fuchsian
[Ro
known
of
D.
interesting
A
beyond
explicitly
numerically
groups
from
of the
although
eigenvalues
trary c o f i n i t e
clear
on
really
eigenvalues
sequence
D c H
The
of
Not
forms.
[Pa 2],
however.
Eisenstein the
the
I],
called
complexity.
even
[Pa
a fundamental
Roelcke
decomposition
class
generated
spectrum
continued
to be o f u p m o s t
is k n o w n
[E3],
briefly
2] a n d
are concerned
spectral
domain
.
self-adjoint
of automorphic
interest,
with
determined.
group
[E2],
the continuous
[Se
the
for a c e r t a i n
finitely
groups
analytically
turn out group
the
on an appropriate is e s s e n t i a l l y
problem
only
([Eli,
are
These
A
is to d e t e r m i n e
the g r o u p s
groups
left
that
n o t of a r i t h m e t i c
kind,
lems
[Ro I]
solved
kind
of groups
of the
in
operator
the eigenvalue
class
finite
a linear
now
was
second are
and
[He 4]), known".
the
none of
For
arbi-
a single
eigenvalue
has
l e d to t h e
very
even
few MaaB
wave
forms
([DIPS],
[PS]],
geometrically
[PS 2]),
"nice"
values.
The
less
say that
to
still
more
PSL(2,~)
Since
latter
has
the
eigenvalues
to h a v e
takes
group
knowledge despite
can
(6 > O)
with
of a r i t h m e t i c a l l y
have all
respect
infinitely cocompact
to the
interesting
for
elliptic
many
or eigen-
groups.
Need-
eigenfunctions
numerical
of
for
Irl ---~ ~
(4)
g(u)
be the F o u r i e r
eigenvalues methods.
attempts
that
is for
and
out
groups.
of
scope
results
Formula. Let
on
This
us a s s u m e
is a c O c o m p a c t
dis-
let
2 In = ~ + r n
-A • S u p p o s e
that
IIm rl < ½+6} and e v e n
Trace
r < PSL(2, IR)
elements,
are
Asymptotic
the S e l b e r g
for c o c o m p a c t
-< 12 -< ...,
is h o l o m o r p h i c
h(r)
from
form
the m o m e n t
{r 6 ¢:
(3)
individual
be o b t a i n e d
without
h:
which
to a s y m p t o t i c
simplest
eigenvalues
(2)
classes
of c o u r s e
some
on the
0 = l ° < 11 be the
known
includes
recourse
its
simplicity
crete
our
results
one
for
class
are
large
[St]).
precise
formula
groups
defective
(see
although
the
function
, ¢
and
satisfies
the
growth
condition
= O ( ( 1 + I r 1 2 ) -I-6)
uniformly = ~
~
in the
strip.
Let
h ( r ) e - i r u dr
transform
of
oo
h . Then
the
Selberg
Trace
Formula
states:
co £~(F)
(5)
E h(rn) n:O
-
4~
{P}~
sum on the r i g h t - h a n d of the h y p e r b o l i c
tanh ~ r
dr
log N (Po) X x , g(log {P}T N(P) =-N(P)-~
+
The
f rh(r) -~
side
extends
elements
over
P 6 ~ ~
all
{I} .
N(P))
~-congugacy N(P)
classes
denotes
the n o r m
of
P , that
is,
N(P)
cz+d,
det
Finally,
Po
that
p = pm o
with
in the
above
is,
integrals
D. A. the
is the
Hejhal
articles [Wa].
case
The
proof
P
o
6 ~
trace
gives and
by E l s t r o d t
larger
absolute
[E4],
Hejhal
~ I
proof
formula
of the
of all
integral
operators
manner:
[O,~[
For
A
, ~ , the
with
P,
sums
and
convergent.
formula
see also
the
in
survey
[Ve I] and W a l l a c h
are
associated every
the
trace
on the b a s i c
operator
of
value.
All
absolutely
Venkov
rests
eigenvalue
associated
maximal.
(5) are
[He 3],
eigenfunctions
~:
element
of its a p p l i c a t i o n s ;
trace
following
m
formula
differential
function
with
and
of the
continuous
of the
hyperbolic
eigenfunctions
in the
square
a detailed
some
of the
invariants
to the
(~ ~ ) = I)
primitive
[He I]
cocompact
is e q u a l
fact
that
the
simultaneously with
compactly
point-pair supported
function
~( Iz-z'12 ) k(z,z') is a p o i n t - p a i r
::
invariant,
k(Mz,Mz')
The
is l o c a l l y a linear
that
is,
(M 6 PSL(2,1R)).
= k(z,z')
::
finite
and
H
~
(Kf) (z)
if
'
then n
tions:
k(z,Mz')
T-invariant
in b o t h
variables
and
hence
defines
1
n
H
:= f K ( z , z ' ) f ( z ' ) d ~ ( z ' ) F
= i 4+r n2
Kf n = h ( r n ) f n l
Z MCT
operator
K:
of
Im z'
series
K(z,z')
Now
\I{ [
is an e i g e n v a l u e where
, is c o n s t r u c t e d
the
by the
even
of
-A
entire
following
with
eigenfunction
function
chain
of
fn
'
h , independent
integral
transforma-
5 oo
I
, Q:
, ¢ , Q(x)
[0,~[
:=
S
dt
~(t)
x tg~:~ (Abel's
integral
transform) ,
Q[
;g:
IR
, ~ , g(u)
:= Q ( e u + e - U - 2 ) ,
g I
~ h:
¢
, { , h(r)
:=
?
e iru g ( u ) d u
.
--oo
The
kernel
K
has
the
eigenfunction
expansion
co
K(z,z')
Under
certain
uniformly
on
=
mild
X h(rn)fn(Z)fn(Z') n=O additional
assumptions
, hence
f K(z,z)d~(z) Y
~{xIH
integration.
This
.
yields
on
the preliminary
Y , this
series
can be computed
trace
converges
by termwise
formula
co
(6)
X n=O
h ( r n)
= f K(z,z)d~(z) Y
Representing
K(z,z)
tegration
summation
and
right-hand
side of
by the
the
arguments
bitrary
of
The
proof
still
continuous
spectrum
term derived series
has
Selberg
also
A
formula
setting.
is a p p l i e d
refrained
(M E T)
with
contains to t h e
eigenpackets from
and
last
interchanging
integral
calculations.
of
the
trace
the properties if
T is n o t
parabolic discrete
associated
K(z,z)
suitable
formula
for a r -
(2),
(3) a n d (4).
cocompact and
In t h i s
with
on the
into the Then
elements one.
in-
the
right-hand
but A
has
case
a
a
Eisenstein side of the
(6).
trace
formulae
He proved
the
of the
trace
to vector-valued with
the
by some
difficult
in a d d i t i o n
discussed
f o M = x(M) f has
also
from certain
trace
geometrical where
T
more
k(z,Mz)
the p r o o f
(h,g)
to be s u b t r a c t e d
preliminary
formula
functions
Then
X MET
transform
complete
is c o n s i d e r a b l y
cofinite.
can
trace
approximation pairs
one
series
a unitary
from publishing
type
formula
functions character
a proof.
In
[He 2]
(6)
in a m o r e
for c o f i n i t e on
~
general
groups
satisfying
X , b u t u p to n o w h e Hejhal
proved
the
trace formula for c o f i n i t e groups in the following more general work. weight
Instead of
he considered the d i f f e r e n t i a l o p e r a t o r of real
2k :
2[ ~2
A The c h a r a c t e r plier system define
A
frame-
X
X X
~2,~
has now to be r e p l a c e d by a s o - c a l l e d u n i t a r y m u l t i of weight
2k
on the subgroup
c o n t a i n i n g the e l e m e n t
w i t h values
(cf. section
F
of
SL(2,IR)
1.3).
It is c o n v e n i e n t to
c o r r e s p o n d i n g to
F
and
-I = (-I O) . It is known that Ak is an esO -1 linear o p e r a t o r on a dense s u b s p a c e D k of a
sentially self-adjoint Hilbert space
i2ky
H k . The e l e m e n t s of
Hk
are functions d e f i n e d on
in a f i n i t e - d i m e n s i o n a l v e c t o r - s p a c e
]H
V and w i t h the trans-
formation behaviour
f{az+b~ :\c~_~]
exp(i 2 k a r g ( c z + d , )X((ca b ) ) f ( z )
(z 6 IH, (c a b)6
F,
arg: C -
]-0%0]--,
]-~,n]>
.
There exists a close c o n n e c t i o n with the s o - c a l l e d c l a s s i c a l entire automorphic
forms.
the function
f
If
g
is a c l a s s i c a l entire a u t o m o r p h i c
d e f i n e d by
is an e i g e n f u n c t i o n of
-A k
f(x+iy)
with e i g e n v a l u e
trace formula is similar to the case is higher at some points.
= ykg(x+iy)
form,
belongs to
k(1-k).
then
Dk
and
The proof of the
k : O , the t e c h n i c a l e x p e n d i t u r e
An e i g e n p a c k e t part arises if and o n l y if the
u n d e r l y i n g m u l t i p l i e r system
X
is singular
(cf. section
1.5).
Selberg noted a striking analogy of his trace formula with c e r t a i n "explicit formulae"
in analytic number theory.
"explicit formulae"
the n o n - t r i v i a l
are i n s e r t e d into a h o l o m o r p h i c F o u r i e r t r a n s f o r m of the primes.
h
On the one hand of these
zeros of the Riemann z e t a - f u n c t i o n
function
h . On the other hand the
is a p p l i e d to the l o g a r i t h m s of the powers of
P r o c e e d i n g from this a n a l o g y Selberg
i n t r o d u c e d a zeta-
function
associated
those
the
as
of
follows.
Riemann
the
F
and
the
trace
1 (s-½)2+r2
-
X
which
zeta-function.
Consider
h(r)
where
with
parameters
The
formula
satisfy
hyperbolic
elements
logarithmic
Z(s)
F
which
derivative
=
zeta-function
to
arises
with
~ {Po}F
'
Re
side of the trace formula t h e r e
the
similar
I
right-hand
as
properties
Selberg
( -a~ ) I 2 + r 2
s,a
of
has
as
s
>
I
,
Re
appears
a
>
I
.
Then
a contribution
a function
of
s
can
on
of
the
the
be w r i t t e n
of
the
product
~ m=O
det
((Po)N(Po)-s-m) id v - X
hyp., prim. This {s
product
6 ¢:
Re
function and
converges
s > Z(s)
satisfies
zeros
of
½+Jr n ,
{s 6 ~:
many
of
s = Z .
are
trace
In =
non-trivial
zeta-functions
There are
zeros
the
Moreover,
the
definition
striking and
the
analogy
L-series
Z
with
of
of
the Z
whole
of
-A k . the
standard
in n u m b e r
trivial
All
but
critical
Riemann
closely
the
of
numbers
on
in
s-plane
the
zeta-function.
the
arising
are
sets
that
a series
eigenvalue
analogue
Riemann
to t h e
exists
of
compact
yields
precisely
is a n
is,
of
on
immediately
½} , t h a t
expansion
in
zeros ¼+r2n
uniformly
continuation
equation.
non-trivial
the
and
formula
a meromorphic
a functional
product
perties
has
Z . The
Re
for
Euler
. The
½-ir n , where
finitely
true
1}
absolutely
line
Hypothesis
resembles All
these
properties
of
is
the prothe
theory.
THE CONTENTS OF T H I S VOLUME
A more J.
direct
Elstrodt
with
trivial
approach
in
[E4]
for
multiplier
to
the
the
Selberg
case
system
of
a
zeta-function fixed
of weight
point
O . For
was free the
suggested cocompact
analogous
by group
situation
in three-dimensional
and M e n n i c k e
[EGM] explained
of this approach tary m u l t i p l i e r contents
hyperbolic
a corresponding
for an arbitrary
procedure.
cofinite Fuchsian
system of real weight
of this volume.
space Elstrodt,
The papers
2k
[Ro I],
of
-A k
p(-~k )
of the self-adjoint
and that the resolvent
operator
group with a unipart of the
[Ro 2] by W. Roelcke and [Eli,
show that a complex number under appropriate set
The e l a b o r a ~ o n
is an essential
[E2] by J. Elstrodt form the basis of our considerations.
resolvent
Grunewald
Both these authors
assumptions
extension
belongs
to the
-~k: Dk ---~ I{k
has an integral
representation
of the form
(-~k-~)-If = f G k ~ ( , z ) f ( z ) d ~ ( z ) F For
z } z'
mod
F
convergent matrix (cf.
p.
operator
the kernel series,
26 , (1.4.7)).
Gkl(Z,z')
the summation
The integral
is stated in Theorem
is d e t e r m i n e d
being extended
by a normally
over all
M 6 F
representation
of the resolvent
(p. 27).
from some defini-
1.4.10
tions and simple c o n s i d e r a t i o n s
(f E H k)
Apart
needed later on, the second important
result reported on in the first chapter
is a theorem by Roelcke on
orthogonal
f 6 Dk
expansions
of the functions
plete system of o r t h o n o r m a l of
eigenfunctions
-~k ; see Expansion Theorem
1.6.4
f
n
(p. 37).
with respect (n h O)
to a com-
and eigenpackets
In section
2.1 we trans-
form the integral (I-~) f tr(Gkl(Z,z')Gk~(Z',z))d~(z') F
into the sum of the series
(7)
I nZOE in-I
and a c o n t r i b u t i o n resolvent
I ) 2 InL~ ifn(Z) i of the Eisenstein
equation yields
series.
On the other hand Hilbert's
9
f
)
F =
lira tr(G..(z
Z t---~Z
Integrating
(8)
F
X -I
Z
as the trace
I
z')-Gk~(Z
we o b t a i n
of the
iterated
trace
resolvent
formula
integral
is c o m p u t e d
in sections
tively.
The
eigenpacket
part
derivatives
since
certain
I 2s-I
p.
Z
elements
Z' Z (s)
denotes
56, C o r o l l a r y
gamma
function
The c o m p u t a t i o n considerably results
has the
resp.
The
of
do not exist.
resp.
the
identity,
F , respec-
After
the
and some calfunction
are
at as l o g a r i t h m i c a . The c o n t r i b u t i o n
form
contributions
derivatives the Barnes
complicated
of our c o m p u t a t i o n s
s
2.4 after
jointly w i t h the
(Re s, Re a > I)
zeta-function
of the e l l i p t i c
more
the r e s o l v e n t
to the
elements
be i n t e g r a t e d
in
the
Z' Z (a)
the S e l b e r g
out to be l o g a r i t h m i c
2.3 and
on the h y p e r g e o m e t r i c
functions
1 2a-1
2.2.6).
2.2,
terms w h i c h may be looked
of h o l o m o r p h i c
of the h y p e r b o l i c
where
formulae
appear
2.1,
integrals
p = a(1-a)
that
of
part,
sums c o r r e s p o n d i n g
sums m u s t
states
version
(1.4.7).
the p a r a b o l i c
the single
I = s(1-s),
there
series
four
and
latter of these
where
useful
into
the e l l i p t i c
substitution culations
split
2.1.2)
the e i g e n p a c k e t
by the
the hyperbolic
quite
and
represented
has been
The p r e l i m i n a r y
of the d i f f e r e n c e
being
integrand
kernel.
(p. 46, T h e o r e m
to the i n t e g r a l
lira tr
This
z'))
n
(8) is equal
kernels
t
I -p
n
of the r e s o l v e n t
Z
KA
(7) over
n~O
series
\
(p. 50,
of the other
of e l e m e n t a r y G-function
functions
in P r o p o s i t i o n
2.3.4
part.
turn
involving
the p a r a b o l i c
the h y p e r b o l i c
12.2.5,
elements
in the case of the
and e s p e c i a l l y
than
Proposition
the
identity.
terms
is
We state the
(p. 61),
C o r o l l a r y 2.3.5
10
(p. 68) 2.4.22
for the
elliptic
(p. IO4)
Resolvent
Formula
an i m p o r t a n t
special
the
achieved
clude
well-known
entire
in the
This
coeompact
approach
apparently
The
direct
sary
very
the
must
very
entire
and w h i c h
notations
the
Riemann
2k
> 2
2.5 we o b t a i n
Formula, Trace
of the
systems
X
the
2.5.2),
by c o m b i n i n g Formula
spaces
and n e c e s s a r y
zeta-function
we c o n -
of c l a s ~ c a l conditions
of w e i g h t
2k 6 IR
I
zeta-function bears
arise frc~ the general
another
approximation non-hyperbolic
elementary
the
arguments
by w h i c h
to o b t a i n
the o r i g i n
are
are
the
of the
some
Trace
Formula.
otherwise
avoided.
automatically Selberg
neces-
Moreover,
our
yields
zeta-function
E
that
we m e n t i o n
that
exactly
functional
Selberg
a function
explicit,
of w h i c h
simple
as
contributions
To be m o r e
zeros the
advantage,
factors
in o r d e r
clarifies
function
(a-½)2+r 2
satisfies
The
Trace
Corollary
p. 108, T h e o r e m
Resolvent
multiplier
chosen
properties.
~(s)
of w e i g h t
Selberg
be m u l t i p l i e d
function
2.5.7,
dimensions
I
of the
simple
the
(s-½)2+r 2
-
technical
the a p p r o p r i a t e
for
the
(p. IO2),
In s e c t i o n
Selberg
From
2.4.21
case.
approach
computation
Z
before.
arbitrarily
makes
of the
of u n i t a r y
to the
h(r)
which
case
forms
existence
case.
(p. 106, T h e o r e m
formulae
automorphic
for the
in P r o p o s i t i o n
for the p a r a b o l i c
Trace
results
and
the n u m b e r s
enjoys E
is an
½ ± ir
n
equation
= Z(1-s)
Z,
H
are
analogous
zeta-function
~
and
to the
its a s s o c i a t e d
s
(s) : ½ s ( s - 1 ) ~
usual
2 F(~)~(s)
notations entire
~, ~
function
for
11
The parabolic 3.2,
and
the
terms
investigation
yield
interesting
order
precisely
at m o s t
2
converges
this
tic
for
2,
the number ~(s, X)
for t h e of
the
N(T)
denotes
phic
forms
under
will
be d i s c u s s e d of
and
EF
P
introduced 3.2,
are
entire
in s e c t i o n
3.3 a n d
is a n e n t i r e
p. 126, C o r o l l a r y
is t h e
exact
groups. , +~
of
3.4 w i l l
function
of
functions
of order
3.2.13).
The
abscissa
renders of t h e E
3.2.12).
An estimate
of the
It r e m a i n s
series
on
Let
r n 6 ]O,T[
series.
func-
asympto-
N(T) , and
scattering
Then
of
the a r g u m e n t
eigenvalues:
so-called
an o p e n
of convergence
to t h e W e y l - S e l b e r g
such that
of the Eisenstein
denote let
matrix
which
the W e y l - S e l b e r q
states:
T
~' - w-7/~(½+it'x)dt = o~-( -F ~) -T
the d i m e n s i o n consideration later
the
G6ttingen
T
½+ir n
- ~1
d
case
I
determinant
formula
neral
~
distribution
zeros
asymptotic
in h i s
example,
(p. 125, C o r o l l a r y
for
by m e a n s
the g r o w t h
For
P(s)
in s e c t i o n s
3.2.11,
or not
is d e f i n e d
where
EP
for a l l c o f i n i t e
denote
(9)
of
whereas
Re s > I
a r g H (½+iT) formula
for a product
1-s n
whether
series
tion
results.
(p. 125, T h e o r e m
I nAO I ~O n
question
account
range
and where
on
lectures:
the
space
R(T)
(see p. 138, T h e o r e m
terms
no m e a n s
o f the
T 2 + R(T)
left-hand
of t h e
is an e r r o r
3.3.13).
s i d e of
"Unfortunately
V
automor-
term
Commenting
(9), S e l b e r g
however,
we have
that upon
remarks
in t h e g e -
of separately estimating the two terms on the left-hand side
of [(9) ] so that the asymptotic formula for the distribution of the eigenvalues cannot be given. Only in scrne special cases w h e n expressed
in t e r m s
of
functions
theory
c a n w e do this,
o n the
left-hand
basis
of
this
side
result
that
a n d in all of
[(9)]
are known
these
is
for c o n g r u e n c e
the function from
special
O(R logR) subgroups
as
analytic
cases R
~(s,x)
the , ~
of the m o d u l a r
can be
number
second ."
ri
term
O n the group,
12
Selberg this
conjectured
so-called
[PSI]
,
but
there
further
Since tion
Selberg
[PS2])under
theses, ment
that
the
still
on this
form
is a l w a y s
conjecture
certain
function
in the
N(T)
recently
assumptions,
remain
topic
E
was
such
difficult
at the
of a W e i e r s t r a H
disproved
problems.
section
of o r d e r product
A strong
f o r m of
([DIPS],
as e x t e n d e d
open
end of
is at m o s t
dominant.
Riemann We
hypo-
shall
com-
3.3.
2 , it a d m i t s
multiplied
by
a factoriza-
eQ
with
a po-
lynomial
Q(s)
of d e g r e e of
Q
tion
at m o s t
in s e c t i o n
ficient est
= a2(s-½)2
a2
with
coefficient
2 . We d e v e l o p 3.4. the
There
of the R i e m a n n
+ aO
certain
polynomial
X
For
I
example,
of the
highest
governs
canonical
if
F
coef-
the h i g h factoriza-
is c o c o m p a c t
or
we h a v e
d~(9)
r 2 n
for the c o e f f i c i e n t s
which
analogous
is r e g u l a r ,
a2 : lim < E T~ ~ n _> O
analogy
constant
in the
zeta-function.
system
formulae
is an a m a z i n g
Euler-Mascheroni
of the
if the m u l t i p l i e r
+ a I (s-½)
~
)
logT
.
n
Re r
full
The
resolvent
Selberg the
Trace
fourth
Formula (2),
results
Selberg the
last
of
of
no
chapter.
less
from
turn Trace
than
3. 4.8
the
(h,g) the
Thus the
out
to
Formula.
This
satisfying Trace
resolvent
Resolvent inherit
approach
Trace just
to the
is e x p l a i n e d
the g e n e r a l
Resolvent the
(p. 157).
usual
zeta-function.
In p a r t i c u l a r ,
functions
information",
Selberg
in T h e o r e m
to the
of r e s i d u e s .
zeta-function
general
and
is d e d u c e d
calculus
to a "loss
yields
Formula
and
(4)
summarized
method
for p a i r s
(3),
of the
are
the
Selberg
above
Formula
method Formula as m u c h
Trace
conditions
by
does
in
simple not
resp.
use
lead
the
information
as
13
One
is t e m p t e d to try to p r o v e the K u z n e t s o v - B r u g g e m a n sum f o r m u l a
an a n a l o g o u s manner.
It seems h o w e v e r that h e r e
in
s e r i o u s p r o b l e m s of
c o n v e r g e n c e a r i s e that c a n n o t e a s i l y be overcome.
I am g r a t e f u l My t h a n k s
M~nster,
to P r o f e s s o r
Dr.
J. E l s t r o d t
for his h e l p f u l
are also due to P r o f e s s o r Dr. H.-J.
January
1987
Nastold
advice.
for his
support.
J~rgen Fischer
i,
i,i
NOTATIONS
In this
volume
positive
Z
denotes
INo:=
integers,
rational,
real
on the
terval IH
left
ends
we
:= {z 6 C:
resp.
the IN
and c o m p l e x
[a,b],]a,b[, ]a,b], [a,b[ val
BASIC FACTS
ring
U {O}.
~,
numbers.
For
for the
the r i g h t
admit
of all IR,
a,b
end-points
is c o n s i d e r e d
IN
stand
is the
for the
(a ~ b)
the open,
a = - ~ , b = + ~ . The
Im z > O}
{
real
closed, with
integers,
upper
fields
. For
of
we w r i t e
the h a l f - o p e n
a,b
set of
inter-
the o p e n
in-
half-plane
as a h y p e r b o l i c
plane
with
the
dxdy line e l e m e n t idzl and the c o r r e s p o n d i n g area e l e m e n t de = 2 Y y (x = Re z, y = Im z > 0). Iz,wl b e i n g the h y p e r b o l i c d i s t a n c e of any two p o i n t s
z,w
6 IH,
o(z,w)
we d e f i n e
:= ¼(eiZ'wl
I
Obviously, We k n o w
o(z,w)
that
+
IN xiN
o:
+ e-lZ'Wl
Iz-wl 4ImzImw
-> I , and
, IR
by
+ 2)
2
z-wl 2 4ImzImw
o(z,w)
= I
iff
z = w .
with
SL(2,IR)
:= {(a b) : a , b , c , d
6 IR,
ad-bc
= I}
,
the g r o u p
PSL(2,IR)
:: L ~fM: (r U {co}
I ~ fM(z)
:= --
~ (r U {oo} ,
,
if
z [ ¢ w { - d}
,
if
c
#
0
,
z
=
-
,
if
c
#
0
,
z
=
~
d
c
or
c=O,z=~,
15
is t h e There and
set is
of a l l
a one-to-one
those
subgroups
abbreviation,
we
M
by
6 SL(2,IR)
1,2 In
automorphisms
correspondence of
SL(2,1R)
denote
the
of
between
that
hence
Mz
upper
the
contain
automorphism
M , too,
the
subgroups
of
element
-I
the
belonging
:= fM(z)
half-plane.
to
PSL(2,~) . For
a matrix
(z 6 ~
U
{~} ) •
COFINITE GROUPS,_FUNDAMENTAL DOMAINS
the
whole
SL(2,IR) F
analytic
for
that
volume
we
containing the
the
gamma
the
on
the
which
the
upper
Mz
the
have
F
is a d i s c r e t e
-I . not
subgroup
F)
domain
with
will
half-plane,
(M 6
a fundamental
that
element
function
corresponding
ously
F c IH
assume
cause
T
of
i.e.
parallel
any
of
in
use
is n o
point
that
is
It
symbol
is k n o w n discontinu-
z 6 IH in
an
of
the
operates
accumulation
]H,
of
confusion.)
PSL(2,IR)
there
a point
(not u n i q u e )
(The
subgroup
IH.
for F
has
e-measurable
set
properties
(1.2.1) (i)
o
MF
IH
=
M6F
From
now
crete
on
~(F
(cf.
[Si],
on we
subgroup
domain Note
(ii)
F
that the
1.2.1 modulo
the
F
of
choice
= O
42,
for
Satz
assume
SL(2,IR)
a finite
the
Notation.
p.
always
with by
N MF)
F
all
4,
[Ro
to
be
hyperbolic
the
Two
(z ~ w m o d
points F)
iff
z,w
6 C
there
p.
{±I} 302).
a cofinite -I
area
fundamental
6 F ~
I],
containing
PSL(2,IR)-invariance of
M
and
~(F)
of domain
~
group, having (cf.
U {~}
are
exists
some
of
a dis-
a fundamental [Ro
, ~(F) F
i.e.
I],
does
not
p.
300).
depend
F .
said
to b e
S 6 F
equivalent
with
w = Sz.
16
It is known that any c o f i n i t e group has o n l y a finite number F-equivalence classes of elliptic number of cusps.
fixed points.
F-equivalence classes of p a r a b o l i c
F \IH
F
in
is discrete,
elliptic element 29
(-~ 6 IN,
w h i c h in
~ -> 2).
SL(2,IR)
cos~
-sin~ )
sin~
cos~
Z(R)
o t h e r w i s e to a
is a cyclic s u b g r o u p of
F
A g e n e r a t i n g element is the m a t r i x
F ) of any
R ° 6 Z(R)
is c o n j u g a t e to the r o t a t i o n
Obviously,
the number
~)
solely depends on the
R
lying in
p
F - e q u i v a l e n c e classes of elliptic fixed points
let
vj
c o r r e s p o n d to the
j-th e q u i v a l e n c e class
the h y p e r b o l i c area of a fundamental domain of
IH.
of
F
We number in
IH
and
(I _< j _< p). Then
F
satisfies
([Sh],
[He 2], p. 2):
(1.2.2)
~(t')
= 2rr 2 9 - 2
+
~- ( 1 -
j=1 ~ 6 IR U {~}
is a cusp of
the subgroup of finite order. SL(2,IR)
1.2.2
9.
of finite order
the
If
T-times
(with respect to
F - e q u i v a l e n c e class of the fixed point of
p. 42,
so-called
the genus of w h i c h is denoted by
the c e n t r a l i z e r
R 6 F
(finite)
iff there exists a com-
is c o n f o r m l y e q u i v a l e n t to a compact,
F
be the
of
IH. In this case the orbit space
p u n c t u r e d compact Riemann surface, As
T
fixed points,
F does not contain p a r a b o l i c elements
pact fundamental domain of
Let
p
PSL(2,IR) F{
F
with stabilizer
a s s o c i a t e d with
is g e n e r a t e d by
-I
If
F
a fundamental domain
F
of
F
F - e q u i v a l e n c e classes of
which in
I/
then we can c h o o s e
w i t h the following properties.
cusps of
such that
T
]~ .
c o n t a i n s p a r a b o l i c elements, ~
F
~I,...,~T
of the
and m a t r i c e s
-I T3• := A3• UA•3
={},
is cyclic and of in-
U := ~
There exist a c o m p l e t e system of r e p r e s e n t a t i v e s
A I , . . . , A T 6 SL(2,IR),
F~ := {M 6 F : M {
and a m a t r i x
is c o n j u g a t e to the t r a n s l a t i o n
Remark.
+T
~j
together with
-I
17
generate
the s t a b i l i z e r s
(i)
[O,I[
(ii)
A-I([o'I[3
F~j , a n d t h e r e
× ]yj,~[ c A j F c
× ]YJ'~[)
[O,1[
(iii)
F ~
T AT I ([O,1[ U j:1 3
of
]]{.
1,3
THE
For
z 6 { ~
LINEAR OPERATORS
{O}
let
trices
A =
x
j%l,
]yj,~[)
-l[S,k],
arg z
pal b r a n c h of l o g a r i t h m ,
x ]O,~[
n A11([O,I[ if
YI'''''YT
× ]yl,~[)
= ~
by
,
j,l 6 {I ..... T},
is a r e l a t i v e l y
the
such that
,
MULTIPLIER
denote
determined
exist
compact
SYSTEMS
imaginary
p a r t of the p r i n c i -
-n < arg z ~ n . For the m a -
a2) C b2)(ci c2)
< al a3
,
B
=
,
a4
b3
subset
AB=:
C
b4
and
=
c3
z 6 IH
c4
we h a v e c3z+c 4 a3Bz + a 4 - b 3 z + b 4
(1.3.1)
Hence,
there
(1.3.2)
(independent
w(A,B)
of
z ),
such that
= a r g ( a 3 B z + a 4) + a r g ( b 3 z + b 4) - a r g ( c 3 z + c 4)
3 {w(A,B) I ~ ~ , h e n c e
w(A,B)
c a n o n l y t a k e the v a l u e s
.
Complete found
an i n t e g e r
2~w(A,B)
Obviously, -1,0,1
exists
lists of the v a l u e s
in [Ma 2], p.
These theorems volume,
entail
for e x a m p l e :
of
115, T h e o r e m
some
w(A,B) 16.,
important
for
A , B 6 SL(2,IR)
[Pe I], p.
44, Satz
rules occasionally
can be
4.
used
in this
18
I .3.3)
"Cocycle
Relation":
w(AB,C)
I .3.4)
w(I,A)
1.3.5)
w(-I,A)
=
w(A,I)
= w(A,BC) =
0
= w(A,-I)
w(A,A_I].
(I .3.6)
(1.3.7)
+ w(A,B)
,
,
= w(A_I,A)
= ~ I ,
[
a)
w(DI,A)
+ w(B,C),
= w(A-I,DI)
0
if
,
a 3 = O,
otherwise
= 0 , Dl
a4 < O
,
,
:=
I
, I > 0
,
o
(1.3.8)
b)
w(A-1Dt,A)
a)
w(A-1'%8
= w(A-1,DtA)
= w(A,A-1),
) : w(A-1%e A'A-I)'
%e
I > 0 ,
cos@
-sinS]
ksin8
cosS/
:=
o
b)
(1.3.9)
w(%@,A)
w(A-1ux,A)
Definition.
is c a l l e d
1.3.2 be
O -< 181
_< n
= w(A-I,uXA)
For
as
k 6 IR,
:= e x p ( 2 ~ i k
a factor
Remark.
denoted
-< ~
,
,
r
ak(A,B)
ok
,
a)
b)
1.3.1
= w(A,A-1%@A)
_< f~r
Let above.
system
the Then
= w(A,A-I),
A,B
6 SL(2,1R)
put
w(A,B))
of w e i g h t
coefficients we
x 6 ]19 .
have
k
of
for
k
the
matrices
A,B,C
6 SL(2,1R)
real:
(c3z+c 4 ) (a3Bz+a4)k
(z 6 m ) .
= Ok(A,B ) • (b3z+b 4 )
(For of
z £ C ~
logarithm:
{O}
the
power
z a = exp(slog
zs z),
is d e f i n e d -~
< arg
via
z ~ ~
the .)
principal
branch
19
The
remark
follows
immediately
from
(1.3.1),
(1.3.2)
and Definition
1.3.1.
Throughout sional the
volume
(d < ~)
first
I .3.3
this
we a s s u m e
unitary
that
k 6 ~
C-vector
space,
abbreviation
we
<
and
(V,< , > ) i s
, > being
a
d-dimen-
semilinear
in
argument.
Definition.
is(Z)
For
:= j s ( z , k )
:= e x p ( i 2 k
set
arg (cz+d))
a The
linear
f: ~{
• l[s,k]
operator
.~ V
As we a l w a y s will
It f o l l o w s
not
from
js(z)-lf(sz)
:=
consider
(1.3.10) Hence,
a fixed
entail
Remark
1.3.2
the o p e r a t o r s
Definition.
X:
functions
Let
F
~ H(V)
dropping
k
in the
that
(S,T 6 SL(2,IR), z 6]]{).
= O2k(S,T)JsT(z)
. I [S,k]
fI[ST,k]
k 6 IR,
(S 6 F)
: O2k(S,T)
U(V)
satisfy:
(fl[S,k]) l[T,k] ~V
be the
a function,
ring
V , k 6 IR.
A map
of all
(z 6 IH).
parameter
(f: IH
of
space
misunderstandings.
Js(TZ)JT(Z)
(1.3.11)
1.3.4
on the
by
fI[S,k] (z)
notation
is d e f i n e d
1
with
the
properties
a)
X(-I)
= e - i 2 ~ k id V
,
b)
X(ST)
= ~2k(S,T)X(S)X(T)
of u n i t a r y
S,T
6 SL(2,IR) ) .
endomorphisms
20
is c a l l e d sion
a
-2k
Note.
(unitary)
in P e t e r s s o n ' s
Considering
will
make
1.3.5 X
For
k 6 ~ F .
The
following 1 ], p.
!.3.6 a)
result
534,
If
If
cient
F
[He 2],
contains
system F
on
does
not
F
pp.
of d i m e n -
just
k
on.
system
the
of
of w e i g h t
2(k+l),
even
of multiplier
2k
1 6 ~ .
unitary
systems
characters
is k n o w n
334-335):
for t h e
system
k6
conditions
existence
f 2g
-
2k
1
2@-2
stated
F
d×d-multiplier
scalar-valued
multiplier
d×d-character
on
defined
are
system
system
it s e e m s
not
multi-
a necessary (i.e.
and
suffi-
d = I )
is:
if
p = O
if
p Z I
in s e c t i o n
also
1.2.)
sufficient
for
the
(d a 2) , s i n c e
the
product
of weight
is a u n i t a r y
a unitary
6 IR.
of a scalar-valued on
b)
exists
elements,
Z,
are
in c a s e
of a unitary
F . Conversely,
2k
I l.c.m.[ml,...,mp T ~ ,
p , ml,...,~p
F
there
weight
parabolic
of weight
2n
(The n u m b e r s
elements,
for e v e r y
contain
--~-~Z
on
are
instead
later
of a n y w e i g h t
existence
parabolic
X
condition
multiplier
The
systems
on the
simpler
2k
is a m u l t i p l i e r system
(resp.
F .
of w e i g h t
appear
2k
Proposition.
plier b)
a multiplier
of w e i g h t
on
systems
X
the m u l t i p l i e r
on
([Pe
multiplier
Obviously,
is a l s o
system
notation)
a lot o f e x p r e s s i o n s
Remark.
iff
multiplier
2k
and
dxd-multiplier to be clear,
any even system
whether
existence of a
unitary
of w e i g h t
every
unitary
2k
21
d×d-multiplier we cannot b)
are
least
conclude
also
system.
system
on
F
has
from Proposition
necessary
In s e c t i o n
the
X
for t h e
2.5
the
a representation 1.3.6
existence
Resolvent
that
like
the c o n d i t i o n s
of a unitary
Trace
this.
Formula
Thus
in c a s e
dxd-multiplier
will
entail
that
at
conditions
"
(F----T ~
'
P
k 6 2~ ~0(F)
are
necessary
weight The
on
computation
plier
1.3.7 F
2k
for t h e
and
I ~ l.c.m. [m] ..... ~p]
existence
of unitary
if
p > I
'
dxd-multiplier
systems
of
F . rules
for
w
yield
the
following
assertions
on multi-
systems:
Let
Proposition.
X
be a m u l t i p l i e r
system
of w e i g h t
2k
on
S 6 SL(2,IR) . T h e n ~ 2 k (STS -I ,S)
S X
(T)
is a m u l t i p l i e r
:=
system
SI S2
(x)
x ( S T S -I )
~2k(S,T)
x
=
of weight
2k
on
S-IFs
and
SIS 2 (SI,S 2 6 SL(2,IR)) , -]
×s
1.3.8 D1
=
×(s)
Corollary.
(~ > O)
, %8
- ×
• x(s)
Assume (lel
x ( S M S -I)
< ~)
that or
if
M 6 P Ux
S 6 F
is c o n j u g a t e
(x 6 IR) .
= X ( S ) X ( M ) X ( S ) -]
in
SL(2,IR)
Then we have
for all
to S 6 F
22
1.3.9 has
Corollary.
a positive
If
R 6
If the
trace,
x(p n)
b)
a)
F
THE
= X(p) n
matrix
P 6 F
(n 6 ~)
is c o n j u g a t e
EIGENVALUE
or p a r a b o l i c
then
Icos~
-sin~l
ksin~
cos~j
to
X(R j) = X(R) j
1.4
hyperbolic
(~ 6 IN,
~ ~ 2),
then
(j = I ..... ~)
PROBLEM
OF A U T O M O R P H I C
FORMS,
THE
RESOLVENT
OF THE
is a c o f i n i t e
group,
DIFFERENTIAL OPERATOR -~k_
As
above
(V,<
we
, >)
tiplier
throughout
a dimensional
system
sponding
If
assume
of w e i g h t
to the
f1'f2:
scalar
, V
72{
that
C-vector
space,
X:
2k 6 IR.
Let
I I
product
are
F < SL(2,IR) r
, U(V)
a unitary
be the n o r m
on
V
mul-
corre-
< , > .
functions
with
the
following
transformation
behaviour
fjl [S,k]
(I .4.1)
then
is a
If we
fix an o r t h o n o r m a l
F-invariant
a canonical
isomorphism
with
in
duct.
values
Therefore
on we a l w a y s
V = cd
V
= x(S)fj
and
it d o e s
basis
S 6 F , j : 1,2
complex-valued
of
between in
for all
V
the
with spaces
cd , e q u i p p e d
not mean
function
respect of
with
an e s s e n t i a l
to
functions
on
,
IH
~ , there defined
its c a n o n i c a l restriction
exists on
inner if f r o m
pronow
identify
and
<x,y>
=
~ry r r=]
\x
(x 1 . . . . , X d ) , y
(yl,...,yd)
6¢ d •
23
By this f:
convention
IH
, V
If t h e and
, being over
not
depend
following
1.4.1
any
IH
~ V
satisfying
fundamental
F-invariant, fundamental on the
etc.
of a function
of
is i n t e g r a b l e
domain
choice
domain
of
of
(1.4.1) F , the
over
F
domain.
e-measurable
scalar
iff
F . The value
fundamental
are
it is i n t e -
of t h e This
product
integral
justifies
the
definition:
Definition.
e-almost
differentiability,
componentwise.
fl,f2:
is an a r b i t r a r y
does
are defined
functions
F
grable
measurability,
Let
everywhere
Hk
equal)
be the
space
e-measurable
of
(equivalence
functions
f:
IH
classes ---~ V
of
satis-
fying
a)
fl[S,k]
b)
llfll2 := ~ < f , f > F
Note.
In t h i s
for t h e f .
volume
equivalence
A class
function
the
for all
de < ~
is s a i d
S 6 F ,
.
equivalence
relation
f 6 Hk
if it h a s
(necessarily
1.4.2
= X(S)f
"equal
class
of a function
e-almost
everywhere"
to b e a c o n t i n u o u s
a continuous
(differentiable,
f:
~£
is a l s o
, V named
(differentiable .... ) ...)
representative
with
the
unique).
Remark.
Hk
is a H i l b e r t
space,
equipped
scalar
pro-
duct (f,g)
:= S < f , g >
de
(f,g 6 H k)
F Clearly
Hk
integrable
is i s o m o r p h i c measurable
to the Hilbert
functions
mapping
space F
into
L2(F,m) V .
of s q u a r e -
24
I .4.3
Definition.
We put 2[ 22
Note.
At
this
(1.22))
which
Roelcke
([Ro
ferential The
same
duced
point does
I],
we
not
p.
operator applies
22
adopt
the
coincide
296,
with
(1.19)).
of D e f i n i t i o n
to the
in D e f i n i t i o n
notation
stroke
1.3.3
the
corresponding
In R o e l c k e ' s 1.4.3
would
operators
and
from Elstrodt
.l[S,k]
to the o p e r a t o r s
p.
definition
notation
have
([El],
the
301, of
formal
to be n a m e d
A2k
(S 6 SL(2,IR))
Ak
defined
is,
for a t w i c e
dif.
introin
(I .4.3).
Ak
commutes
ously
with
differentiable
(1.4.2)
Formula Maa8
. I [S,k]
(S E SL(2,IR)),
function
Ak(fI[S,k])
(1.4.2)
is p r o v e d
differentia]
(I .4.3)
=
f:
by m e a n s
Kk
:= k + l y ~
Ak
:= k + l y ~
+ y~
[El],
Lemma
3.1.
(1.4.2),
a twice
p.
. I [S,k]
301,
Note
Ak
and
the
different
preserves
continuously
satisfies
(S 6 SL(2,IR))
the
commutation
the
(z-z)~-{
: k +
(z-z)
.
relations
for
the
f
relation
: Kk_iAk
resp.
notations
[Ro in
+ k(1-k) 1],
function
(S 6 r)
pp. 305,
[E]]
transformation
differentiable
= X(S)f
= k +
by the
(1.20)-(1.25),
fl [S,k] also
we have:
~
- y~
-A k = A k + I K k - k(k+1)
(Cf.
By
of
continu-
operators
the o p e r a t o r
(1.4.4)
~ V
(Akf) l[s,k]
~
with
IH
that
resp.
behaviour f:
IH
306, (3.1)-(3.5), [Ro
I]
.)
(1.4.1),
---~ V
with
i.e.
25
(Akf) l[S,k ] = x ( S ) A k f
In p a r t i c u l a r
this
is v a l i d
.
if
f 6 Hk
is t w i c e
continuously
diffe-
rentiable.
1.4.4
Notation.
Let
ferentiable
functions
If
not
F
does
is r e d u n d a n t F
in
Akf p.
IH,
here,
whence
1.4.5
Satz
be the
f 6 Hk
contain
. Obviously, 309,
Dk
as
parabolic
condition
The
elements, F
in
1.4.1
b)
twice
continuously
dif-
Akf 6 H k •
case
is a d e n s e
3.2., R o e l c k e
Theorem.
with
in this
Dk
set of all
the c o n d i t i o n
has
linear
a compact
follows
fundamental
from
subspace
Akf 6 H k domain
the c o n t i n u i t y
of
H k . In
[Ro
of
I],
shows:
operator
-Ak:
Dk
, Hk
is e s s e n t i a l l y
self-
adjoint.
1.4.6
Notation.
extension
The
of
Let
-A k
eigenvalue task
1.4.7
Notation.
the
with
problem
is the
Let I 6 ~
(-~k -1)-I is a b o u n d e d
certain
scribed the able
kernel
linear
::
Gkl
...... ~ Hk
as
the
forms
spectral
p(-~k )
denote
the
unique
self-adjoint
its domain.
of a u t o m o r p h i c
be the
in the h y p e r b o l i c
decomposition
resolvent
of
set of
plane
-~k
-~k ' that
is
for w h i c h
(-~k - I i d a )
-I : Hk
) Dk
operator.
restrictions
resolvent
Dk
Dk
to d e t e r m i n e
set of all
Under
-~k:
on
(-Ak-l) -I
. Elstrodt
i , Roelcke
([Ro
by an i n t e g r a l
improved
this
result
2],
Satz
operator
7.1.) with
dea suit-
in so far as he showed
26
the
validity
meters,
of
the
representation
for
every
in p a r t i c u l a r
[El],
Satz
5.6.,
kernel
as
follows:
(I . 4.5)
F
is t h e
function 6 IN
[E2],
with and
o
function
F
almost
Satz
For
ks(O )
where
-a
integral
s £ ¢ \
:=
gamma
6.1.,
-s
parameters or
is r e p r e s e n t e d
Kor.
-b
by
the
-c
power
(1
"
for
4.6)
z,w
6 IH
H(z,w)
NOW,
for
s 6 ¢ \
z,w
6 IH,
1.4.8
Proposition.
endomorphisms variables s 6 C \
The
proof
can
be
of (z,w,s)
V
- n:
n
resolvent
b
admitted > c . For
I
put
,
hypergeometric only Izl
if <
I
the
(b)
p.
n
n
F (a+n)
. z n--~
37
with
and
the
s >
I,
r(a)
(a)n:--
following)
Re
6 ]]90 } ,
I
::
s(1-s),
F , put
:: ½
E
x(M)
'
JM(W)
H(z,Mw)
h
respect
, the
series
with
z,w
- n:
n 6 IN O } ,
of
this
assertion to
in
arbitrary
6 IH
to
the
(1.4.7)
£ IH, Re
(O(z Mw)) s
z,w
With
{Ikl
transferred
the
d >
(cf.
let
= 4 IIz-~l m z I2m w
o(z,w)
I)
and
the
M6F
where
}
2
(Im
~-~_
Gkl(Z,W)
(I .4.7)
o
para-
:= ( w - z l k
{Ikl
z ~- w m o d
£ IN
of
series
n (C)n
= n=OE
obtain
is
6 IN °
and
o
I <
set
F(s+k,s_k;2s;1)
6 ¢,
([MOS],
Furthermore,
n
F(a,b;c;z)
(a) F(a,b;c;z)
-k-n:
larger Re
We
and
6 IN
a
with
6.2.).
{k-n,
a,b,c
a > c ,
1
r ( s~ +- k )~ ry( s - k )
function
for
,
JM
as
operator
'
Definition
norm
converges
z ~ w mod
in
'
in
normally
the in
1.3.3.
ring the
F ,
s > I .
the
case
d 6 IN.
d =
I
(see
[El],
p.
318)
of
27
By aid of w:
(1.3.10),
SL(2,IR)
Def.
1.3.1
× SL(2,IR)
1.4.9
Remark.
z % w
mod
For
and the
, {-1,0,1},
Re s > I,
F , S £ F
the
computation
equations
a)
Gkl(Sz,w)
= js(z)x(S)Gkl(Z,W)
b)
Gkl(Z,Sw)
= Gkl(Z,W)Js(w)-Ix(s)
c)
Gkl(Z,W)
Hence,
for
z 6 ~{ Gkl
f 6 H k , the
is t h e r e s o l v e n t
Theorem
function
kernel
(integral
s 6 C , Re s > I ,
domain
(tA
I = s(1-s)
, z,w
6 IH,
are valid:
,
-I
,
:= t r a n s p o s e
Gkl(Z,
of
is
)f
A ) .
F-invariant
whenever
is fixed.
1.4j0 Let
= tGk](W,Z)
for
one verifies:
Ikl - s ~ INo ,
following
rules
of
F . Then
sought
for:
representation
Ikl
- s ~ INo,
i 6 p(-~k ) , and
of t h e
resolvent)_m.
I = s(1-s) for
,
f 6 Hk
F
a fundamental
we have
(-Xk-~)-If = S ~k~(,z)f(z)d~(z) Both
sides
same
elements
coincide
modulus
this of
equation
literally
of the from
o f an e n d o m o r p h i s m
A resolvent
kernel
in
The
[He 5].
to S e l b e r g .
to
Dk
and they are functions
on
not only IH
they
the also
point.
the proofs
over
belong
H k , b u t as c o n t i n u o u s
at e v e r y
For a proof be taken
of
of
corresponding [El]
(Satz
statements
5.6.),
[E2]
(Kor.
V
meaning
its o p e r a t o r
representation
similar
to
construction
o f the r e s o l v e n t
(].4.7) kernel
for
d =
I
can
6.2.),
the
norm.
can also
be
found
c a n be t r a c e d
back
28
1,5
For
EISENSTEIN SERIES
every
(S £ F) bolic
unitary have
the modulus
elements
eigenvalue
1.5.1
multiplier
T 6 F
of
x(T)
Notation.
tives
of the
system I . In the
will
~I'''''~T
such that
are
and
the
-I
T~ ~ I
of
and regular
1.5.2
Remark.
only on the Thus
of the
degree
the
T
:= A
Note
system
If t h e
group
F
of
certain
significance
if
I
x(S) para-
is a n
cusps
system
of
F
-I UA ] 3
and choose
F~j
:= {M 6 F: M ~ j : ~ j }
(I -< j ~ T)
Let
m
denote 3
I of
of
of representa-
x(Tj)
X ;
X
. y*
:=
is c a l l e d
T E m. j=1 3
is
singular
if
by Cor. class
1.3.8
of
T
u p to o r d e r
]
the
eigenvalues
, b u t n o t on and
T•
its
of
x(Tj)
depend
representatives.
is i n d e p e n d e n t
of the
~i,...,{ T
on
as they d e s c r i b e
-~k
1.5.3
that
contains
system
of
T~ = O .
are unique
of t h e
tance
singularity if
choice
multiplier
eigenvalues
sections
stabilizers
eigenvaluc
F-conjugacy
mj
following
of
the
]
called
the
be a c o m p l e t e
classes
A I ..... A T 6 SL(2,IR)
the m u l t i p l i c i t y
r
.
Let
by
on
b e of a s p e c i a l
F-equivalence
generated
X
parabolic
elements
and
F , Eisenstein
series
the c o n t i n u o u s
p a r t o f the
X
a r e of
is a s i n g u l a r fundamental
spectral
impor-
decomposition
"
Definition.
of weight
2k
on
gether
with
-I
Assume
that
I
eigenvector.
Let
{
be a c u s p o f
F . Choose generates the
A 6 SL(2,1R) stabilizer
is an e i g e n v a l u e
Then
for
z 6 IH,
F ,
of
s 6 {
X such
group
x(T) with
a multiplier that
T
:= A
system -I
F{ = {M 6 F: M ~
with
v
Re
s > I
UA
to-
= ~} .
a corresponding
29
E ( z , s ; v , A , k , x ) :=
is c a l l e d
½
an E i s e n s t e i n
multiplier
system
X
series
and
extends
over
a complete
of
with
respect
F6
According
to
(z,s)
fixed.
Therefore
to
6 IHx
Eisenstein
plete
system
and
change
is t e r m w i s e
verifies
, there
also
yields
(1.5.1)
only that
s
~ , the
sum
"
of right
converges
E M£F~\F
cosets
of the
A
by
the E i s e n s t e i n
E(,s;v,A,k,x)
right
uXA
cosets
I [S,k]
{s
..." F M
A
being Re s > I}.
of the
r~
x 6 IR
When
v
choice of
absolute-
6 ~:
computation
with
of m o d u l u s
series
on
of the
of the
series.
a factor
uniformly 6 > O ,
function
independent
the E i s e n s t e i n
appears
(Im AMz)
cusp
v . The
for e v e r y
by m e a n s
] (z) . S u b s t i t u t i n g
-uxA
series
a holomorphic
of r e p r e s e n t a t i v e s
of
-I
JAM(Z)
for the
of r e p r e s e n t a t i v e s
Re s ~ I+~}
it d e f i n e s
the v a l u e s
-I v
F .
{ s 6 ¢:
easily
2k
the e i g e n v e c t o r
system
to
series
F , as one
~2k
for
x(M)
of w e i g h t
[Pe 2] the E i s e n s t e i n
ly in
The
E (A,M)-I M 6 F { \F a2k
with rules
does
respect for X,
not
is r e p l a c e d
I .Simple
com-
by
computation
satisfies
= X(S) E(
,s;v,A,k, X)
, S 6 F •
Moreover,
E ( z , s ; V , A o , k , x ) = O2k(A,N)
if
N 6 F , A ° = AN , T O = N-ITN
stein
series
corresponding
modulus
I .
Applying
the M a a 8
ries
, v 6 k e r ( x ( T o ) - i d V)
to e q u i v a l e n t
differential
E(z,s;x(N)v,A,k,x)
operators
cusps
agree
(1.4.3)
, i.e.
up to a f a c t o r
to the
yields
EkE(
,s;v,A,k,x)
AkE(,s;v,A,k,x)
the E i s e n -
=
(k+s)
E(
, s ; v , A , k + 1 , X)
=
(k-s)
E(
, s ; v , A , k - 1 , X)
Eisenstein
of
se-
30
Now,
(1.4.4)
entails
(1.5.2)
(cf.
-AkE(
[Ro
2],
According
to
tiable some
p.
[Ro
i E @
I], f:
has
-I
I:
: s(1-s)
pp.
every
UA
V
Let
and basis
of
that
such
3OO-301
a Fourier
an orthonormal x(T)
s >
,s;v,A,k,x)
]]{ ~
Propositi0n_ t T = A
Re
E(
, s ; v , A , k , X)
twice
continuously
292).
function
~.5.4 that
for
satisfying
expansion
~
-I
be
of
the
a cusp
generate
(vl,...,Vd)
of
and
F
and
stabilizer
of
V
Vp
= If
with
form:
A E SL(2,IR) group
consisting
~ Bp = O
: e
-Akf
following
the
2~iBp x(T)Vp
(1.4.1)
differen-
F~
of
such Choose
eigenvectors
for
I -< p -< m ( { )
,
for
re(I)
,
with
Bp E ]O,I [
< p -< d
where m([)
f:
]~ ~
satisfy
V
is
f
coordinate
qP
:= < V p , q >
i)
~P
ii)
there
-= O
that
s E C
for
twice all
continuously S { F , -Akf
differentiable = If
with
some
and
•
= j A ( Z ) -I
functions
: IH
for exist
( U ( I m Az)
up
:: < v
P satisfying:
~
m(~)
,u>
+ q(Az))
, -, ¢ ,
" ]O,~[
< p -< d;
coefficients
b
po
, c
po
E (~
(1
s
p
s
m([))
to
I E C ,
a representation
f(z)
with
to b e
: X(S)f
with
has
k e r ( x ( T ) - i d v)
assumed
fl [S,k]
k = s(1-s) Then
:-- d i m
such
31
I
s +
1-s
bpoY uP(y)
CpoY
= i
if
s , ½ I
if
s = ½
,
y > O
.
!
+ CpoY21og
bpoY
Y
2~i(n+Bp)X qP(z)
:
E n:-~ n~O
is a F o u r i e r
series
It c o n v e r g e s
uniformly
(Yo > O,
The
function
By v i r t u e f
u
ing
to
One
can
morphic
of
of
(1.5.1)
respect
show on
that
and
=
Re
z
,
y
:
Im
coefficients
half-plane
IR
z
,
~,n(Y,S).
× [yo,~[
{ , also
128,
130,
every
z 6 ]]£. All
the
tinuation holomorphy
{
[Se
on the in
F
zeroth
meets
s E {
it has
can
Fourier
the
with
a Fourier
be e x t e n d e d
;v,A,k,x)
of
I],
poles
of
line
(z,s)
respect
(z,s)
assumption
Re
s > I
expansion
accord-
even
z
v
lying
C ~
to
Re
x {s C C:
]½,1]
this
Re
property (cf.
p.
no
. There
293, s 6 ¢
property
{s C C:
s : ½} , h e n c e
an en-
has
Moreover,
it has
mero-
E ( z , s ; v , A , k , X)
[Ro 2],
in the h a l f - p l a n e
{s C C:
s
(cf.
exists
which
g(s)
[N]).
iff
interval
enjoys
and that
[LI],
;v,A,k, X)
to a f u n c t i o n
. There
C IHx{
[Se 3],
in the
E(z,s;v,A,k,x) with
for e v e r y
independent
E(Zo,
contained
;v,A,k,x)
of
E(z,
function
of
is c o n t i n u o u s
the
,s;v,A,k,x)
{s 6 ~: Re s = ½} , such
n
and
hence
cusp
named
of o r d e r
simple
is c a l l e d
E(
;v,A,k,x)
g: C ~ line
pp.
1.5.4
1.5.4.
a pole
E(z,
on e v e r y
(1.5.2),
1.5.4,
E(z,
is a r e a l - a n a l y t i c
are
complex-valued
absolutely
to e v e r y
Proposition
on the
[He 2],
x
f .
function
zeros
certain
in P r o p o s i t i o n
in P r o p o s i t i o n
and w i t h
tire
with
,
I S p ~ d).
coefficient
on
qp,n (y,s)e
Re
is for
s > ½]
is no pole
E ( z , s ; v , A , k , X)
s = ½} . The m e r o m o r p h i c (1.5.2)
[Ro 2],
§10).
at
of
its p o i n t s
conof
32
1.5.5 will
Convention. always
denote
lence classes matrices
If
F6j
normal
basis
contains
a complete
of cusps of
such that
groups
~
-I
F . AI,...,A ~ 6 SL(2,IR)
and
Tj : A~IuAj
(I ~ j ~ 7). For every (Vjl,...,Vjd)
x(Tj)Vjp
e
Vjp
of
where
From now on the Eisenstein
From
Ejp(Z,S)
[Ro 2]
1.5.6
%1
generate
j 6 {1, .... T}
V
of
F-equiva-
are appropriate the stabilizer we choose
an ortho-
with
~jp = O ,
I _< p <_ mj
8jp £ ]O,1 [,
mj < p _< d .
I
series
(p. 294, Lemma
,s): ]O,~[
the cusp
~i,..., ~T
E(z,s;Vjp,Aj,k,x)
will always
be
.
_P_roposition.
Ujp,l(
elements,
system of representatives
2~iSjp
named
parabolic
]0.2.)
we adopt
The zeroth Fourier ~V
the following
result.
coefficient
of the expansion
of
with respect
Ejp(Z,s)
to
has the form I pjp,l(s)y 1-s + 6jl VlpY s ,
if
mI > O
if
mI = O .
Ujp,l(Y,S O
Here
Pjp,1
is a meromorphic
least at the points {s E ~: Re s : ½}
Moreover, nate
,
function
of holomorphy (6jl: Kronecker
the following
Dirichlet
of
on
~
Ejp(Z,
which
is holomorphic
), especially
at
on the line
symbol).
series
representation
of the coordi-
functions
• jp,lq(S) is known:
:--
(j,l=1,..,Y ; p:1 ..... mj ; q=1 ..... m I)
33
1.5.7 q 6
Proposition. { 1 , . . . , m I]
with
For
there
Cn+ I > c n > O
every
exist
j,l
6
{I .... ,T} , p 6
a sequence
(n 6 IN)
and
(Cn)n61N
=
{I ..... mj } ,
(Cn(j,p,l,q))n6iN
a complex-valued
sequence
(an)n6iN
a =
(an(J'P'l'q))n6iN
'
such
that
X n:1
c
n 2s
converges
absolutely
if
n
a s 6 ¢ , Re
s > I , and
~jp,lq(S)
F ( s - ½ ) r (s)
:
F(s-k)
Re
See
1.5.8
[He
every
jp
2ns
n:l
if
cn
we
~(s)
:=
the
line
63,
p.
368
(5.22).
is a p o i n t
is d e f i n e d
by
(~jp,lq(S))
,
index,
I,...,T;
lq
of
the
holomorphy
column
p : I .... ,mj;
of
index,
q :
all
both
I .... ,ml).
in
~jp,lq'
the
lexicographical
Further,
for
these
set
~(s)
is a n
the
which
~(s)
(j,l :
p.
('.'Scattering M a t r i x " ) .
s 6 ~
being
order
It
2],
Notation
T*×T*-matrix
s
F(s+k)
x
s > I
Proof.
For
•
:: d e t ~ ( s )
immediate
consequence
of
Proposition
1.5.7
that
~(s)
has
representation
{ r(s-½)r(s)
(1.5.3)
~(s)
(b n 6 @, if
Re
By
[Ro
(j :
s > 1 , the
2],
S~tze
I, .... ~;
~*
~
=
gn+1
Dirichlet
10. I.,
bn 2s ~n
> Bn > O,
series
10.2.,
p = 1,...,mj)
s n:1
the
are
being
n 6 IN) absolutely
Eisenstein
linearly
series
independent
convergent.
Ejp(
,s)
for
s % ½
if
34
all
the
E
(z,
)
are
holomorphic
s
at
(z 6 IH
arbitrary),
and
3P they
satisfy
(I .5.4)
the
functional
Ejp(Z,l-s)
equations
ml
T Z i=I
=
Z q=1
~j
(z 6 IH,
for
all
s 6 ~
E(z,s)
in w h i c h
denote
the
tEjp(Z,S)
equivalent
to
E(z,I-s)
Applying
(1.5.5)
(1.5.6)
terms
the
I ..... ~;
: ¢(I-s)
• E(Z,S)
line p :
p =
are
I ..... mj)
holomorphic.
vectors
of
which
1,...,mj)
Then
Let
are
the
(1.5.4)
is
yields:
¢(1-S)¢(s)
= IT~
~(1-s)~(s)
:
Moreover,
by
(10.30)
(1.5.7)
t¢(~)
2],
Elq(Z,S)
I ..... T;
involved
(j =
twice
[Ro
j :
T~×d-matrix
functions
(1.5.5)
the
(l-s) p,lq
, hence
I
: @(s)
,
we
$(s)
have
: ~(s)
,
l ~ ( ½ + i t ) I 2 = ~(~+~it)~(~-It)~ '
:
especially
As
every
within
function
the
val
]½,1]
~l{s
6 ¢:
is
finite.
is
a zero
a
finite
Ejp(Z,
)
(z 6 If{, j =
half-plane
{s
6 ~:
, by
of
Proposition
Re
s >
Due of
virtue ½}
also
to
(1.5.6)
n-th
order
subset
of
them
lie s
of
I
Re
in is
s >
contained
has
1.5.6
a pole
in
I .... ,Y;
~}
]½,1]
~ . Hence
(t 6 IR).
of
only
all
. The
the
[O,½[
,
1,...,mj)
poles
the
poles
number
n-th
all
p :
order
zeros lie
of
in
of
of
in
the
inter-
of these ~ ,
poles iff
1-s
~ , apart
from
{s 6 C:
Re
s >
½}.
85
By
(1.5.7)
the o r d e r s
By a n a l o g y the
with
following
1.5.9
[He 2]
@
in
(p.
proposition
Proposition.
G } O , such
of
128,
s
-
IF(s)]
Theorem
There
F(s) G(s)
exist
[He 2 ] ,
1,6
SPECTRAL
large
p. 296, p.
374
DECOMPOSITION
system
contain
elements),
parabolic
the
phism not
function
integrand
Gkl(Z,z')
depend
theorem
11.8.)
F,G:
C
,¢ ,
X
where
e
is h o l o m o r p h i c
and
(s 6 ¢)
constant
C > O .
E~] -)
-Ak " ORTHOGONAL
is r e g u l a r
SERIES
(especially
EXPANSION
if
F
does
,IN D k
not
for
0 ~ 1 < Ikl-~}
> I , the n o r m
is a b o u n d e d Here
Theorem
,
:= {(Ikl l)(]-Ikl+l): 1 C % ,
Re(½+~)
functions
s 6 ¢
OF
If the m u l t i p l i e r
~ E~
1].6., p. 130,
entire
for all
~ e x p ( C l s l 4)
sufficiently
(Cf.
coincide.
holds.
IG(s) I ~ e x p ( C l s l 4) some
s
that
~(s)
for
and
on
in the
of
z C IH
is the of
V
the c h o i c e theory
of
;
(cf.
square
of
by v i r t u e of the
[E2],
the o p e r a t o r of R e m a r k
fundamental
integral
Kor.
equations
8.4.b)). norm
1.4.9
domain (cf.
of the the
Z
of
[RSzN],
endomor-
integral
does
F . From 97.)
now
a
38
follows
the
existence
eigenfunctions real
a n d of
of a c o u n t a b l e
of the o p e r a t o r
finite
Every
function
-~k:
multiplicity,
non-zero-eigenvalues
orthonormal Dk
the
system
~ H k . The
(fn)nAO
eigenvalues
s u m of r e c i p r o c a l
squares
Note.
f £ D~.
uniformly
A countable
infinite
or
finitely
many
Laplacian open.
finite
has
an e x p a n s i o n
absolutely
of
is a l w a y s
set.
Roelcke
linearly
for e v e r y
3.3 w e r e
the
form
cofinite
Sarnak
([DIPS],[PSI],[PS2])
of the
f =
E (fn,f) "fn nZO
the
in
recent
work
(cf.
which
to be e i t h e r
question
will
This
would
question
in-
is still
at the
be a t r i v i a l
Iwaniec,
both
exist
the h y p e r b o l i c
be d i s c u s s e d
of D e s h o u i l l e r s , that
a countably
if t h e r e
of
[Ro 3]).
conjecture
suggests
]H.
end of conse-
Phillips
conjectures
and
might
well
false.
X
is a s i n g u l a r
llGkl (z,)II Re(½+~) stein
series
1.6.1
multiplier
is an u n b o u n d e d > I
tribution
D
group
Roelcke's
of it.
If
understood asked
by S e l b e r g
quence
be
sets
independent e i g e n f u n c t i o n s
true,
But
on c o m p a c t
set
If a c o n j e c t u r e
section
are
converges. f%
converging
of
(cf. Ejp
[E2],
Definition.
Let
a dense subset of
H, A:
of e l e m e n t s
has
the
following
a)
vo = O
b)
lim
c)
Av~
and
Korollar
H D
vI 6 D
of
F , i.e. z 6 IH
8.4.b)).
for
every case
give
I ~ Ek , the E i s e n -
an a d d i t i o n a l
operator.
A family
an e i g e n 2 a c k e t of
A
if it
Stieltjes
I 6 IR,
I 6 IR,
integral
being
understood
decomposition
con-
"
properties: for all
norm
Hilbert-space,
a symmetric is c a l l e d
T • -> I , the
for
In this
f 6 Dk
be a s e p a r a b l e , H
for all
, the
on
p : I ..... mj)
formula
vI 6 H
three
llv -viii= 0 1 = f ~dv. O
function
(j = I,...,T;
to the e x p a n s i o n
(vl)16iR
system
sums.
as the
H-limit
of
37
1.6.2
Notation.
orthogonal
1.6.3
Two e i g e n p a c k e t s
if
(vl,w ~) = O
Theorem.
A system
p : I ..... mj)
of
-Ak:
Wjp,l(z)
(vl)16iR , (wl)16iR
for all
I,U 6 IR.
of e i g e n p a c k e t s
~k
' Rk
eigenpackets
I °t
(Wjp,i - Wjp,~,
Wlq, l - Wlq,v)
= ~ 2~(tl-t
if
I : ¼+t~,
p 6 {I ..... mj],
Proof.
By
See
), ,
(j,p)
otherwise
them vanishes
able
Satz
that e v e r y
]2.].,
TheQrem.
the e i g e n p a c k e t s
Wjp,l
form
From
formula
system
in
for
I _< ¼
for
I a ¼, t := ~
~V ,
~ O .
(l,q)
j,1 E {I ..... T] ,
[Ro I], Satz
eigenpacket
identically.
orthonormal
are c o m p l e t e
: IH
q 6 {I ..... ml].
[Ro 2],
Expansion
=
Z = ¼+t2u; t I ,tU >- O,
we get an e x p a n s i o n
1.6.4
Wjp,l
,
[Ro 2], Satz 12.2., this o r t h o g o n a l
in the sense
by
(j = I,...,T;
satisfy
[O with
(Wjp, l)16iR
is d e f i n e d
~O Ejp (z ,½+ir)dr
These
are c a l l e d
for
system of
[Ro I], Satz
The o p e r a t o r (fn)n~O
of e i g e n p a c k e t s
-~k
f 6 9k
which
5.7. and
function
f 6 Dk
to
[Ro 2], Satz case,
-~k: 9k ---~ Hk
of e i g e n f u n c t i o n s
is m a x i m a l
is o r t h o g o n a l
in the s i n g u l a r
(as in Theore~n ].6.3),
H k . Every
[]
5.7.a).
which
j = I,...,T;
7.2.
too:
has a c o u n t together
with
p = 1,...,mj ,
has an e x p a n s i o n
of the
38
f(z)
=
X (fn,f) .fn(Z) n>_O
a. : IR ~ 3P c o n s t a n t s by
- ajp(V)
:
(Wjp,l
bjp(1)
- bjp(H)
:
llWjp,l
are
be u n d e r s t o o d X (fn,f) n_>O
- f
Note.
T~ = O
If
In b o t h
1.6.5 the n
the
the
Proof.
Only
an e i g e n v a l u e that
space
the
direct
space
and
Let
the
,f)
of
case
-~k " T h e r e
by the
absolutely
~ O , thus multiplier
space
p = I, .... mj)
the d e r i v a t i v e s theorem.
The
on c o m p a c t
Expansion
we h a v e
be the
sets
Theorem
are
to
series in
IH.
1.6.4
the
following
eigenvalues
result.
corresponding
to
as in Expansion Theore~ 1.6.4. Then e v e r y and
E 1-2 n n A O 1%O n
of c u s p r3p( of the
converges.
to be t r e a t e d . so E C
[Ro 2],
f E Dk
residues
up
system.
case
remains
to
uniquely
,
Radon-Nikodym
exists
. According
w
determined
integrals,
multiplicity
the s i n g u l a r
• d b j p (l)
Wjp,~ll2
fo,fl,f2,..,
finite
sum of the
(j = I,...,T;
-
regular
of e i g e n f u n c t i o n s
spanned
, IR
Io,11,12,...
system
I = So(l-So)
the
] [f]
where
3P,P
uniformly
of a r e g u l a r
singular
with
of the
we put
case
orthonormal occurs
sense
converges
Theorem.
: IR
Lebesgue-Stieltjes
in the
z 6 IH,
dbjp(1)
3P
ajp(1)
integrals
includes
~ , b
,
dwj~), i (z)
] ~ dajp(1) E f dbj (l) j Ip=I~ p
functions
to a d d i t i v e
The
m,
T X
] [f] (z) =
with
+ ]If] (z)
p.
with
,So)
with
to the
Eisenstein
302,
,s);
series.
1
be
such Satz
eigenvalue
eigenvalue
:= r e s ( E j p (
I 6 IR
Re s o A ½
290 a n d p.
belonging forms
Let
and
s = so )
11.2., 1 the
is
39
A ~
for
the
is a f u n c t i o n
eigenvalue
g £ Dk
with
the
properties a)
-Akg
: ig
b)
for e v e r y exists z
The
vanishing
all
cusps [Ro
cusp
~ > 0
x = Re
By
f
,
~ = A-I~ such
as
y = Im z
of the
zeroth
Satz
8.1.,
tends
Fourier
space
is f i n i t e - d i m e n s i o n a l ,
thus
value
property,
i
also has
this
to
F
uniformly
in
in the
expansions
at
b).)
of c u s p
the
there
infinity.
coefficients
to c o n d i t i o n
the
of
Ig(A-Iz) I : O(e-~Y)
that
is e q u i v a l e n t
2],
(A 6 SL(2,I]<) )
space
forms
for the
eigenvalue
of e i g e n f u n c t i o n s
i.e.
1
occurs
with
for the finite
i eigen-
multipli-
city. As
already
contains
stated
only
(j : I,...,T; Hence, rip(
only ,s o )
E 1-2 n n k 0 I #0 n
in s e c t i o n
finitely
many
1.5 poles
p = I ..... mj), for a f i n i t e
vanish follows
the h a l f - p l a n e
of the E i s e n s t e i n
in p a r t i c u l a r ,
number
{s 6 ¢:
so
with
identically.
Thus
the
convergence
from
Satz
8.1.
[Ro 2],
This
the
in
not
of the
proves
Ejp(z,
lie
s o >_ ½
of
s ~ ½}
series
all of t h e m Re
Re
)
]½,1].
all
the
series assertion.
D
2.
THE TRACE OF THE ITERATED RESOLVENT KERNEL
The aim of the second chapter is a special case of the Selberg Trace Formula which is a relation between the eigenvalues of the operator -Ak
and the elements of the group
vent kernel
((1.4.7), Theorem
r . Our knowledge
1.4.10)
and Expansion Theorem
a base for the proof of the trace formula. Re(½+Vn~-~) > 1, R e ( ½ + ~ )
> I
about the resol-
For
I,~ 6 C ~ E k
1.6.4 are with
we consider the integral
(l-~) S tr(Gkl (z,z') Gkp (z' ,z)) d~(z')
,
z 6 IH
F
(tr A :: trace of the square matrix
A ).
We transform this integral by means of Expansion Theorem
1.6.4 into
m.
E -j--l< n_>O In-I
ln-I_ g ) Ifn (s) 12 +
in section 2.1.
E3 ~1 7 ~ T~ ~ j I p=] -~\¼+t--i
i . ~I ~)" (z'21+'It)12dt ~ n 2IEjp
On the other hand, we have from the resolvent equa-
tion: (I-P) S tr
n->O ~n -~
t r < G k l ( Z , Z ' ) - ~ p (z,z')) .
1.6.5 we see that
~n-~ Ifn(Z)i
tr(%(zz) Z~-~ Z
m, -
Z
j=Zl p=1 is
~-integrable
resolvent kernel.
over
-F •
~+0-~
] >IEjp(Z,½+it) ~+t2-~
The integral
The transformation
is the trace of the iterated
of the integral of the right-hand
side of this equation apart from the contribution (1.4.7)
I2 dt
forms the contents of sections
2.2 - 2.4.
of
±I
to the series
41
2,1
REPRESENTATION
2.1.1
Theorem.
I = s(1-s),
OF THE TRACE BY THE EIGENVALUES
Let
U = a(1-a)
. Assume of
1.6.4 with corresponding eige~nvalues
=
Z
S tr(Gkl(W,Z') F {
I
-~k
s,a 6 C , Re s, Re a > I, Ik l - s ,
mal system of eigenfunctions
(l-U)
OF
that
(fn)n>_O
-Ak
according
is a maximal
r
orthonor-
to Expansion
(In)n>O • Then for all
Gku(Z',Z))
Ikl-a { INO
z,w 6 IH
Theorem we have:
do(z')
I_ )' In U
n_>O \ In-I m.
j I p=1
Proof.
Let
function
-
G ku (I)
Gku
¼+t-21-U ) .<Ejp(Z, ½+it), Ejp(W, ½+it)> dt
~+t2-1
denote the l-th column vector of the Fix
z 6 IH. As a function
d×d-matrix
of the first argument,
G(1) ( ,z) is an element of H k as G(1) ( z) I [S,k ] = X(s)G(1) ( ,z) kp ' kU ' kU _(i) (S 6 F) by Remark 1.4.9 a) and ~ku ( ,z) II is finite by [El], p. 323, Satz 5.5. norm.)
Hence,
(The modulus
of a matrix
(-Ak-l) -I G(1) ku ( ,z)
stands for its operator
is an element of
Dk ' and Theorem
1.4.10 implies: S Gkl(W,Z') F By Expansion
G(1) (z', z) do(z') ku
Theorem
1.6.4,
(-Ak-l) -I G (1)kU(w,z) =
: (-Ak-l) -I G(1) (w,z) ku '
this function
has the expansion
Z {f , (-~k-l) -I G (1)kU( , z ) ) n_>O\ n + j [ (_~k_l)-I
G(1) k~ (,z) ] (w)
Note that fn' (-~k -I)-I G(1) ku ( , z ) ) : I [f G(1) ) In-I \ n' k U ( ,z)
w 6IH
(I) ( , z ) ) ((-Ak-]~) -I fn' G ku I [G(1) ( z ) - in_l \ ku ,
'fn)
• fn(W)
fixed.
42 According to Remark I .4 9 c) . is the
l-th line vector of
Gku( . Gk~(Z,
,z) . = tGk~(Z
--
) , and
l-th component of the column vector (cf. Theorem 1.4.10).
fn '
)
hence,
{G(1)
\ k~ ( ,z),
G(1)k~(,z) is the
f Gk~(Z,Z') f (z') d~(z') = I f (z) n ~n-D n
F
Thus
(_~k_l)-IG(1) (,z)>.fn(W) kp
:
I ~n k
I ~n- ~
fn, (i) (z) • fn(W)
By Expansion Theorem 1.6.4, ] [ ( - ~ k - l ) -I G ku (I)
('z) ] (w)
T
=
mj ~ da. (~+t 2) Z f --]P j=1 p:1 ¼ dbjp(~+t 2)
dwjp,¼+t2(w)__ dbjp(¼+t 2) dbjp(¼+t 2 )
X
with Wjp,¼+t2(w)
b
t : f Ejp(W,½+ir)dr, O
t -> O ,
(¼+t 2 )
= const.
+l[Wjp,~+t2112 : const. +2nt
a, (~+t 2 ) 3P
= const.
+<Wjp,~+t 2 ' (-Ak-l) -I G(1) ( ' z)) ' ku j : I,...,T; p : I ..... m. . 3
JP
(Wjp,_~_+t2, (-~k-l)-IG (IkD)( , z ) )
(Theorem 1.6.3)
(I) ( ,z) ) : ((-Ak-7)-lwjp, ¼+t2, G kD
l-th component of the vector
]: Gk~ (z, z' ) ((-Zk-7) -I Wjp,¼+t2(z')
d~0(z ')
= (-Ak-~) -I (-Ak-l)-1 w.3P, 4+t ~ = (z) From
-AkEjp(Z,½+ir) Wjp'¼+t2(z)
= (¼+r2)Ejp(Z,½+ir)
(r 6 129)
follows:
t : Of ¼+r2-~I (_Ak_~) Ejp(Z 1+ir d ~ t = (_Ak_7) Of ¼+r2-7 I Ejp(Z,½+ir)dr
.
is the
4S t Interchanging E
JP
(z,½+ir)
concludes in
(-Ak-])
Dk
in
c
Dk
trouble
Since
by the c o n t i n u i t y
Wjp,¼+t 2
t I 0 ~+r2-~
that
(-Ak-])
hence
'
O 6 IH×IR.
(z,r)
without
is j u s t i f i e d
with
Ejp(Z,
m a y be replaced
belongs
½+ir)dr
by
to
of Hk
is c o n t a i n e d
(-~k-~)
. Therefore
t (-Zk-7) -I W j p , ¼ + t 2 (z) : ~ I Ejp(Z,½+ir)dr O ¼+r 2-~ Carrying stead
out this
of
1
(_Ak_Z)
step once more with
and
E
3P
(z,½+ir)
I (_~k_~)-1 Wjp
t : ~ I 0 ¼+r2-7
~+t2(z)
t +~ I O ~+r2-I
ajp(¼+t 2) = const.
m .
jp, (i)
being
the
l-th
and
.
I
¼+r2_~
in-
Ejp(Z,½+ir)
yields
' Hence,
Z
one
component
I ¼+r2-~
• E
JP
(z,½+ir)dr
• ___j___1 • Ejp, (i) (z,½+ir) ¼+r2-~
of the v e c t o r
function
E
.
dr
,
JP
Consequently, ] [ {-~k-l) -] G (I) ( z) ] (w) k~ ' T
m. J
I
~J
Z j=1 p=1
Computing
~
the c o l u m n
1 6 {I .... ,d}
dxd-matrix
made
I Z I -i nhO n m. % J I + ~Z Z ~ j I p=1 E(z,s)
• Ejp, (i) (z,½+it)
• Ejp(W,½+it)dt.
in this way S Gkl(W'Z') G(1) k~ (z',z) d~(z') F and s u b s e q u e n t l y c o n s i d e r i n g the trace of the
up by these
S tr
I
¼+t2-p
vector
for every
If
I
0 ¼+t2-1
column
vectors
yields:
Gkz(Z',Z) ) d~(z') I I -V n
" < f n <~Z ;w'' f~ n> '~ '
~ I j ~ O ¼+t2-1
is the m a t r i x
I ¼+t2-H
defined
• <E
before
(z,½+it),Ekp(W,½+it)>j 3P (1.5.5),
then
dt
44 m,
T J j=IZ p=IE <E:p(Z,½+it)3 , E pj (w, ½+it)>
= tr<E (w, ½+it)
(1.5.5)
for all By
tE(z,½+it)
/ tr(~(½+it) k
)
tE(z,½-it)
- E(w,½-it)
t~(½+it)
>
t 6 IR.
(1.5.6)
and
(1.5.7),
~(½+it)
is unitary,
hence the last term
equals m,
tr E(w,½-it)
. tE(z,½-it)
=
E j=l
E <Ejp(Z,½-it),Ejp(W,½-it)>. p=l
Finally we get
(
tr Gkl(W,Z')
Gk#(Z',Z)
) d~(z')
¼+t2-1
~+t2-p
Y
:
i
1
E I -I n>-O n
< f n (z)'fn(w)>
m.
+
<Ejp(Z,½+it), E j p ( W , ½ + i t ) > dt j=I p--1
This yields
According
(2.1.1)
-
the assertion.
to
[E2], Satz
(x~)
7.1.,
(7.13),
the resolvent
equation
yields
s %x(w,z )%~(z ,z) ~(z) : l~ (Gkx(W,Z) %(w,z )) Z'~ Z
for all
IkJ-a ¢
z,w E IH,
]J : a(1-a),
Re s, Re a > 1,
Ikl-s,
o
In connection kernel
I = s(J-s),
with Theorem
Gkl(Z,W)
Ikl-s { INO, This result
, in
2.1.1
this formula
(1.4.7) defined
can be continued
decomposition
deformation of
-~k
of
I = s(1-s)
to a m e r o m o r p h i c
is also stated in Hejhal's
of "continuous
for
F "
is treated.
shows how the resolvent
paper
with
function of [He 5]
(in a certain
Re s > I, s E ~ •
where the effect
sense)
to the spectral
45
For
our
Then
following
(1.4.7),
expansion
considerations
we r e s t r i c t
to the
X ( I ) J i ( w ) = x ( - m ) J _ i ( w ) : id v
of
F(a,b;a+b;x)
for
x ~ 1
and
(cf.
case
w = z .
the k n o w n
[MOS],
p.
44)
asymptotic imply:
(2.1.2) lim (Gkl(Z,Z')-Gk~(Z,Z')):-~(~(s+k)+~(s-k)-~(a+k)-9(a-k) z~ z
+ lim
½
E
z' ~ z
usual
By
[El],
normally
9
denotes
section on
O(z,Mz')
and
majorant
of
z',
it is t h e r e f o r e for
= a(1-a)
s,a
If
z' 6 U
z 6 IH
test,
]M(Z')
~(z,Mz') -s is n o t
F
there
> I , such
all
H(z,Mz')
converges
uniformly at
every
z 6 IH
Re s, Re a > I,
.
Ikl-s,
Thus, the
by
series
Re s > I ,
(O(z,Mz')) '
in a n e i g h b o u r h o o d
z' : z .
a neigh-
that {I,-I}
h
converges
exist
M 6 F ~
e-almost
function.
an e l l i p t i c
s
continuous
6 C,
for
of
C(U) and
E MEF
of the g a m m a
Hence,
(2.1.2)
IkI-a ~ ]N O
of and
z (2.1.1)
, I = s(1-s),
:
(2.1.3)
(i-p)
= -d
+ ½
o-almost
Together
series
a constant
E x(M) M6F~{I,-I}
as a f u n c t i o n
for
(hs (o(z,Mz') )-ha (O(z,Mz') ) ) •
derivative
Poincar&
Re s > I
for all
2 C(U)
the W e i e r s t r a S
entail
the
if
z
of
½
logarithmic
F , by the d i s c o n t i n u i t y
of
U
the
1.4.,
]~d×IH
point
bourhood
and
H(z,Mz')
' id v
M6F~{I,-I ]
As
fixed
x(M) ]M(Z')
)
with
f tr(Gkl(Z,z')Gk~(Z',Z))d~(z')
• ~
(~(s+k)+~(s-k)-~,(a+k)-~(a-k))
I tr x(M) jM(z) H(z,Mz) M6F~{I,-I}
every
Theorem
z 6 IH.
2.1.1
this
formula
implies:
(ks(~(z,Mz))-k a(~(z,Mz)))
46
(2.].4) I X In- 1 n_>O + ½
I ) . ifn(Z)12 = -d • ~<~(s+k)+9(s-k)-9(a+k)-~(a-k) In -
E M6 :'~{I
tr x(M) jM(z) H(z,Mz)
)
)
,-I }
m.
:
for
IEjp(Z,½+it) 12 dt
w-almost
Integrating
2.1.2 Let
Z
every
over
Theorem
F
z 6 IH. we obtain
(preliminary
version of the resolvent
s,a 6 C, Re s, Re a > I, Ikl-s,
= a(1-a),
Ikl-a ~ IN o,
trace formula).
I : s(~-s),
in =: ¼+r 2 , n _> O •
Then
:((s-½)i2+r 2
n>O
(a-½) 2+
r:)
n
d
• ~-~--(9
(s+k) +~ (s-k) - ~ (a+k) - ~ (a-k))
n
+ S [½ Z F M6F~{I,-I}
tr x(M)
jM(z)
H(z,Mz)
)
m,
-
E
Z
j=] p:1
Proof.
-
From Theorem
s-½)2+t2
(a_½)2+t 2
1.6.5 follows
(z,½+it) 12dt
de(z)
convergence
of the
the absolute
series I
E !-I n_>O(ln The monotone
convergence
Z
__/I
ln-~ ) = n_>O
I
In- )
llfnll2
the interchange
of summation
and integration.
[]
Now the remaining the hyperbolic, puted separately
integral over
elliptic
F
splits into the contributions
and parabolic
in the following
elements of
sections.
of
F . These are com-
47
2,2
THE C O N T R I B U T I O N
Every hyperbolic
OF THE H Y P E R B O L I C
matrix
P 6 SL(2,IR)
ELEMENTS
with positive
trace
is conjugate
to a dilatation
DN(p)
with a unique
:: ( N(P)~o
factor of dilatation
pends solely on the is frequently
2.2.1
N(p)O_!z )
N(P)
F-oonjugacy
class
called the norm of
{P}F
Notation.
For
classes of hyperbolic ~(x)
x > I
let
elements
of
::
> I . For {P}F
:: {S-Ips:
N(P)
de-
S 6 F} , N(P)
w
,(x) F
P 6 F
be the number of
F-conjugacy
of norm not exceeding
x .
1
z
{P}r I
2.2•2
Lemma.
Proof•
If
F
fundamental
~(x) = O(x)
system cusps,
let
domain
in
If
matrices
For
to Remark
Y A 2
put
1.2.2
AI~ I = ~
(2
not a fixed point of
from
s ~ Yl
and
F , a complete
r-equivalence yl,...,y Y > O
is the only parabolic
classes of are chosen
AIF , hence
fixed point of AIP
has a positive
to the real axis.
~> := A2A71 A2A~I.
of the
domain
(j : I ..... T)
in the closure of
distance
it has a compact
2.5.
A I , . . . , A T 6 SL(2,IR)
contained
Euclidean
elements
IH, and the simple proof may be drawn
of representatives
T = I , then
AIFA~I
~ .
T A 1 . Assume that a fundamental
~]'''''~T
according
x ~
does not contain parabolic
[He I], p. 7, P r o p o s i t i o n Hence,
as
Observe
that
In this case let
c % O
since
~
is
s := m i n { Y l , Y 2 , c - 2 y ~ I}
48
The
closure
of
set
of
hence
IH,
Kj
:= {z
6 AjF
:
> ~ ( K 4)J _> ~ {[[ O\, I
Now It
fix
any
hyperbolic
is w e l l
known
IogN(P)
=
Iz,Pzl
(axis)
ax(P)
Claim•
There
that
ax(Q)
Proof
of
Sz
6 F
Ist Then
2nd
Q 6
and
claim. let
Q
case s
6 lIP
fixed
Sz
> Yl +I
claim
the
two
fixed
points
of
AIQA~ I
z'
E ax(P)
with
U-mAISZ ' 6 we
have
For
3rd
Sz'
~ =
there
[O,1[
I
may
case.
integer
claim
is n o w
occur
a 3rd
case:
icAiSz+di2
of
, and
N(P) the
> I .
equation
hyperbolic
line
P.
j E {1,min(2,T)
Select
} , such
S E F , such
that
= ax(Q)
by
m = O
one
,
c KI
end
j : I.
point,
Consequently,
~ Yl +I not
real
exceeding and
there
that
is o n e
exists
.
Sz'
Then
Re A I S z ' . 6 A~IUmKI
proved
(by c h o i c e
of
< g .
Im A I Sz Im A 2 S z
unique
norm
.
As
z' 6 a x ( P ) ,
N AIIumK I .
the
Im A I S Z
least
< Im AISZ'
x ]y1,Y1+1]
6 ax(Q
and
.
of
largest
and
is m e t
at
the
sub-
S Yl +I
has
be
is a c o m p a c t
z 6 IH} the
6 S ax(P)
ax(P)
m
I}
trace
points
£ ~
a x ( A I Q A ~ I) = A I S
Let
on
z 6 ax(P)
c K I . The
Yl
+
.
:= SPS -I
Im A I S Z
positive
lies
' m
Suppose
s ~ Im A I S Z
AISZ
z
the
{P]F
¢ ~
< yj
= inf{Iz,Pzl:
iff
both
0 AjIumKj
the
case.
is v a l i d
exist
with
logN(P)
joining
z
× ]yj,yj+l ]~/ > O , j = I ..... T .
P 6 F
that
~ <_ Im
I > - c2im
AISZ
> + c
>- y2
-> ~ .
) . For
< _>2
49
Now one of the first two cases applies with
j = 2
instead of
j : I.
Thus the claim is proved. It follows that for every hyperbolic m 6 ~
and
j 6 {1,min(2,%)}
the two sets
Kj
and
P 6 F
there exist
, such that the hyperbolic
U-mAjQA~IumKj
Q 6 {P}F ' distance of
satisfies:
IKj , U-mA:] 1 UmK j I : IA~ 1 UmKj, Q A~ t UmKj I S log N(Q) = log N(P) as
A71UmK. 3 3
contains at least one element of
For two distinct hyperbolic the associated
ed above do not coincide
if
dj
z
J
6 K.
= sup{l
Then
]
d
]
J
dj ,
log x}
diameter of
K. :
J
and
min(2,T) f E cardtW 6 A,FA71:
j=l
]
3
} Izj,WKjl
-< log x + dj
min(2,T) [ J=]E cardlW 6 A.FA-]:3 ] WKj c B(zj, min(2,T) E j:1
N W'Kj)
IKj,WKj I
.
Izj,WKjl s IKj,WKjl +
_<
construct-
]
be the hyperbolic
,wl : z,w
~(x) -<
u-m2A J2 Q 2A-]U J2 m2
[ 6 A.FA71: ~ min(2,~) E card~W
and let
{PI}F , {P2}F
Jl = J2 " Hence we have
j:1
Fix
ax(Q)
classes
U -ml Ajl Q.A71U ! ]I ml '
elements
~(x)
F-conjugacy
,
2~(B(zj,
} log x + 2dj)
log x +2dj)) ~(mj)
as
~(WKj
= O
for
W,W'
6 AjFA~ I , W % +W'
Kj
is a subset of the fundamental
domain
AjF
of
(Remember that AjFA~ I .) The last
term is known to be equal to min(2"T) j=IE
2 ~(Kj)
( ) 2~ cosh(log x + 2dj)-1 : O(x) as
This proves the lemma.
X
, ~
co
.
[]
50
2.2.3
Remark
and Notation.
is t h e
unique
number
N(P)
Assume
that
> I , such
P
that
6 F
is h y p e r b o l i c ,
N(P)
either
I
=
A P A -] : DN(p)
O A
6 SL(2,rR) . A l o n g
there
exists
or
_½
APA-
1
= -D N(P)
with
some
N(B) with
a unique
F , the
number
group
No(P)
AFA
-I
is d i s c r e t e ,
> I , such
that
-I
too,
and
hence
DN
(p) O
generate Po
the
:: A - I D N
centralizer (p)A
Z(DN(p))
together
with
of
-I
DN(p)
with
generates
the
respect
to
centralizer
ArA
-I
Z(P)
o of
P
with
element
of
conjugacy P 6 ~ and
F
exactly
2.2.4
F •
There
exist one
to
is
n 6 IN
The
p.
Po
with
is c a l l e d P ,
N ( P o)
exactly
[Ro 2],
Remark.
H:
to
associated
class.
there
According
respect
one
= No(P)
For
every
hyperbolic
either
(7.11),
primitve
a primitive .
primitive
such that
263,
{Po)F
the
p = pn o
hyperbolic
hyperbolic hyperbolic
element or
Po
Fmatrix 6
F
_p = p n o
we have:
function
IH×IH
, (~ ,
{w-z~ k :: \ ~ /
H(z,w)
(cf.
(1.4.6))
satisfies
H(Mz,Mw) for
all
This
M 6 SL(2,IR),
remark
is u s e d
o f the h y p e r b o l i c
2.2.5 2k ,
= ]M(Z) z,w
in t h e
terms
a fundamental
I = s(1-s).
Then
the
6
JM(W)
trace
that
domain function
-I
]]1.
following
to the
P r o p o s i t i o n _ t. A s s u m e F
H(z,w)
of
X
computation
of t h e
contribution
formula.
is a m u l t i p l i e r
F , s 6 ¢,
system
Re s > I,
Ikl-s
of w e i g h t ~ ]N o ,
5i
G~lh} : ]H - - ~
, Gk~hyp(Z) := ½ X tr x(M) jM(z) H(z,Mm) hs(g(z,Mz)) M6F M hyperbolic
satisfies:
G k l h y p l F 6 L 1 (F,~0) , and I
f Gklhyp(Z)d~°(z) F
E
- 2s-I
N(p) -s -I
tr k (P .log N(P o)
1 -N (P)
tr P > 2
Here
the sum r a n g e s
m e n t s of
F
solely on
{P}F
Proof.
{P}F '
to
on
Po
F-conjugacy
trace.
{P]F ' {Po}F
classes
of h y p e r b o l i c
For an a r b i t r a r y
is the p r i m i t i v e
P. AS a l r e a d y
stated,
By C o r o l l a r y
representative
hyperbolic
the n o r m s
N(P),
1.3.8,
ele-
element
N ( P o)
trx(P)
> I
P
corredepend
also depends
only.
Mz :
(1.3.10), Hence,
the
with positive
of the c l a s s sponding
over
(-M)z
for all
M 6 F , z 6 IH. By D e f i n i t i o n
1.3.4
and
x(M)JM = x(-M)J_M
for
z 6 ~I
E {P}]'
Gklhy p(z) =
E tr x(S-Ips) " (z) H(z,S-IpSz) • hs((~(z,S-Ipsz)). S6Z (P)kF JS-Ips
tr P > 2 H e r e an a r b i t r a r y class
{P}F
the e l e m e n t s
" X ..." SEZ (P) \F the r i g h t
Yp
::
representative
extends
cosets
of
is s e l e c t e d
of w h i c h
have p o s i t i v e
over a complete
Z(P)
with
from e a c h
respect
trace.
F-conjugacy The
sum
s y s t e m of r e p r e s e n t a t i v e s to
of
F . The set
~_~ SF SCZ(P)\F
is a f u n d a m e n t a l
domain
Hence
that
it f o l l o w s
of
Z(P),
and we h a v e
~(z,S-Ipsz)
= ~(Sz,PSz).
52
f
Z
1 "
(z) H(z,S-Ipsz)
F s6z(P)\I J Js-Ips :
ks(a(z,S-IPSz))
! dc0(z)
f [kS(O(z,Pz)) [ de(z)
Fp N o w choose
A 6 SL(2,IR)
such
that
f Iks<0(z,pz))l d~(z)
:
is a f u n d a m e n t a l
diagonal value
matrix
of the
of
last
integral
fundamental
generated
by
-I
of
a(Dz,N(P).Dz)
does
domain
and
not change
of
Z(DN(p))
DN(Po ) ,
Z(D N(P))
.
: DN'p't) ' N(P)
A Z ( P ) A -I = Z(DN(p))
D 6 SL(2,IR)
arbitrary
tal d o m a i n
domain
-I
> I
Then
f Iks(O(z,N(P)'z)) ] d~(z) AFp
Fp APp
APA
hence
As
for every
: o(z,N(P)z),
if
A~p
z 6 IH, the
is r e p l a c e d
. The g r o u p
by any
Z(DN(p))
IR × [I,N(P o) [
is
is a f u n d a m e n -
Therefore
f[ks( {z,Pz))
d {z) :
f
Fp
( iN
I R × [ I , N ( P o) [
d (z)
S\4N(P) (Ira z) 2/
N(P o ) 7 -co
f
k ( (N (P) -] ) 2x2 + (N (P) +I ) 2y2) s 4N(P)y 2
I The
N(P)+I
substitution
latter
term
in the
x - N ( P ) - ~ y-t
inner
I " y dy
co ~
•
the
implies
N(P)
Z CI
last
term does
ks
((N(P)+I) 4N(P)
(I+t2))I
dt
--co
I
for every
the e x i s t e n c e hyperbolic
of a c o n s t a n t F-conjugacy
C I > ] , such
class
{P}F
log N(P o)
_< C3
• N(p)-Re C2,C 3
co f -oo s
(
log N(P o)
depending
1 iRe s
4N(P)
(N(P)+t) 2
solely
-
i+t2/
, on
s,k .
dt
that
" Hence,
not e x c e e d
CI+1 Ci_i
the constants
transforms
2 N(P)+] N(P)-I
2.2.2
integral
into
N(Po)
Lemma
dx y
•
C2
the
53
As
Re s > 1 , we c o n c l u d e X
N(P)
-Re
s
f r o m Lenm~a 2.2.2
• log N ( P o)
that
converges.
{P}r tr P > 2 The d o m i n a t e d
convergence
theorem
now entails:
G k l h y p l F 6 L 1 (F,o)
,
and f Gklhyp(Z) Y
X {P}F
d0~(z) =
X S6F\Z(P)
tr X (S-IPS)
tr P>2 • S " (Z) H ( z , S - I p s z ) F JS-Ips From Definition X(S-1ps)
1.3.4,
j
(1.3.10)
(z)
h
(d(z,S-Ipsz))
and R e m a r k
H(z, S-IPsz)
d~0(z)
s 2.2.4
follows:
= X (S) -I X(P)X(S) jp(SZ) H(Sz,PSz) ,
S-Ips z 6 IH. With
the f u n d a m e n t a l
Gklhyp(Z)
domain
do(z)
Fp
X
=
defined
tr X (P)
{P}r
F
above,
we thus h a v e
f jp(Z) Yp
H(z,Pz) hs(d(z,Pz)) d~(z)-
tr P > 2 Now
fix
f
Yp To p r o v e N(P)
P 6 F
, tr P > 2. It r e m a i n s
jp(Z) H(z,Pz) k s ( d ( z , P z ) )
this,
> I
again
Note
O(z,Pz)
that
select
to s h o w that
dc0(z)
_
A 6 SL(2,IR)
for all
= o(Az,N(P).Az)
1 2s-1
log
with
N(p) -s
N(P ° )
1-N(P) -I
A P A -I = DN(p)
=: D ,
z 6 IH
,
O2k (A-1 ,A) jm(Z)
H(z,Pz)
= JD(AZ)
H(Az,N(P)'Az)
d2k (A-1D,A) O2k (A -1 ,D)
as a c o n s e q u e n c e By
(1.3.6)
and
of R e m a r k
(1.3.10)
2.2.4,
and D e f i n i t i o n
(1.3.7), d2k (A -1 ,A) :
°2k (A-Im'A) d2k (A-I'D)
I ,
moreover
JD --- I
1.3.1.
54
Therefore we obtain jp(Z) H(z,Pz)
ks(g(z,Pz))
de(z)
Fp :
~ H(z,N(P) .z) ks(g(z,N(P) .z)) d~(z) A]:p
Here, as well as above, the fundamental domain be replaced by
IR × [I,N(Po) [.
AFp
of
Z(DN(p))
may
Hence, the last integral eauals
N(P o) f
f <-~P)Zl)x+i(N{P)+1)y/ -"
I
By the substitution
N(P)+I N(P)-]
{ t+i [ <-~7/
log N(Po)
: N(P)+I N(P)-I
~
N(P)+I N(P)-I y
x
" ks
+(N_P)+I)2y 2 4N(P)y
~dy y2
"
• t, this term is equal to
{(Nm)+1) 2 hs < 4N(P)
(I+t2))
dt
4N(P) )s F(s+k)F(s-k) -(N(P)+1) ~ • log N(P o) • 4~F(2s)
(1-it)2k (1+t2)s+ k
( F
4N(P) s+k,s-k;2s;
-
If the hypergeometric
2 (N(P)+I)
function
is represented
I ) dt -1+t 2
by the corresponding
power series, the integrand may be integrated term by term. This is ensured by the dominated convergence theorem.
Hence,
S jp(Z) H(z,Pz) hs(O(z,Pz) ) de(z) Fp _ N(P)+I ( 4 N ( P ) )s N(P)-I (N(P)+1)2 • log N(P o) ~ F (s+k+n)F ( s - k + n ) I < 4N(P) )n ~ (l-it) 2k n:oX 4~F(2s+n) " n-~. " (N(P)+1)2 -~ (1+t2)s+k+n
dt
55
2k
~
co
(]-it) (I : ~ + t - ) -- -- dt
f
f
:
(1+it) -(s+k+n) (]-it) -(s-k+n)
dt
--co
: 2 f O
(cos ~)2s+2n-2
= ~.22-2s-2n
. cos(2k~)
F(2s+2n-1) F(s+k+n)F(s-k+n)
F(s-½+n)F(s+n) g-~ " F ( s + k + n ) F ( s - k + n )
:
Inserting
f Fp
this v a l u e
jp(Z)
d~
by Gau8'
into the
H(z,Pz)
By
[MOS],
([MOS],
p.
9)
formula.
s e r i e s we get:
hs(g(z,Pz))
de(z)
F (s-½)F(s) 4~F (2s)
\ (N(P)+1) 2/
F
(
s-½,s;2s;
4N(P) ) (N(P)+I)2 "
p.38,
F ( a , a + ½ ; 2 a + 1 ; z 2) = 2 2 a ( 1 + ~ - z
thus
t = tan ~)
duplication
N(P)+I . (4N(P) is. I~N(Po).V~N ~
(subst.
the last t e r m is e q u a l
1 2s-1
,
to
N(P) -s -I
log N(P o)
The p r o o f
)-2a
1-N(P)
[]
is f i n i s h e d .
As a c o n s e q u e n c e
of L e m m a
2.2.2
the s e r i e s N ( p ) -s
(2.2.1)
X
tr X (P) log N(P o)
{P}r
I-N(P)
-I
tr P > 2 converges
uniformly
Hence
series
the
As a l r e a d y p 6 F element
with Po
stated
absolutely
represents in R e m a r k
tr P > 2
there
and e x a c t l y
one
on c o m p a c t
sets
a holomorphic and Notation exist
function 2.2.3,
exactly
n 6 IN
in
with
{s 6 ~: Re s > 1)
.
on this domain.
for e v e r y h y p e r b o l i c
one p r i m i t i v e
hyperbolic
p = pno ; by C o r o l l a r y
1.3.9a)
56
X(P)
: X(Po )n
Thus co E n=1
E {Po}F The n o t a t i o n
"
the s e r i e s
(2.2.1)
equals N (Po)-ns
tr X (Po)n
,
log N ( P o)
Re s > I
I - N ( P o )-n
E
..."
indicates
that
the sum e x t e n d s
over
all the
{Po}F F-conjugacy
classes
As a f u n c t i o n
of p r i m i t v e
of the v a r i a b l e
hyperbolic s , the
elements
function
Po { F .
(2.2.1)
has
the p r i m i -
rive co Z n=1
Z {Po}r
tr X (Po)
co
As this
E E m = O n=1
E { mo}r
E tr log
converges
{s 6 ~: Re s > I}
2.2.6
log A
ahyp(S)
is h o l o m o r p h i c , {s 6 ~:
and
tr X (Po)n nl N ( P o )-n(s+m)
uniformly exp(tr
is d e f i n e d ,
Corollary.
The
:=
N ( P o )-ns -n I-N(P o)
co
E {Po}F
series
such that
n I n
the p r o d u c t
absolutely
we o b t a i n
Ehyp:
~ det I d m:O converges
Re s > I
,
on c o m p a c t
log A) = det A
function
~ {Po)F
X ( P o ) N ( P o )-s-m
sets
for e v e r y m a t r i x
from Proposition
X(Po)N(Po)
A
2.2.5:
{s 6 ¢: Re s > I}
uniformly
in
, {
,
-s-m
on c o m p a c t
sets
in
Re s > I} , and
S Gklhyp(Z) F
I d~0(z) : 2 s - ~
-hyp (s) -hyp
~ N (Po) -ns X X tr X(Po )n log N(P o) • 2s-I {po]F n:1 I-N(Po )-n I
Re s > I,
Ikl-s { INo,
I = s(1-s).
57
By a n a l o g y tion, by
~ -hyp
the
Euler
is c a l l e d
product
expansion
the S e l b e r q
of the
zeta-function
Riemann
and
zeta-func-
frequently
denoted
Z .
2,3
THE C O N T R I B U T I O N OF THE ELLIPTIC ELEMENTS
During F
with
the
to the
hand
computation trace
in the
there
following
of the
contribution
will
occur
two
lemmata.
of the e l l i p t i c
an i n t e g r a l
which
elements
is c o m p u t e d
of
before-
.R
2. 3.1
Lemma.
Let
k 6 IR,
s 6 C,
Re > ½
, O < e < ~
•
Then
2 ei2kq°(cos
%o)2~-2d
e
Proof.
Assume
for the m o m e n t
ei2k~(cos
~)2S-2d~
: lim tfl
@ = lim t21
2 2-2a
that
(2~2) tm ~~
E m=O"
F(1,~+k;2a;1+ei28).
{ IN.
2 2-2s ~ e i 2 ( k - a + 1 ) ~ 0 e i2(k-~+l+m)~
(1+tei2~) 2s-2
d~
: lira i21-2~ Z r(2-2a+m) t~1 m= O r (2-2a)
(-1)rotm " I m' k-~+1+m
= lim t~l
(k-~+1)0 F ( 2 - 2 a , k - s + 1 ; k - d + 2 ; - t e
i2 ]-2s
last
I " k-~+1
nominator
transformation of the
m-th
dqo
O
(
ei2
+ e i~(k-s)
The
s-k
-- I 1 -2a
is a c h i e v e d term
by the
ei2(k-s+]+m)e _ei(k-s+l+m)u
i28)
F(2-2s,k-s+1;k-~+2;t))
by m u l t i p l y i n g factor
numerator
F(k-~+1+m)
.
Both
and the
de-
58
hypergeometric
functions
occurring here are c o n t i n u o u s
as
t~1
,
and we
have:
F(2-2~,k-s+1;k-~+2;1)
Moreover,
by
[MOS],
p.
£ (k-s+2) F (2~-I)
=
([MOS] , p. 40)
F (c~+k)
47, we h a v e the t r a n s f o r m a t i o n
formula
F ( 2 - 2 ~ , k - ~ + 1 ; k - ~ + 2 ; - e i2e)
F(k-~+2)P(2a-1) F(2_2¢,k_~+1;2_2a;1+ei2e) + (1+ei29)2a-1 k-~+1 F (a+k) I-2~
= F(k-a+2)F(2~-1) F (~+k)
F(I ,s+k;2s;1+e i28)
. (-ein(k-~)).e-i2(k-~+1)@ + (1+ei2e)2s-1 k-~+1 F(1,s+k;2s; 1+e i28 ).
1-2o~
Hence, i1
f ei2kq°(cos ~) 2 a - 2 e = i21-2s
= i(cos
The a s s e r t i o n
2.3.2
Lemma.
O < 8 < # . I(k,s,8)
I
1-2d
8) 2s-I
for
Let
d~
1+ei2e (
2s-I )
e i(2k+1)8
~ - k 6 IN
k 6 IR,
e
i2(k-s+1)8
F(1 , s + k ; 2 ~ ; 1 + e i2e)
__/_I I-2~ F ( 1 ' a + k ; 2 a ; 1 + e i 2 8 )
now follows
by c o n t i n u i t y .
s 6 $, Re s > max(½, Ik 1 ) , I = s(1-s),
Then := r ( s + k ) F ( s - k ) r(2s)
V~ cos 8 + i ~ - x
cos 8
O
( 1 - x cos20) -½ x s _32 F ( s + k , s - k ; 2 s ; x ) d x
E
]
2s-1
2iei2k8
m=O
/e-iS (2m+I) \ ~ k--i-m
ei8 (2m+I)) s+k+m
Note I.
If
I(k,s,B) =
1 2s-1
where again Note 2.
8 =
For
with
v j
v
E IN, j E {I,.
. .,v-I]
, then
i8 (21+1) s+k+l
2ie
I
1=0
)
(
~
-ie (21+1) )
-e
rT
$ = -
r
8
=
0 we have
Proof of the lemma.
T(k,s,O)
n+i(+(s+k)-$(s-k))
=
Suppose first that
0 < 8 <
5.
I £ we represent
F(s+k,s-k;2s;x) by the corresponding power series with
0
as its
centre of expansion, the dominated convergence theorem allows to compute the integral
cos 9 := VF
cos
I (k,s,8) by termwise integration. The substitution
e
yields TI -
2 r(s+k+n) r (s-k+n) I(~,S,O)= 2 t (cos 0 )I-2s-2n / ei2k9 (cos q )2s+2n-2 d9 r(2s+n) n! n=O 8 m
the last equation is ensured by Lemma 2.3.1.
From the integral representation of the hypergeometric function ([MOS], p. 54, 2.5), the functional equation and the duplication formula of the gamma function follows: I(k,s,8) = -2iei (2k+1)8
n-f 22s-2
(dominated convergence theorem). By CMOS], p. 38, F(s-f,s;2~;4t(l-t) ) = (?+It-$1) 1-2s
80
Hence, )@ = - 2 i e i (2k+I
I(k,s,@)
I 2s-1
! I • (~ t s+k-1 (1-t)-s-k(1-t(1+ei2@)) - I d t + ~ t-s+k(1-t)s-k-1(1-t(1+ei2@)) -I dt> .
o
The
½
two
power the
latter
series
first
of
e i28
integral
E
are
resp.
ei2@m 2-s-k-m
i2@m
and after
I s+k+m
the
substitution
-e -i20
e -i2@m
E m:O
. Then,
after
the
integrands
substituting
into
x
:= 2t,
dx
I - F(s+k+1+m,s+k+m;s+k+1+m;½) s+k+m
e
E m=O
e -i2@
by e x p a n d i n g
I s+k-1+m -s-k-l-m ~ x (I- 2) 0
m=O
=
computed
equals
ei28m2_s_k_m
I m=O =
integrals
x
:= 2(I-t)
x s-k- 1+m
the
second
) - s + k - I -m d x
(I-
one
equals
• 2-s+k-m
0 oo
=
-e
-i28
e-i2@m
Z
m=O
Inserting proof
of
these
terms
the assertion
The proofs
the
terms
2.3.3 lizer
into
the a s s e r t i o n
< 8 < ~ , observe now
of
the
Notation. Z(R) of an
the
for
that
follows
of both
I s-k+m
If
2v
elliptic
formula
for
O < 8 < ~ . To p r o v e I(k,s,@)
: e
2~ik
I(k,s,@)
yields
this
for
also
I(-k,s,~-8)-
For
the
@ - 2
by continuity.
the notes
series
last
require
~ (e m:O
element
an a p p r o p r i a t e
-i8(2m+I) s-k+m
(V 6 IN,
O
~ ~ 2)
e
rearrangement
i@(2m+1)
)
s+k+m
is t h e
"
order
R 6 F , the u n i q u e
of
of t h e
R o 6 Z(R)
centrawhich
61
COS is c o n j u g a t e
tive
elliptic
Obviously
~ ~
-sin
sin ~
cos
to
element
all
in
corresponding
the e l e m e n t s
elliptic e ~ m e n t s
~ 1
whenever
to
of the R
SL(2,IR) , is c a l l e d
primi-
R .
F-conjugacy
is.
the
There
class
{Ro} F
is a o n e - t o - o n e
primitive
correspondence
O
between and
the
the
F-equivalence
F-conjugacy
elliptic
element
classes
classes
R 6 F
of e l l i p t i c
of p r i m i t i v e
has
a unique
fixed
elliptic
points
elements.
representation
j 6
element
F-conjugacy
2.3.4
classes
Proposition.
on
F ,
Y
I = s(1-s).
that
a fundamental
Then
the
X
domain
with
elements
R
there of
is a m u l t i p l i e r of
and
exist
only
finitely
F •
system
F , s 6 ~ , Re
of w e i a h t
s > m a x ( l , Ikl),
½
X trx(M) M6F M elliptic
jM(Z) H(z,Mz) hs(O(z,Mz))
satisfies:
Gklell
F 6 LI(y,~)
f G (z) de(z) y klell
,
and
I - 2s-I
X {R} F
[ tr
X (R)
ie i2k@ 2sin9
0<8<~
1
v(R)X -1 ( e i 8 ( 2 1 + I )
(R) 2
The
sum
"
Z
..."
i:O
extends
over
all
~ (s+k+l, ~--~)
-i9(21+I)
F-conjugacy
classes
O<8<~ of w h i c h
are
conjugate
in
SL(2,1R)
,s-k+l,
- e
{R} r elements
Ro
some
function
U {~}, ~lell(Z):=
Gklell: ]H - - ~
to
(2v : o r d ( R o ) ) ,
of e l l i p t i c
Assume
every
O
corresponding
{I ..... ~-I,m+I ..... 2 v - I }
many
2k
elliptic
IH
As
R : Rj
-
the p r i m i t i v e
in
to a r o t a t i o n
{R} F
the
62
0 cos sin
|
@
cos@
bitrary ing
\ !
-sin@
representative
solely
on
By C o r o l l a r y {R} F
with
some
of
8 6 ]O,~[
{R} F , ~(R)
{R} F ) of the p r i m i t i v e
1.3.8,
tr X(R)
also
. Whenever
denotes element
depends
R
half R°
on the
is an ar-
the o r d e r
(depend-
corresponding F-conjugacy
to
class
only.
Proof.
As
are not
elliptic
~lell(Z)
x(M) j M = x ( - M ) j _ M
=
fixed
Z
points
Z
(M £ F), we of
have
for all
z 6 IH
which
F :
tr X(S-IRS) JS_IRs (z) H(z, S-1RSz)
ks(~(z, S-IRSz))
•
{R} r S6Z (Rl\r 0<8<. The
inner
sum runs
through
right
cosets
ment
R 6 {R} F . As
1.3.4,
of the
(1.3.10)
X(S-IRs) J
and
a complete
system
centralizer
Z (R)
in the p r o o f
of Prop.
Remark
that
2.2.4
of r e p r e s e n t a t i v e s
of an a r b i t r a r i l y 2.2.5,
of the
selected
it f o l l o w s
I (z) H(z,S-IRSz) = x(S) -I x(R) x(S) JR(SZ) H(Sz,RSz), S-RS
from
eleDef.
z61H
.
Consequently,
Gklell(Z)
whenever Now
fix
:
Z tr X (R) Z {R} F S6Z(R)\F
z 6 IH R . The
is not
an e l l i p t i c
fixed
k
(o(Sz s
RSz)) '
point.
set FR
obviously
" ) H(Sz,RSz) JR(SZ
:=
is a f u n d a m e n t a l
k___2 SF S6Z (Rk F
domain
of
Z(R)
,
and
S r lJR(SZ) H(Sz,RSz)ks(O(Sz,RSz)) 1 d ~ ( z ) = S Iks(O(z,Rz)) I d~(z) F S6Z (R)\F FR The
last
integral
exists.
.
R.
63
P r o o f of this claim. order
2~
Let
corresponding
~(RIFRu N uRmFR ) = O invariant,
for
R° to
be the p r i m i t i v e R .
elliptic element ~-I 1 ]]q = U R~FRv ' i=O
Then we have
1 # m . Moreover,
as
~(
, )
of
is a p o i n t - p a i r
we have
Iks(~(z,Rz))l d~(z)
=
~
FR
Iks(O(z,Rz)) [ de(z)
R~F R
Consequently,
fl ks(~(z,Rz))l
FR In the last
~8 = (
R
integral
cos
8
-sin
e
sin
8
cos
8
I
de(z)
= ~ ~
can be r e p l a c e d
which
ensues
from the p o i n t - p a i r
a := cos
8 , B := sin
o(z,%Sz)
invariance
to
property
for a b b r e v i a t i o n ,
t(x)
R
in
of
, ).
Putting
we have
x = Re z
y = Im z ,
.
we get:
]~ []~S((~(z,Rz)) ] do~(z)
2 7
7 1
V y=O t=B(y+~)
I =~ ~B t 2B
tks(t2
ks
dt dy
¢@2 .
~
i
@ 2 dy dt
I
Y=Yl
y/
( ~t) 2_i - (y-~-6) t 2~
=: J(t)
where
This
SL(2,IR).
o(
'
:= ~I B ( 1 + x 2 + y 2)
.
by the r o t a t i o n
is c o n j u g a t e
= ~2(1+x2+y2)2+4s2y2 4y2
By the s u b s t i t u t i o n I
8
Iks(O(z,Rz)) I de(z)
t Yl = 2B
/ (~)2-1"
'
Y2 = 2--~ t + /(~)2_1"
"
64
By m e a n s
of some e l e m e n t a r y
for all
t > 2B • Moreover,
hS t2 a2 I <) -~ -
-vZ 2
substitutions
~ ~ x %7 o
dt=
one c o m p u t e s
(I-~2x)
hs{
)
(substitution
_
I
~
~ xRe
s -
3
(i_ 2x)-~
r(s+k) F(s-k) 4~F(2s)
~Ao This
integral
gularityfor elements,
The
exists,
as
F(s+k,s-k;2s;x)
integral
Gklell(Z)
it is o b v i o u s
] Gklell(Z) F
de(z)
=
do(z)
4 2 ) t2+4a
i dx.
S jR(z) FR
sin-
F-c~njugacy classes of elliptic 6 LI(~,~)
is d e t e r m i n e d
tr x
x -
has only a l o g a r i t h m i c
Gklel I I
that
J(t) =
dx
F(s+k,s-k;2s;x)
x / I. There exist only finitely many hence
that
.
as follows:
H(z,Rz)
hs(O(z,Rz))
dc0(z)
0<9<~
Fix ment
R
and let R°
2m
denote
corresponding
S JR (z) H(z,Rz)
the order
to
of the p r i m i t i v e
elliptic
ele-
R . We have
hs(~(z,Rz))
I d~0(z) : ~ ~
jR(z)
H(z,Rz)
2.2.4
and
hs(d(z,Rz))d~J(z),
FR
as one e a s i l y integral FR
verifies
on the
is r e p l a c e d
NOW s e l e c t For all
by m e a n s
left-hand by
of R e m a r k
side of this
equation
R~9 , 1 = I,...,m-I
A 6 SL(2,IR)
satisfying
(1.3.10)
remains
that the
invariant
.
ARA
-I
= %8 , O < e < ~ .
z 6 I]{,
~(z,Rz)
= o(Az,%sAz)
, ~2k (A ,A -1 )
jR(Z)
H(z,Rz)
: ]%8 (Az) H(Az,%sAz) ~2k(A-1%Q 'A) d2k (A-I'%8)
when
65 as a consequence of Remark 2.2.4,
(1.3.10), Definition
1.3.1 and
(1.3.3). From (].3.3),
(].3.4) and (1.3.8)
we conclude that
~2k ( A ' A - I )
O 2 k (A-I ~8 A ' A - I )
: ~2k (A-I 98 ,A) O2k (A-I '%0 )
I .
O2k(A -I ,%8 )
Thus, jR(Z)
H(z,Rz) hs(g(z,Rz))
* (z) H(z,~Sz) ks(a(z,%Sz)) de(z). do(z) : ~I ~ 398
FR With
~ = cos 8 , B = sin O , we have 2k "
(z)
-
1%8
(Bz+cO
[ 8z+c( [ 2k '
H(z,~0z)
:
ink (Bz+s) 2k e
I~z+~
12k
(B(1+x2+y 2)-2iay)2k (~2 (1+x2+y2)2+4 2y2)k
~2(1+x2+y2)2+4~2y2 a(z'%Sz) = " 2 4y Substituting again
I
~
,
x = Re
t(x) : ~ ( 1 + x 2 + y 2)
j%8 (z) H(z,%Sz)
hs(O(z,%Sz))
z
,
y = Im
z
we find
do(z)
co
= 2eink 1 7 S I (t-2i~)2k k {t2+4~2~ q y--O t=~(y+~) (t2+4s2) k s \ - - ~ /
dt dy.
2By 7( )2 ] (y
t
2"
According to the results already obtained Fubini's theorem may be applied,
and the last term equals
66
i.k e
~ ~
7 (t-2ia)2k 2~
I 4V~
(t2+4a2) k hs
i2k@
The last e q u a t i o n This
proves
In the tives
_l
(aV~ + ivY)
follows
3
e-i8(21+1)
-
I to L e m m a
2.3.2.
be a c o m p l e t e
system
,,s-k+l, ~--L---;
"
the p r o p o s i t i o n .
following of the
let
RI,...,R Q
F-conjugacy
classes
(~j £ IN, ~3'
By C o r o l l a r y
1.3.9
these
4 2 ) t2+4a
2k (1-S2x) 2 .xS-~ F(S+k,s-k;2s;x)dx
ei@ (21+I) ~---g--~ , ,s+k+l,
from Note
2~ I,...,2~p
With
x =
O
E " 2~2sin 8 I=0
ie
(substitution
dt
I
F (s+k) F (s-k) F(2s)
I - 2s-I
(~)
b),
notations
~
2)
of p r i m i t i v e
denote
X(RT)
the a s s e r t i o n
elliptic
the o r d e r s
= X(Rj) m
of r e p r e s e n t a -
of
elements
of
F;
RI,...,R p
(j = I ..... p, m = I .... ,mj).
of Prop.
2.3.4 may be stated
as
i2~km FS Gklell(Z)
de(z)
_
I 2s-I
p
vj -
E j:l
E m=1
vj
I
t[tr X (Rj) m
ie 2 sin __~m 3
• I 2
~
3
-
Z i=O
i ~m ~j (21+1)
I (
e
• ~' X(Rj) ~3 = x(Rj 3) = X(-I)
Since
~m ,s+k+l, -1--(21+I)~ 9,'s-k+l, --<))] ~ (----~--) - e 3 3 ,
= e
-2,ik
id V ,
2~i v(k+~jp has
d
eigenvalues
ajp 6 {O ..... ~j-1} Thus
for all
of the form , p = I ..... d,
e
the e n d o m o r p h i s m
)
]
with
j = I ..... p .
j 6 {I ..... p} , 1 6 { O , . . . , ~ j - 1 }
we have
X(Rj)
67
i2~km ~j
vj - I Z tr X (Rj) m m=1
=
d
Vj-I
I p=l
I m=1
- i ~ m (21+1)
ie 2 sin ~m V. 3
e
3
i 2+ nIm ()a jv p. e
]
i2~m 1-e
d = E p=1
Let
v.-1
~ lim E t ~ I n=O
ajp(1)
3 E q=O
]
i2~m
v.-1
_ 3 t q+nv] I m =I
6 {O,...,mj-1}
- - -V.( a 4 ]
e
be the r e s i d u e
JP
+l-q-n~ 4)
of
J
l + a j p m o d vj •
Then
the last t e r m e q u a l s
d E p=1
lim t2 1
1 < 1-t 3
tajp(l )
~.-I ~. - 3[ 3 q=O
> tq
½d(~j-1 ) -
=
dX p=1
a. (I) 3P
,
since ] I m=1
1 e
)
J
a
~ -I 3 )m E tr X (Rj m=1
if
i2~km ~. ie 3 ~m 2 sin-3
e
for
mod ]P
~
i~m(21+1) v. ]
being
considerations
formula
q ~ l+a
~. 3
otherwise
that
6 {0 ..... ~j-1}
From these cation
,
=
,
One also computes
~jp(1)
I
m+qn ill J~
- -!d(~j-1)
the r e s i d u e
follows
([MOS],
p.
of
+
l-ajp
by an a p p l i c a t i o n 14):
d ~jp(1) E p=1
m o d 9j
of the m u l t i p l i -
68 2.3.5
Corollary~
For the holomorphic
Sell: e \ ]-~, Ikl ] ~ P -'-'ell(S) ::
U
j:1
~ ,
d(1-~)s [ ~j J
(F(s-k)F(s+k))
l:0
If
@.(1)3 ::
j
d E ~. (i), ~j(1) p:1 DP
Re s > max(l, Ikl), i = s(1-s)
f Gklell(Z)
= -2s-I j=1
:=
j
d Z ~ (i) p:1 3P
holds:
, then
de(z)
~[~) • i=O 3
(~j (1)-½d(vj-1)) 3
+ % ~ (s +~k)+ (l ,~ j J
2s-I
-½d (I-~) J
I ---a. (i) - -I ~~ (i) F( 3s-k+l'%)j ---~) F(~)s+k+1%)J J )]
%).-I 3( U
with
function
j:1
(1)-}d(%)j-1)) )]
~
3
+
3
i --~
%) ,--I
%) ]
i:O
<sj ( i ) ~ [sk+l ~ j
s+k+l>]
+ ~j(1)9[--~) j
I -ell (s) 2s-I ~ell(S)
(Note: log F
The powers of the gamma function which is holomorphic
on
{ ~ ]O,~]
are defined via the logarithm and real-valued
on
]O,~[
.)
69
2,4
THE C O N T R I B U T I O N
The notations
fixed in Convention
For every parabolic exist unique termined
OF THE PARABOLIC
element
ELEMENTS
1.5.5 are also used in this section.
M £ r ~ {I,-I}
with
j 6 {I ..... T} , n C ~[ \ {0}
uniquely
modulo
F j , such that
tr M = +2
and an element
there
S 6 F
de-
M = S-ITns] " By Corollaries
].3.8 and 1.3.9, d i2~nBjp ~ e p=1
tr X (M) : tr X (Tj)n =
Noting
½
again that
E
M6F
X(-M)j_ M = x(M)]M
tr x(M) jM(Z) H(z,Mz) (h
(M 6 F)
(0(z,Mz))-
we get
ka(@(z,Mz)) )
s
M parabolic T :
z j=l
d
z S6F c \ r ~j
i2~n~jp
z z e p=l n£Z--{O}
]
S-
iTns(Z) H(z s - I T n3s z )
(h s(o(z,S-IT~S z))-h a(o(z,S-]T~sz)
where
S
runs through
right cosets
of the stabilizer
The contribution resolvent
E
that
is
system of representatives
group
of the parabolic
kernel,
f [½
]:
a complete
(z 6 IH),
)
r'
cj
terms
with respect
to
of the F .
to the trace of the iterated
the integral
trx(M) jM(Z) H(z,Mz) • (hs(0(z,Mz))-ha(J(z,Mz)) >
M6r M parabolic m, -
E
g
j:l p:1 (cf. Theorem
2.1.2),
-~
(s-½)2+t 2
I (a_½)2+t 2 1 • ~I Ejp(Z, ½+it) I2 dt] de(z)
will now be computed
in two steps.
In the first
Z0 step we consider the "regular part"
T d i2~nB~pj E Z 5E e " (z) H(z,S-ITjsz)n F j=1 S6F~jkF p=m +13 n6Z\{O} /S-ITns3 (ks(°(z'S-ITnsz))-ha(°(z'S-IT3sz)))3 de(z)
and in the second step the "singular part" m.
E Z F j=1 p=1
E ~ [S6F~3\F
Z n6Z\{O}
(z) Js-ITOs 3
H (z,S-ITnsz) 3
(ks(O(z,S-IT~.Sz))-ka((5(z,S-IT~Sz)))
41 ! < (s_½~2+t 2
12+t2 > • IEjp(Z,½+it)12 dtl din(z) . (a-½)
In the second step it will turn out that the terms arising from Gkl-Gk~
can only be integrated together with the contribution of the
Eisenstein series; the individual integrals do not exist.
Step I
2.4.1 F
Lemma.
Let
X
be a mUltiplier system of weight
a fundamental domain of
2k
on
F ,
F , s 6 ¢, Re s > max(l, Ikl), I = s(1-s).
Then the function Gklpar,reg: ~4 Gklpar,reg(Z)
~¢ , :=
T d E E I E e i2~nBjp ]S_ITns(Z)H(z,S-I~j Sz) ks(O(z,S-IT3sz)) j=1 S 6 F ~ F p=mj+1 n6Z\{O} 3
71
satisfies:
Gklpar,regl~
6 L 1 (F,~),
; Gklpar,reg(Z) F
and
dc0(z) = ~
[- (dT-T~) log 2 - log
T d sin H H j=l p = m .+I 3
~Bjp
j=1 with
Bj :=
Proof.
Fix
Similarly
d I p=mj+l
Bjp
•
j 6 {I ..... T}
with
as in the proofs
from D e f i n i t i o n
~s ~
1.3.4,
mj
< d
of Propositions
(1.3.10)
and R e m a r k
and
p 6 {mj+1 ..... d}.
2.2.5 2.2.4
j ~s~Z~ ~z s ~z, = ~s, ~ , 3
and 2.3.4
it follows
that
~n~SZ~ ~ z ~sz,
(s £ F).
3
Moreover,
x(s-IT~s) = x(sl-lx(T~)x(s) by C o r o l l a r y
1.3.8,
so that
J -I (z) H(z S-ITnsz) S Tns ' 3 3
= j n(SZ) T 3
H(Sz,T3Sz) (S 6 F, n 6 Z ~ {O},
Now ~T: = ~ 3 S6F
is a f u n d a m e n t a l
domain
sF
~j of
~F
F~j , and
z 6 IH).
72 i2~n~jp Z SCF
{j
Y e \F nE~\{O}
JS_ITns (z)H (z,S-1 ~jSz)hs (~(z,S-I~jsz) ) J
i2~n~jp < ~T f
.
n6~[~{O}Y e
. /Tn(z) H(z,TTz)hs(O(z,T~z))
3 Choose
d0~(z)
c]c0(z)
] A 6 SL(2,IR)
I
I)
such that
= U = AT A -I
I
]
Remark 2.2.4, (1.3.10), Definition 1.3.1,
(1.3.3),
(1.3.4) and (1.3.9)
imply that
3 moreover, (n C ~ ~ {O}, z C IH).
Jun (Az) : I
Thus the last integral equals i2~n8jp f AFT. ]
E e nEZ~{O]
H(z,z+n) h (a(z,z+n)) s
de(z)
i2~n~jp e [O,I[×]O,~[
H(z,z+n) hs(a(z,z+n))
Observe that the fundamental domain be replaced by the fundamental domain
AFT3
of the group
[O,1[ x ]O,~[
AF{jA -I
The last term equals
f O
E nC~{O}
ei2~n~JP ¢2i-~-~k h ( 4 _ ~ \2iy-nJ s\ 4y2
The existence of this integral has to be shown.
]
may
of the same group
by virtue of the invariance of the integrand with respect to (m C Z).
de(z)
nC~{O}
d~ y2
~U m
73 For
N 6 ~, y > O
we apply Abel's partial summation and get:
N i2nnBj {2iy+nhk {4~n 2] E e P \2iy-n/ hs\ 4y2 / n=1
N 1-e
{2iy+nhkh{4~h
[nE=1 \2iy-n/
i2nBj
~2iy+n+1hk (~+i)2> s\ 4y2 /- \ ~ ] hs(4y2+4y2
Pl
+ k {4y2+lh
(4y2+(N+1)2
2 ~ {2i-~k h { 4 ~ 12iy+n+1~k 4y2+(n+1 1_ei2~Bjp, In=El \2iy-n/ s\ 4y2 /- \2~n--~/ hs( 4y2 )2>
{4y2+I] +
ks\ 4--~--] + CI hRe s \ 4y2 /
the constant
C I depending solely on
grable over
]O,~[ with respect to
tegral is ensured. For every
n 6 IN
2iy+n~k <2iy-n/ < -
I
If this function is inte-
~dy , the existence of the last inY
we have
h {4~t~hs\ 4y2 /
{2iy+n+11)k hs(4y2+(n+I)2) \2iy-n4y2
{2iy+n~ k /2iy+n+1) k \2i--~/ - \2iy-n-1
+
s,k .
{ 4y2+n2 2>s - \4y2+(n+1)
4y2 hRe s + { \4y2+ (n+1) 2/
k f s< 4y2 ]
s\ 4y2 ] F (s+k) F(s-k) 4~ F (2s)
F<s+k,s-k;2s;
F<s+k,s-k;2s; and
4y2 4y2+n2/ 4y2 4y2+(n+1)2/ '
74 _ (2iy+n+1~ k / 4 y n 2 + 2 ~S ,2iy-n-1/ I + II - \4y2+ (n+l) 2j
/ ~ k k2iy-n/
1 _
=
(I+ 2l +i y~-ziy+n~ ,k n-lJ
= O(~) ,
Moreover,
as
n
I + 11
2n+I 4y2+(n+1
- (I
, ~ , uniformly in
y > O .
a simple application of the mean value theorem yields for
y>O: , IFIs+k's-k;2s;
- \4y2+n 2
................
4y 2 4y2+n2) - F(s+k's-k;2s;
4y2+ (n+1)
sup q6 [n, n+1
4y 2 2n+II ........ LS+kl Is-kl
(4y2+n 2) (4y2+ (n+1) 2)
I2s I
4y2+4y2 ~(n+ii2]
F (s+k, s-k; 2s; t)
sup
t=
4y2 4y2+q2
F( +k+1,s-k+1;2s+l;
q6[n,n+1]
4y2+~ / ([MOS], p. 41).
The function [0,1]
(cf.
t i , (l-t) F(s+k+1,s-k+1;2s+1;t)
is continuous on
[MOS], p. 44). Hence there exists a constant
pending solely on
q6[n,n+1] <-C 2 (1 -
s,k,
such that
4nF(2s+1) 4y2 ~-I 4y2+n2 4y2+n2 / = C2 n2
From these estimations we get:
4y2+ 2/
C2 > O
de-
75
O
/2iy+nhk k { 4 ~ h -
C2i~+n+1> k
\2iy-n/
\2iy-n-1
S\
4y2 /
I - O
s\
4y 2 /
4y2
2n+I ~ (4y2)Re s " n2 SO (4y2+(n+1)2) I +Res dy
y
1 1 3 I 1 J IxS-~(1-x)-~ F(s+k)F(s-k) F(s+k,s-k;2s;x) = n O 4.F(2s) 2n+I n -(n+1)
+ 2C2
2
as
= o (~I2) n
n
y
dx .O(1)
7 t2Res dt O (t2+I) I +Re s
....., ~
,
since both the last integrals exist.
The monotone convergence theorem now yields the existence of the integral
0
( 2 i y + n f k /4y2+n2h
s\ 4y2 /I +CI k ~ s
/2i~f
k {4y2+(n+1)2)l
\ 4y 2 /] y
Moreover, i2.nSjp ~ e O n=1
(2i_~y~hk k C 4 ~ h dy \2iy-n/ s\ 4y2 / Y 2
exists and may be computed by termwise integration (dominated convergence theorem).
Analogous assertions are valid for
76 co
n:-co
Hence,
O~
\2i--~-~n/
s\
4y2
instead of
/
n=1
the existence of
n6~[\{O}X e
P \2iy-n/
hs\
4y 2 /I y2
is shown.
As
j 6 {I ..... T}, p 6 {mj+1 ..... d}
bitrarily,
the first assertion of the lemma is proved:
Gklpar'reg
Now
have been chosen ar-
F £ LI(F,~)
~ Gklpar,reg
(z) dm(z)
is to be computed.
F
Again fix
j 6 {I ..... T}
with
m 3• < d
and
p 6 {mj+],...,d}
.
With the results achieved up to here, we have
X X ei2~nBjP'Js -IT~,SH(z's-IT~ sz) hs(°(z'S-ITnsz3 F S6F j~F n£7/--{0} 3 i2nnBjp =
]
Z
e
- i :
Tn 3
i2 nBjp
[ f X e O O n£~\{O}
X e n6~[\{O]
(z) H{z,T~z) ks(~(z,T~z)) d~{z)
j
FT. n6~[~{O} 3
After the substitution
k {4_ td \2iy-n/
i2~nBjp ~ Of
series has the form
) d0~ z)
{2i~n~k \2iy-n]
s\
dy
4Y 2 / dz -~ y
/4y2+n2] d_X ks\--~y2 ] Y 2
y : ½1nlt ½ (l-t) -~~
the
n-th term of this
77 ei2~nBjp
1
1
3
Inl ~ (V~ + i1~TC~-t)-2ksgnn (1-t)-½t s-Z 0 F(s+k)F(s-k) F(s+k,s-k;2s;t) dt 4~F(2s)
I i2~n6 jp I< = 2s-------1 e 21hi
i<~ ) ) I + ~ (s-k)-9(s+k) sgn n ,
by Lemma 2.3.2, Note 2. Consequently,
z
n6Z\{O}
-
e
1
2s-I
i2~nBjp ~ { ~ h k S \2iy-n] O
-
½log
1-e
Jp
k { 4 ~ _ ~ h dy 2 s\ 4y2 ] Y
-
½log
1-e
i2~6j + ~ i (9(s+k)-~(s-k))
I
2s-I I-log 2 -
-i2~6j P)-iog(1-e
P))]
log sin ,Bjp + < ~ (s+k)-~(s-k)>(½-~jp>]
Summing up these terms for j =I,...,7
and
p = m.+1,...,d 3
we obtain
the assertion:
f Gklpar,reg(Z) de(z) F T d U U sin j=1 p:m.+] ~Bjpj 3 T d (½(aT-T*) - X X j=l p=m .+1 3
-- 2s-------~] [-(dT-T~)log 2 - log
+ (~(s+k)-~(s-k))
°jp)]
78
StepII
2.4.2
Notation.
Let
For a b b r e v i a t i o n
ks,a(O)
Ikl-s,
Ikl-a ~ IN
o
we set
:= k s ( O ) - k a ( O )
f(t;s,a)
It r e m a i n s
s,a 6 ¢, Re s , Re a > I ,
:=
,
I (s-½)2+t 2
~ > I ,
and
I (a-½)2+t 2 '
t 6 IR
to compute:
(2.4.1) m.
f [ z F j 1 p=l
z z S6F~jkFn6Z"-{O}
" (z) H(z,s -I IS-1T~jS ]T~z)
17
4n -~ f(t;s,a)
2.4.3
Lemma.
Let
(o (z, S-IT~sz) ) kS, a
IEjp(Z,½+it) I2 dt
s,a 6 ~ , Re s, Re a > m a x ( 1 , 1 k I) .
]d0J(z) The
function
m.
Z ~
;
Z_ I j I p=l
Z S6F
I j\r n6Z~{o}
4n
belongs
to
LI(F,~).
of f u n d a m e n t a l
Proof. Lemma
f(t;s,a)
integral
z)
(z,½+it) I2 dt
(2.4.1)
z))
,
is i n d e p e n d e n t
z 6F
,
of the c h o i c e
domain.
Remember 2.4.1
The
J I n (z) H(z,S -I s-TjS
(2.1.4),
and let
Theorem
I = s(1-s),
2.1.2,
Propositions
B = a(1-a).
The
2.2.5
functions
and 2.3.4,
79
E n>-O
ln_l
in_]]
F
Gklhyp - Gk~hyp) F
'
and
Gklell - Gkpell) F
Gklpar,reg - Gk]]par,reg) F
are elements of
LI(F,~), the integrals of these functions do not de-
pend on the choice of the fundamental domain have for
~-almost every
F
of
r . By (2.1.4) we
z 6 IH:
m.
E
E
E
j I p=1
E
S6F~j\F n6Z~{O} I
4~
X n_~O
<~nI_
t
~ f(t;s,a) -~
JS_ITns (z) H(z,S-1T3Sz) 3
l~s,a((~(z,S-1T3Sz))
IEjp(Z,~+it)1 2 dt]
I ) Ifn(Z) 12
in- p
i<
+ d • ~-~ ~(s+k)+~(s-k)-~(a+k)-~(a-k)
)
-(Gk~hyp(Z )- ~ h y p (z)>- (~lell(Z) - ~pell(Z) ) - • []
This implies the assertion.
By virtue of this lemma we may again assume without loss of generality that
F
is a fundamental domain according to Remark 1.2.2 with
~i,...,~i , AI,...,A T , yl,-..,y T
2.4.4
Lemma.
For every
fixed.
j 6 {1 .... ,T}
let
~. denote a complete 3 system of representatives of the right cosets of F~j with respect to F
80
which
contains
the
identity
I
Re s , Re a > max(l, Ik [ ) . T h e n
as an element.
Further
the f o l l o w i n g
integrals
let
s,a 6 C ,
exist
(abso-
lutely):
(2.4.2) m.
3
]
x
x
F j=1 p:1
3
I 4~
~J f(t;s,a)
lEjp(z ,½+it) f2 dt]
de(z)
,
(2.4.3)
X
X
F sCWj\{I}
]Tn(SZ)
n6~{O}
H(Sz,TTSz)
k s,a(O(Sz,TTSz))
j = 1,...,m;
The
integral
Proof.
Fix
(2.4.1)
is equal
j 6 {I .... ,m]
f x
x
~
F S6Wj\{ I} n 6 ~ { O }
S
Z
We
find
X
f
s6w \{I} n6~\{o}
SF
S6W.~{I} 3
A.SF 3
that
n6Z\{O}
~(IR
to the sum of these
with
mj
h I
and
terms.
7
hs
(@(z TTz)
,a
de(z)
'
ks
,a\
n
~
4y2 )
J
Y
AjSF)
= O
by m e a n s
3
argument:
Every
z 6 IH
[O,1[
× ]yj,~[
with c AjF
3
sz) ~(sz,T~sz ks,a(~(sz,T Sz)) d~(z)
s
× ]yj,~[
p = 1,...,m
SEW \ { I } following
,
p 6 {1,...,mj}.
"]
3
:
de(z)
J
Im z > yj by some
is m a p p e d U1
with
into the s t r i p 1 6 ~ .
The set
of the
81 A F U ~ ] SEW.\{I} ] AJ FA~13
generated
A.F ]
at m o s t
The
integrand
every
A.SF ] by
-I
in the
U , and
e-measure
the L e b e s g u e
x+iy 6
domain
of the
~
u
measure
A.SF}
subgroup
A~SFj
s~wj-{1}
of
intersects
zero.
last t e r m does not d e p e n d
on
x ;
for a l m o s t
of the set
is e q u a l
to
I .
3
S6W.
D
Hence,
and
in a set of
y > O
{x 6 IR:
is a f u n d a m e n t a l
the l a s t t e r m d o e s not e x c e e d
YJ k {4y2+n2~ k {4--~%1 neff--{ O } % (t S\--~y2 ] + a\ 4y2 ],
E n6~{O}
the c o n s t a n t s of c o u r s e ) .
Yj 4Pes 2Re s-2 f ......Y O (4y 2 +n2) R e s
C(s)
C(s),
The
E
C(a)
> O
I <
nE~\{O} ~
4Res C(s)
~j A R e a y 2 R e a -2 dy O
o n l y on
is d o m i n a t e d
1 2 Re s-1
2 Y
dy + C(a
depending
last e x p r e s s i o n
d_z
s
resp.
a
(and
2 Res-1 Yj + C(a)
of the i n t e g r a l s
As a l r e a d y m e n t i o n e d S £
4Re a
I 2 Re a-1
2 Rea-1) Yj
in the p r o o f of L e m m a
oo
is proved.
2.4.1,
we have
for all
W. : D
JS_ITns 3 Therefore the
(2.4.3)
k,
by
<
T h u s the e x i s t e n c e
,
(4y2+n2)Re a
z) H ( z , S - ] T ~ 'Sz)
the e x i s t e n c e
sum of the t e r m s
follow
from Lemma
=
JTn (Sz) H ( S z , T ~ S z ) ]
of
(2.4.2)
(2.4.2)
and
2.4.3.
a n d the c o i n c i d e n c e
(2.4.3)
(n 6 Z \ { O } ,
of
z 6 ~) .
(2.4.1)
with
(j = I ..... T; p : I, .... mj) D
82
2.4.5
Notation FY
and Remark.
:=
nT 9=i
F n
By the dominated
We put for
A- 1 ( ~
× ]O,Y])
J
convergence
(2.4.4)
Y > O :
theorem,
(2.4.2)
to
m.
lim Y~
S X= X FY j I p=1
I n6Z~{O}
j n(Z) Tj
H(z,T
Lemma.
For every
Re s > max(1,1k I)
Y > O
Z)
(O(z,T3z)) ks'a
I ? f(t;s,a) 4 n -~
2.4.6
is equal
and every
IEjp (Z
½+it)1 2 dt]d~(z)
s 6 ¢
with
the integral
m.
T 3 Z Z Z j=1 p=1 n6Z~{O}
~ FY
n ks(O(z,Tjz))
de(z)
exists.
Proof.
We have ]" FY
ks (o(z,T3z))
d~0 (z) =
S A. F Y 3
dv
< OS O; according
2.4.7
s\
4y2
Corollary.
there exists
1 O(--~)n
/I dx "-~y =
to the proof of Lemma
For every
I~s(O(z'unz)) I de(z)
as
n
2.4.4.
Y > O ,
s,a 6 ~ , Re s , Re a > max(1,1k I)
also m~
T 3 I S ~ Z ~ f(t;s,a) Fyjlp=l -~
Proof. Lemma
The assertion 2.4.4.
is an immediate
IEjp(Z,½+it) I2 dt de(z)
consequence
of Lemma
2.4.6 and []
83
In order
to compute
familiar
expression:
2.4.8
Lemma.
(2.4.1)
Let
is equal [
Y~lim
the integral
~ T*n~1
(2.4.1)
we transform
it into a more
s,a 6 { , Re s, Re a > max(1 , [kl) •
The integral
to
({2i~k
Y Of \\2iy-n/
(2iy-n~k~ k + \2iy+n/ )
{4~n2] ~
s,a\
4y2 ] y2
m.
J1 7
-
f E Z ~-~ f(t;s,a) FY j=1 p=1 -~
Proof.
By Notation
(2.4.1)
is the sum of
j 6 {1 .... ,T}
with
FY n6Z~{O}
mj >_ 1
and the integrals and
p 6 {I .... ,mj}-
3
E JTn(z) F n6Z'-{O} j
H(z,TTz)
ks(C~(z,TTz))
H
1
1=I l*j
Claim.
denotes
the indicator
function
The second term on the right-hand
to zero as
Y
Fix
Y > O ,
Then
by
(z) d~(z)
IAj I (z) - (IRx]O,Y])
_
(Z)
dc~(z)
The modulus
,
A11 (IRx ]O,Y])
of the set
M .
side of this equation
tends
~
Proof of the claim. dominated
(2.4.3).
A. (IR×]O,Y]) 3
I-
]M
the integral
JTn(~) H(z,T~Z) ks(O(z,T]z)) .~_~
n£Z~{O}
where
2.4.4,
]
3
: S z
-~
(2.4.4)
2.4.5 and Lemma
d~(z)
~ (z) H(z,T~.z)ks(~(z,T~.z))d~(z)
>-
s
and Remark
12 dt
IEjp(Z,½+it)
of the term under consideration
is
84
T X
Z
ks ( d ( z , T
f
1=1 n 6 ~ { O }
z))
(z) de(z)
• 1 _
~
AII(/Rx]Y,~[)
l#j T
Z X l=I n E ~ { O }
ks (a(z,unz))
[
• ] (z) de(z) hjil I (IRx]y,~[)
AjF
l#j
Since ~ Y > Yj:
complies with the conditions
~([AjF N AjAII(IRx]y,~[)]
in Remark 1.2.2 we obtain for
N [O,1[x]yj,~[)
= (0(Al1[A1Y n (]IR×]Y,~[)] R A~I([o,I[×]yj,~[)) < c0(Ail ([O,1 [×]yj,~[)
fi A~ I ([O,1[×]yj,~[))
: O ,
i # j ,
hence, the last term does not exceed T YJ z z f 1=I nCE\{O} O l#j
h { 4y2+n2~ s\ 4y2 J
dy
(Y > Yj )
~-
Y
The proof of Lemma 2.4.4 shows that this sum is finite. Now the dominated convergence
Therefore
theorem yields the claim.
(2.4.1) equals
IT n (z) H(z,T~.z) ks, a(d(z,T~.z)) 3
lim [ X m j( f X Y~ j:1 ~ n[~\{O}
• I
i
+f
z
z
Y S6Nj\{I} n 6 ~ { O }
]
-1 (z) de(z) (•x]O,y]) 3
(Sz) H(Sz,T~Sz)
hs,a(~(Sz,T~.Sz))
Tn 3
m.
f FY
XT j
I
1 7 Z3 ~-~ f(t;s,a) p:1 -~
IEjp(Z,½+it) I2 dt d~(z) ]
de(z))
85
If
Y > m a x ( y I .... ,y%) , the
may
be
multiplied
affecting
the
Lemma 2 . 4 . 4 most
all
by
value
that
of
the the
Im A . S z 3
z 6 F .
The
j-th
set
Ylim ~
(2.4.1)
is equal
_E m.3 fTf j-1
of
factor
~ yj
middle
for u
every
SF
$6~
is
integrand (Sz)
A7 t ( I R x ] O , Y ] ) 3 we h a v e s h o w n i n
as
< Y
fT.:=
the
1
integral,
3 F~j , and
term
the
S 6 ~.x{I} 3
a fundamental
without proof
and
of
~-al-
domain
of
3
to
X n6~[x{O}
n(Z)
•
H(z,T
z) h s , a ( ~ ( z , T
z))
3
3
]
-1
Aj
(z)
d~(z)
(IRx]O,Y])
m,
J 1 oo S X X ~--~ S f ( t ; s , a ) [ E j p ( Z , ½ + i t ) FY j:1 p=1 -~ ~
Similarly 2.2.4,
to the proof
(1.3.]O),
of Lem~ma 2.4.1
Definition
% n (z) H(z,T3z)
1.3.1,
we now c o n c l u d e
(1.3.3),
hs,a(O(z,T~z))
I2 dt de(z) ] .
(1.3.4)
= H(Ajz,unAjz)
from R e m a r k
and
(].3.9)
that
hs,a(O(Ajz,unAjz))
3 (n £ ~ -- {0}, Hence,
the
first
term
in the b r a c k e t s
T X m f H(z,unz) j:1 3 AjFT ' ] The
fundamental
by the
domain
fundamental
we see that
the last
T~ f X O n[~-{O} Summation
term
\2iy-n/
and i n t e g r a t i o n
h ,a(O(z,unz)) s
AjFT. 3
domain
[...]
of
equals
• IIR
]O,y] (z) de(z) ×
A F~ A71 j 7
:
[-I,U]
may b e
replaced
.J
[O,1[
x ]0,~[
is equal
hs
z £ IH).
,a\
Noting
that
to
4y2
] Y
2
may be i n t e r c h a n g e d
because
T* =
Z m j:1 3
86
Y
(Y > 0 ,a\
O
4y2
mentioned
This
Our
/I y
fixed),
proves
next
the
task
lemma.
is
n~l
to
evaluate
Of \ \ 2 i y - n /
the
+
integral
\2iy+n/
/
"
s\
/ Y2
4y2
Re
!
the
already
above.
(2.4.5)
By
as
n
substitution
transformed
y =
'
s > max(~,
Ik I)
--!
½ n x ~ (l-x)
2 , n >_ I ,
this
expression
into Y
I n=1
3
n
(Vx -
ig]-:x) 2 k
+
(gx +
i ~ - C x ) 2k
I -x)
--l
--
2X
1
~ ks(
Oo
.y =:
I (x)
z
i
z
n m=n
oo
Ym
z
f
m=1
Yn
-
4y 2 4y2+n2
(n
I (x)dx
f
,
the
double
series
converging
Ym+1
m
I (x) d x
-
I
z
n
n=1
Ym+l
1 =
f
I(x)
E
n6]N n_<2y (~-1)½ According
Z n6IN n_
to
[HW],
1 - y n the
>
I),
Ym
CO
n=1
) dx J
where
--
.
0 k
=
is
p.
+ log
347,
T
Theorem
+ O ( T -I
Euler-Mascheroni
I n
dx
422,
as
.
we
T
constant.
have
~ ~
,
for
T > 0 :
absolutely
,
87
Thus
l
I
I
n = Y + log
2 + log
Y +
--i
½log(-I)
+ O(y-1(1-x)
I
2x~)
n6IN
(~-1 )½
n-<2Y
for
Y
-~'
, x 6
~
I
]0,1
~ 4y2+I
]
As F(s+k)F(s-k) F(2s) ([MOS],
p.
44),
F(s+k,s-k;2s;x)
it f o l l o w s
= -log(l-x)
x fl I
1 4y2+I
I
f
for
that
4y2+ 1
1-
+ O(I)
S
I (x)" O(Y -1 (1-x)-½x½)dx = O(Y -I
-log(l-x) • (1-x)-lx Re s-1 dx)
0
O I
I
+ o0<-,
s4¥2+, <,_x>-i
1 <,4 :
>
for
Y
, ~
•
0
I
Moreover,
the
integral
f O
II(x)
log (1 _ I) I dx
exists,
so t h a t
4y2+I f O
I(x)
I
z n6IN
~
dx
n N 2 Y ( ~ - 1 )½ I
I
S I(x) (T+log O
By L e m m a
2.3.2,
2 + log y + ½ 1 O g ( x - 1 ) )
Note
dx
+ o(I)
for
Y ---~ ~
2, we h a v e
I
I
f I(x) O
. (y + log
2 + log Y ) d x
- 2s-I
(Y + log
2 + log Y)
,
Re s > m a x ( i , Ikl)
We
summarize
2.4.9
Lemma.
is e q u a l
to
these
For
results
in
s 6 ~ , Re
s > max(~,Ikl)
the
integral
(2.4.5)
.
88
I
I[
2s-I (Y + log 2 + log Y) + ½ ~ O
(g~ + ig]-c~) 2k +
(V~ - igT=-~) 2k] • (l-x) -½
3
• x-~ ks(~) log(~-1) dx
2.4.10
For
Lemma.
+ o(I)
s 6 ¢ , Re s > max(½, ]kl),
as
Y
the
,~
.
following
equation
is v a l i d : I ½ f [ (Vx + igTC-x)2k + O ] <-y-
2s-1
Proof.
Both
(g~-
2 log 2 + ½~(s+k)
sides
The
computed
substitution
(1-x)-½x -~ k s ( ~ ) I o g ( 1 - I ) d x
+ ½~(s-k)
of the e q u a t i o n
{s 6 ¢: Re s > max(½, Ikl) } valued.
igT:-~) 2k]
Hence
- ~(s)
to be p r o v e d we may
cos <0 := g~
- 9(s+½)
+ 2--~-I)
are h o l o m o r p h i c
assume
transforms
that the
s
on
is real-
integral
to be
into
4 ~ cos(2k<0)" (cos <0)2s-2 O
F(s+k) F(s-k) 4~F (2s)
F(s+k,s-k;2s;cos2<0)
log tan<0 d~)
u
I - 29
This Now
~ E n=O
equation fix
F (s+k+n) F (s-k+n) F(2s+n)n!
is j u s t i f i e d
n ( IN o.
[T
d
ei2k<°(cos
by the d o m i n a t e d
<0)2s+2n-21Og tanq~ d~0"
convergence
Then
Re 2 f e i 2 k ~ ( c o s O
=Re
2 S O
Re 2
~)2s+2n-21og
tan
~O ei2k~(cos ~)2 s + 2 n - 2 - 2 ~
do
(sin O)2u
d$]~=O
theorem.
89 ~ [d(
: Re
21-2s-2n
ei~ (k-s-n+½+~)
F(1-s+k-n) F(2s+2n-l-2~) F (s+k+n-2~)
• F(-2a, l-s+k-n;s+k+n-2a;-1)
+ 21-2s-2n ei~(a+½)
F(1-s+k-n)F(2~+1) F(2-s+k-n+2~)
• F (2-2s-2n+2a, 1-s+k-n ~2-s+k-n+2~ ;-I ) ) ]~:O ,
according to
[GR], 3.892.3.
Re [-21-2s-2n
This term equals
~e i~(k-s-n)
F(1-s+k-n)F(2s+2n-1 F(s+k+n)
+21-2s-2n iei~(k-s-n)
F(1-s+k-n)F(2s+2n-1 F (s+k+n)
+21-2s-2n
F(1-s+k-n)F(2s+2n-1
L
ie i~(k-s-n) " d--d
-21-2s-2n
I"(s+k+n-2c~)
(-2~ (2s+2n-1) )
F(-2c~,l-s+k-n;s+k+n-2c~;-1)
~ 1-s+k-n I F (2-2 s-2n, I -s+k-n ; 2-s+k-n ;-I ) ]
c~=O
+ O
n : -im <~ ~ ei2k~(cos ~)2s+2n-2d~ ) 0 r(s-½+n)~(s+n) -g~ F(s+k+n)F(s-k+n)
+~
9(2s+2n-1
F(s-½+n)F(s+n) F(s+k+n) F(s-k+n)
d d~
I fI t-2~-I ~(-2a) O
The last transformation
(1_t)s+k+n_ I (1+t)s-k+n-ldt ]a=-O
is obtained by combining the first and the
fourth term of the preceding expression 3.892.3.
and taking notice of [GR],
and [MOS], p. 54. The last integral exists only if
Hence here and in the following
~d
[''']~=-O
~ < 0 .
stands for the left-hand-
90
side d e r i v a t i v e putation
of
[... ]
rules concerning
at the point
a = 0 . Furthermore
some com-
the gamma function have been used for this
transformation.
Finally we get:
(V~ + i _ _ .
2k + (V~ - i _ _ .
2k (l-x)
ks( x ) l o g ( - 1 )
dx
0
= - ½Im
F(s+k+n)F(s-k+n) F(2s+n)n!
1 n=O
~ ei2kW(cos 0
w)2s+2n-2dw
(=: 11 )
co
-½r~-½
F (s-½+n) F (s+n) r(2s+n)n,
Z n=O
~(2s+2n-1)
(=:
r 2)
+~-½ ~ r(s-½+n)r(s+n) F (2s+n)n'
n=O
d[
I
Thus
Ii, 12
and
theorem entails
11
=
I
f t -2a-I F (-2a) 0
da
- ½Im
Z3
(1_t) s+k+n-1 (1+t)s-k+n-ldt]a=_O
are to be computed.
(=:I 3 )
The d o m i n a t e d c o n v e r g e n c e
that
n/2 j ei2k~P(cos ~p)2s-2 F (s+k) F (s-k) F(s+k,s_k;2s;oos20) d~ 0
F(2s)
= - ¼ I m SI (V~+i%rf:-~) 2k (l-x)_I 2 x s--3 2 F(s+k) F(s-k) F(s+k,s-k;2s;x) 0 F(2s)
=-
½Im [2~_i <.+i(~(s+k)-~(s-k)))]
I 2S-I
2.3.2, Note 2
• ½ (~ (s-k)-~ (s+k))
For the computation ~(2s+2n-1)
by Lemma
dx
of
Z2
= ~(s-½+n)
Furthermore,
the logarithmic
the integral
representation
we observe that + ~(s+n)
+ log 2
derivative
~
([MOS], p.
14).
of the gamma function has
91 I
~(z)
= -y + S (l-t) -I (1-tz-1)dt, O
Re z > 0
([MOS], p. 16).
Hence, -i
~
F (s-½+n)
I X 2 = -i~ ~ (-y + log 2) n=O _!
_¼~ 2
_i
_¼~ 2
~
F (s-½+n)
~
r ( s - ½ + n ) r (s+n)
Z n=O
l
F (s+n)
r(2s+n)n!
F(2s+n)n!
n=O
F (s+n)
F(2s+n)n!
n
1
(=: 12, I )
_~+
I
~ (1_t) -I (1_t s 2 n)d t 0
(=: 12,2 )
I
J (1-t)-1(1-ts-1+n)dt 0
(=: 12,3)
We have: 12,1 = -In -½(-Y + log 2) I
2s-I
r(s-½)r(s) r(2s)
F(S-½,S;2S;1)
(y - log 2).
i
~ r(s-i)F(s) S (i_t)-I (F(s-½,s;2s;I) - t s-~ F(s-½,s;2s;t) 12,2 = -¼~-Y F(2s) 0 (monotone convergence
2s-I
" i S (l-t) -I 0
1-ts-~( 1 + V ~ ) I - 2 s
1 } x-1 (I-(I-x) S-~(1+x)-S-½) 2s-I O
Fix
dx
at
)
dt
theorem)
([sos], p. 38)
(x := ~ )
z £ ]-1,1[ . Then
iS x-1 (I- (l-x) S-~3 (1-zx) 0s)
dx
0 I
3
= lira b %O
S xb-1(1-(1-x)s-~-b(1-zx)-½-s)/\ dx 0 \ /
= lim b +O
(1 _ F(b) F ( s - ½ - b ) F ( s + i , b ; s - ½ ; z ) ) F (s-i)
= lim b+O
I ~-
r(b)F(s-½-b) F(s-~- ) F(s-½) - F(s+i~
(monotone convergence theorem)
~ r(s+i+n)F(b+n) I F(s-½+n) n=1
• n-T
92
Since
the
series
ever
z 6 ]-1,1[
converges
uniformly
is fixed,
with
respect
it is c o n t i n u o u s
at
to O
b 6 [0,1]
when-
and the last term
equals co
lim b+O
I
I - r (b+1) F (s-½-b) F (s-½)
: ¥ + ~ S-½)
The m o n o t o n e
+ log(l-z)
convergence I 2s-I
X2,2
-
I X s-½ n:1 I S-½
theorem
lim z~-1
S xO
(s-½+n)
zn n
Z
1-z
yields
that
I-(I-x)
(1-zx)
dx
z > -I --
S
I
--
I
2s-I
2s-I
I
2s-I
This
theorem
implies
further
that
I
2s-11 X2'3
I 2s-I
lim b + 0
~ x b-1 0
lim b+O
(I-(I-x) s-1 (1+x) -b-s)
( I _ 2-b F(b) F(s) ) ~ F~bTs[ '
(cf.
dx
[MOS],
p. 7)
= I---~-~ ( s( -IOg ) ) 22 s- Y- -1
Hence,
E2 = X2,1
+ E2,2
+ X2,3
- 2s-I
-y-
3 log 2 - ~#(s) - ~(s+½) +
.
Finally,
X3
~ ¼ -½ = d [
I ~ F(s-½+n)F(s+n) t-2~-1(1_t)s+k+n-1(1+t)s-k+n-ldt . X F(2s+n)n: " S ] F(-2s) n=O O s=-O
93 Summation
and left-hand-sided
differentiation
may be interchanged
because the functions
i
a I
; d~ [F(-2a)1 OS t-2a-1 (]-t) s+k+n-1 (1+t)s-k+n-ldt
are continuous n 6 I~° .
in
[-I,O]
The monotone
and uniformly
convergence
]
bounded with respect to
theorem further
implies that
E3 = ~d [¼ -½ F(-2a)F(2s)F(s-~)F(s)O} t-2~-1(]-t)s+k-1(1+t)s-k-1" F(s-½'s;2s;1-t2)dt]~=-O
d 1 = d--~ ~ "
I 2F(-2a)
S1 t-2~-I (l-t) s+k-1 (1+t) _S_kdt ] O Ja=-O ([MOS], p. 38)
=
] 2s-----1 " ~ "
d~[
r (s+k) F (s+k-2a)
F (s+k,-2a ;s+k-2s ;-I) ]
a=-O ([MOS],
I 2s-1
. ½F(s+k)d~[22a
I F (s+k-2~)
F(-2a,-2a;s+k-2a;½)]~=_ 0
(by a linear transformation, I
I (log 2 + ~ ( s + k ) + ½ lira 2s-1 aIO I
2s-I
54)
p.
F(s+k) F(-2a)F(-2a)
cf.
[MOS], p. 47)
~ F (-2a+n) 2 n_E1 F(s+k-2a+n)
2-n) " ~.
(log 2 + ~(s+k))
Hence, z~ + z 2 + z 3 = 2s<-I ~ (-~-2 log 2 + ~(s+k) + ~(s-k) - ~(s) - ~(s+~) +I---2s-I ) "
D
The proof is finished.
Combining
2.4.11 F
Lemma 2.4.8, Lemma 2.4.9 and Lemma 2.4.~0 we obtain
Lemma.
Assume that
is a fundamental
domain of
in Remark and Notation
2.4.5.
s,a
6 ¢ , Re s , Re a > max(1,1kl),
F Then
according
to Remark
1.2.2,
FY
as
94 m.
f E E F j:1 p=1
E S6F ~j~r
E JS_ITns (z) H(z,S-1~j Sz) ks,a(O(z,S-1~jSz)) n677\{ O } 3
oo
I
~ f(t;s,a) -~
4~
=
lira
~
log
Y
-
log
2a-I
log Y - log
2
IEjp(Z,½+it) 12dt] de(z)
+
½~(s+k)
+ ½~(s-k)
- ~(s)
-~
(s+½)
+
y~co
2 + ½~(a+k) + ½~(a-k) - 9 ( a ) - ~ (a+½) +
m.
T 3 ~ S E E 4 ~ S f(t;s,a) FY j=1 p=1 -~ For the c o m p u t a t i o n ing n o t a t i o n
of the
of R o e l c k e
from the E i s e n s t e i n cients
"near
2.4.12
Definition.
mark 1.2.2, so
that
joint.
E~
([Ro 2], p.
of
Let
sets
still
288)
F
be a f u n d a m e n t a l
domain
and Remark
with
× ]Y,~[),
:= ~ Ejp(Z,½+it) ,
[
Ejp(Z,½+it)
2.4.5
Proposition
By
[Ro 2], L e m m a
in
[He 2], pp.
11.2.
311-312
the followwhich
Fourier
arise
coeffi-
if
of according to PmY > max(Yl,...,yT)
1 = I,...,T , are p a i r w i s e t 6 ~
dis-
put
z 6 FY
- ]A I " (z) -1 if
(remember
]
F " :
F N AII(~
3P
left we a d o p t
the zeroth
j 6 {I ..... T} , p 6 {I ..... mj} ,
(z,½+it)
de(z)
for the f u n c t i o n s
by s u b t r a c t i n g
F Y as in N o t a t i o n
the For
FY-integral
series
the cusps
] IEjp(Z,½+it)12dt
• Ujp,l(Im z 6 F n
AlZ,½+it) , (m
× ]Y,~[)
1.5.6).
resp. we get
Lemma
11.3.a)
and
transformations
as
,
95
2.4.13
Lemma. 3
x
x
IE Y (z,½+it)l 2 d e z)
F j=1 p=1
= 2r~log
where
Y > max(Yl, .... y ) ,
m.
T
I
For
3P , Y - ~' ( ½ + i t ) + 2Re
~ , ~ = det ¢
2.4. 1 4
Corollary. (
lira Y~
=
-
Proof.
[
y2it] tr ¢(½+it)
are d e f i n e d
t 6 ]R ~ {O}
•
in N o t a t i o n
,
1.5.8.
We have
T mj 'Ejp (z'½+It)'2 [ y2it ]) ] _Z E din(z) - 2T~log Y - 2Re tr ~(½+it) • 2 ~ FY j I p=1 (t 6 m
--(½+it)
For e v e r y
~
Y > m a x ( y I ..... yT)
{0}
).
,
m~ Y
J
!+it~J
I I FY j=1 p=1
IEjp(Z,~
_ .~
2
dc0(z)
m. T = I Z__ I3I E 3P (z,~+it) 12 d0,(z) - ~ F j I p=1 f x~ The
last
integral
convergence
The
theorem
T
S F~FY
m
j
E ~ j=1 p=1
assertion
Before
converges
to
(note that
~.
O
as
for all
2
m. T I I 3 IEYp(z, ½+it) 12 de(z) j=1 p=1 Y ~
~ , by the d o m i n a t e d
Y ~ Yo > m a x ( Y 1 ' ' ' ' ' Y T )
T
mj
Yo
IE p(Z,½+it) I de(z) -< S E E IEjp(Z,½+it) P j=1 p=1
now follows
immediately
from L e m m a
12
:
de(z) <
~)
.
[]
2.4.13.
we can c o m p u t e
consider
the t e r m
the limit in L e m m a 2.4.11 it is n e c e s s a r y to [ y2it] 2 Re tr ¢(½+it) . ~ - ] and the zeros of ~ more
in detail. In s e c t i o n that
~0
1.5 we a l r e a d y
has only
finitely
stated
that
l~(~+it) I = I
for all
many
zeros
in the h a l f - p l a n e
t 6 ~R,
{s6~: Re s<½}
,
96
all of t h e m conjugate the
contained
of any n o n - r e a l
representation
solutely
for
bounded.
Especially, real
2.4.15
Notation.
contained
in
the
also
and
Yn
zeros counted
Formula
(1.5.6)
(n ~ N+I)
are
Starting
[He 2], p.
2.4.16
437
~
with
F p.
the
uniformly
~.
of
parts ~
It f o l l o w s
series
there
from
converging
of the
be
the
zeros
with multiplicities.
larger
than
positive
multiplicities.
that
½ ,
zeros
exist
ab-
is
only
of
~
finite-
Let
which
parts.
Moreover,
let
are
ql,...,qN
qN+1,qN+2,..,
imaginary
Pl ..... PM '
the p o l e s 1.5.9
and 156, into
For
the
The ~n
de-
count-
qn '
n ~ I,
:= Re qn - ½
G
of
~ ,
and
from
I-qi .... '1-qN ' counted
products
Proposition
12.5.,
account,
one o b t a i n s
P1' .... P M '
p.
157, the
1-qn
multiplicities.
the r e p r e s e n t a t i o n s
by W e i e r s t r a B
s 6 ¢ ~
with
1-qn '
of the
of o r d e r
4
Proposition following
1-ql,...,1-qN
'
and 12.6.
result.
1-qn ' 1-qn:
we h a v e
I--2S
~(s)
of
the c o m p l e x
n ~ I .
(2.10)
Lemma.
n ~ N+I}
where
~
from Proposition
taking resp.
of
exactly
functions
set of real
1-Pl,...,1-PM
implies
entire
that
by a D i r i c h l e t
the
counted
with
:= Im qn '
~
is a zero
and
~ .
Let
of
of
that
of
zeros
also
[O,½[,
by the m e r o m o r p h y
[0,½[ ,
real
ably many are
zeros
interval
zero
(1.5.3)
Re s > I
ly m a n y
note
in the
=
gl
,~,
~
number on e v e r y
"
M I -s-pro H m= I s-Pro
gl > 0 compact
is d e f i n e d subset
of
N
S-qn
n=1
1_S_qn
by
(1.5.3).
its d o m a i n .
(S-qn) (s-q n) H n~N+1
(1-S-qn) ( 1 - s ~ )
The product
converges
'
97
2.4.17
Corollary.
We h a v e
M (½+it)
= - 2 log
Z m=1
~I +
2Pm-1
N
2Dn
(Pm-½)2+t 2
Z n=1
2 2 ~n+t
2D n
)
2H n
for Z n~N+1 The
sum c o n v e r g e s
Transformations integral
2.4.18
uniformly like
formula
Lemma.
holomorphic
If
and
function, such
that
ii)
h(t)
= O(IRe
every
Y > 0
i ~ 4--~ f
pp.
of
~
201-202
.
and use
of C a u c h y ' s
h:
{t 6 ¢:
IIm t I < ~}
, ¢
is a
IIm t I < ~ , tl -I-6)
the
as
Itl - - ~
, with
some
6 > O ,
integral
h(t)
[ • 2 Re
y2it] 2it
tr ~(½+it)
dt
and
lim y~oo By m e a n s
[He 2],
~ > O
= h(-t),
exists,
in
subsets
yield:
h(t)
for
on c o m p a c t
those
i)
then
t 6 ~ .
2 2 + 2 (t+Yn)2 n n + ( t - y n) ~n +
1
of L e m m a
f
h(t)
2.4.13,
[ • 2 Re tr ~(½+it)
Corollary
2.4.14
y2it] • 2~J dt
and L e m m a
= ¼ tr ~ ( ½ ) . h ( O ) .
2.4.18
we n o w
prove:
2.4.19
Lemma.
is a h o l o m o r p h i c
Assume
that
function
g > O
and
h:
{t 6 ¢:
with
i)
h(t)
= h(-t)
,
IIm t I < g ,
ii)
h(t)
= O((Re \
t)-41 /
as
Itl
)
CO
.
IIm t I < s} -
'
98
Then
for e v e r y
Y > m a x ( y I ..... yT)
the f u n c t i o n
m,
F Y 9 z~
belongs
to
;
Z Z j=1 p=1
LI(FY,~),
oo
IEjp(Z, ½+it) 12 dt
the i n t e g r a l
I
lh(t)~(½+it)
exists
lh(t) I -~
I dt
and m.
lim Y~
=
Proof.
f Z Z F Y j=1 p=1
_
~
7
1
Let
h(t)
is p o s i t i v e ,
(½+it)dt + ¼ tr ~(½)
I
I
(s-½)2+t 2
(a-½)2+t 2
=
~ h(t)dt-loy Y
• h(O)
(t 6 ~ )
and
= O(f(t;s,a)),
The f i r s t
assertion
The other
assertions
h(t)
12 dt d~(z) -T ~. .
.
s,a 6 ~ , ½ < s < a . T h e n
f(t;s,a)
h(t)
f h(t) I . .
= f(t;s,a),
follows
as
Itl
....
immediately
are firstly p r o v e d
from Corollary
for the s p e c i a l
a > s > m a x ( 1 , 1 k I ) . The p r o o f
2.4.7.
case
is o b t a i n e d
in t h r e e
steps:
Ist step. exists
By L e m m a
2.4.11
in c o n n e c t i o n
with Lemma
2.4.3,
there
99 m,
f :: lim Y+ ~
_ ~
I E j p ( Z , ~ +-i t
f(t;s,a)
i Z Z~ F Y j=1 p=1
)
I2
dt
d~(z)
-oo
<2sll
1 ) log 2a-]
y] m,
: lim ~ Y~ ~
2nd step.
For
~ f(t;s,a) -~
Y ~ Yo
iEjp(Z,½+it) [2 d~(z) - 2T ~ logy]dt.
Z Z Y j:1 p:1
> max(Y1' .... YT )
and
t 6 IR
let
m,
"[
For
fixed
2
t 6 IR ~ {O},
y 6 [yo,~[
gy(t)
is a n o n - d e c r e a s i n g
function
m. T 5 Z Z3 F~F Y j=1 p:1
: _ ~~,( ½ + i t )
2.4.13.
f(t;s,a)
Thus,
(gy(t)
for e v e r y
- gy
(t))
Y (z,½+it) IEjp
to
y 6 [Yo,~[
3rd step.
The
first
We a p p l y
step,
t 6 IR
the
is n o n - n e g a t i v e
function and n o n - d e c r e a s i n g
.
the m o n o t o n e
Lemma
2.4.]8
convergence
and C o r o l l a r y
theorem:
2.4.7
imply
co
lim ~ Y~ ~
I
~ f(t;s,a) (gy(t)-gy -~
(t))dt o co
= i - ~ tr ~{½)
• f(O;s,a)
1
4n
5 f(t;s,a)gy --co
in p a r t i c u l a r , Consequently lim f( y-~oo
of
2 do(z)
o respect
y2it -~-~].
, as
gy(t)
by L e m ~ a
3
:= ~ Z Z I Ejp{Z,½+it) I d~(z) - 2~*log Y - 2Re [tr ~(½+it) F Y j:1 p=1
gy(t)
the i n t e g r a l s
lim f( ;s,a) Y~ ;s,a) .gy b e l o n g to
(t)dt O
exist. (gy-gy) and h e n c e also o LI(IR) , and we have
;
with
100
I ~-~
lim y~
- 4~I
]- f ( t ; s , a ) g y ( t ) d t
~
-~
I
7
4n
by C o r o l l a r y
f f(t;s,a) -~
lim g y ( t ) d t y~
~
f(t;s,a)~(½+it)dt
2.4.14,
hence,
oo
1
L by L e m m a
" <0' ½+it)dt
+ ¼ tr ~ ( ½ ) f ( O ; s , a )
= f(t;s,a),
a > s > max(l, Ikl),
-6 (
--co
2.4.18.
In the special tions
f(t;s,a)
J"
4~
of the
However,
case
lemma
if
h
h(t)
are proved.
is an a r b i t r a r y
assumptions
of the lema~a, the
element
LI(IR)
Hence,
of
by L e m m a
because
2.4.13 ~
holomorphic
function h(t)
and L e m m a T
]19 9 t I
the a s s e r -
h(t)
J
m.
z
IR 9 t I
= O(f(t;s,a)) 2.4.18
3
z
function
the
subject
to the
, h(t)-~-(½+it) for
Itl
is an
~~ .
function
y 2 IEjp (z, ½+it) I de0 (z)
F j=1 p=1 belongs
to
L I ~)
for e v e r y
Y > m a x ( Y l , . . . , y T) . For t h e s e
Y
we
obtain m.
zT
I
z3
F Y j=1
co
I
7 h(t) -~
p=1
lEjp(Z,½+it) I2 dt
de(z)
- Y~ ~
I
S h(t)dt --¢o
m,
1
~
4~
r
%
f h(t) [ J
-co
z
z
3
IEjp(Z,½+it) 12 do(z)
- 2~log
I = + ~-~
~ f h(t)
I
f h(t)
(
qg'
- ~(½+it)
+ 2 Re
[
tr ~(½+it)
y2it]h -~]/
m,
T 4~
Y ]
Fy j=1 p=1
-~
as a c o n s e q u e n c e
f
z
3
z
F.~FY j:1 p:1
of L e m m a
2.4.13.
IEYp (z, ~+it)I 2 d ~ ( z ) d t
,
dt
dt
log Y
101
As m.
TE f E] F\9 Y j=1 p=1
lh(t) I
-< lh(t) I f F
for
IE y p(Z ,~!+.It) I2 dc0(z)
m. 3 E
T E
Y IE °(z,½+it)I 2 d0~(z)
j=1 p=1
]P
Y ~ Yo > m a x ( Y 1 ' ' ' ' ' Y T )
LI(IR) , the d o m i n a t e d
and since
convergence
the last
theorem
integrand
belongs
to
implies
m.
I ~f h(t) lim ~-~ Y~ ~ -~ Now application general
bolic
2.4.18
completes
2
d~(z)dt
the p r o o f
= O .
of this
lemma
for []
the a s s e r t i o n s
the s o - c a l l e d terms
2.4.20 mental
of L e m m a
Y (z,½+it) IEjp
h .
Combining obtain
f ZT Z3 F\F Y j=1 p=1
of the
"singular
to the r e s o l v e n t
Lemma. domain
Let of
lemmata part"
2.4.3,
2.4.11
and
of the c o n t r i b u t i o n
2.4.19
we
of the p a r a -
trace:
s,a 6 ¢ , Re s, Re a > m a x ( 1 , 1 k l ) ,
F
a funda-
F . Then
m,
f 7T E F j:1 p=1
S6EF ~ jk
F
n6Z-{O}I
I -f
4~
j_1Tns (z)3 H(z'S-I~jsz)ks'a(°(z'S-1~jSz) )
f(t;s,a)
IEjp(Z,½+it) I2 dt
]
dc0(z)
--oo
I
- T*log 2
2s-I
I 2a-I
-~
-'[*log 2 + ~
9(a+k)+~(a-k)-2~(a)-29(a+½)
o~
+ ~
f f(t;s,a) --co
(½+it)dt
.
+
tr
I -¢(½) ~*
102
The
contribution
iterated
of all
resolvent
parabolic
kernel
arises
elements from
this
of
F
to the
in c o n n e c t i o n
trace with
of the
Lemma
2.4.1.
2.4.21 X
Proposition.
a multiplier
Res,
Assume
system
that
F
of w e i g h t
Re a > m a x ( 1 , 1 k l ) .
2k
is a f u n d a m e n t a l on
F ,
H(z Mz) '
- 2s-11
X p=1
[ -co
2+t2 (s-½)
- dxlog 2 - log
T ~ j=1
_ I 2a-I
dTlog
F ,
E ~,
2 - log
(a-½) 2+t 2
d R sin p : m +I 3
T N j:l
- h
(o(z,Mz)) a
m,
Y j=1
of
Then
f [ ½ X tr X (M) jM(Z) ]: ME F M parabolic
-
s,a
domain
)
]Ej
(z,½+it)12dt P
]
d0~ z)
+ (,(s+k)-~(s-k))(~Id T -
Y B4) j:1 3
~6jp4
d H sin p : m +I 3
~6jp
x +(9(a+k)-~(a-k)) (½dT- Y Bj) j:1
oo
+ ~-~ -~]
(Remember
Just
by the pose
the n o t a t i o n s
as the
sections
we
(s_½)2+t 2 -
2.2
contributions and
logarithmic consider
2.3,
of
(a_½)m+t 2-
section
of the
the
integral
1.5.)
hyperbolic
we r e p r e s e n t
derivative
(½+it) at
the
and
elliptic
right-hand
of a h o l o m o r p h i c
side
elemcnts
of this
function.
For
in the
equation
this
pur-
103
I ~( 4-~ -~f
(2.4.6)
I
¼5t2
(s-½)2+t2
which exists by Lemma 2.4.19.
-~(½+it)dt
Corollary
,
Re s
>
½
,
2.4.17 implies that this inte-
gral equals
- 2 log ~I " 4-~-~
i)
¼+t2
(s_½)2+t 2
M I ~ < I + m=1 Z ~ -~ f (s-L) 2+t2
+~)
2Pm-1
¼
(Pm -½)2+t2 dt < N
4~ -~
(S_½) 2+t 2
I
¼+t 2
dt
"
f
2~n
2
n-1 ~n+t Z
2
( - 2Y]n2
+ 2 2tin
n>N+1_ \Rn+(t-Yn )2
=(I
)
I
M
<
1
Z n=1
Z 2s-I n>N+I
-s-1+qn <
Dn+ (t+Yn)
2S-1))
2 4 1 ~ log ~1 + 2S-I m=Z1 sZ1+Pm
2s-I
~)) dt
Pm /
~n /
I + I s-1+q--------~ 8-I + ~
2s-I qn
2s-I] ~ n "/ "
The last integrand may be computed term by term for the following reasons.
If
½ < s 6 IR, then either all the terms of the integrand
are non-negative
(s A I)
or all of them are non-positive
and the integral is known to be finite. For arbitrary Re s > ½
s 6 ¢
(½ < s ~ I), with
one has to apply the identity theorem for holomorphic
func-
tions after one has checked that the series
(2.4.7)
converges
Z n->N+ I
I 1 S-~ + - I+qn s- I+qn
uniformly on compact subsets of
and hence is holomorphic,
and that
2s-1)
2s-1 qn
¢
7
"
qn
\
{1-qn, l-qn: n Z N+I )
104
I
4--~
I
f
is holomorphic plies
)
2
on
E nAN+I
2
Dn
of 2
E nAN+I
The convergence
lqn I-4
(cf. Cot.
2 (t+Yn)2Dn +
~2n+ (t-Yn)~
{s £ {: Re s > ½}
the convergence
gence of
< n>_N+1
of
)
dt
(2.4.7)
im-
From this and the conver-
2.4.17)
follows
the convergence
of
Dn+y n
the series
:=
Hence,
Z
+
+
the integral
(2.4.6)
Z
is equal
+ ~ +
+
to
M
I 2s-I
I I E log ~I + 2s-I m= I s-1+pm
I
N
/-
1
2s-I > 2s-] n =I z { , ~ -1+qn ~ + 2 (qn-½)a
2s-I I
n~N+1 Z
1 + I 2s-] s---~+qn +~--------~ s- I + 2 (qn-½)2
+
2s-1 ) 2 (~n-½)z
M +S
-
the series
Z -~ + log ~I m=1 Pm converging
In connection
2.4.22
uniformly
with Proposition
Corollary_ t
T* = O
The function
or
on every compact
2.4.21
subset of its domain.
we conclude:
Let
D :=~-(]-~,max(½,1kl)]
If
'
½ tr[I
U {]-qn: n = 1 .....N} U {1-qn,1~:
,-O(½)]
6 IN
replace
max(½
Ik))
n ZN+])) .
by
)k).
105
E
par
: D
*{ , T ½d-B :: 2 -d%s JNI[\~/[{F(s+k)\~ ~-~J/ j4
-:par(S)
d
=
p=m.
½tr[Iy,-O(½) ]
(s-½)
m=1
[[ (I + n>_N+1
is holomorphic; every
compact
the subset
of
its
F(s-k)
hT* ]
\r (~Yi~7½)
I +
exp
½ (~---~-~) qn -~
qn -½]
s-½ h-1 (I
(possibly)
]
[
-s
" BI
I + :----T " U Pm -~] n=1
(sin nBjp) -s] +I
9-½"-I gn ~]
infinite
domain.
e~<_½(s_½) 2 .( I + I ]] \ (qn_½) 2 (%_½)2//
product
The
converges
logarithmic
uniformly
on
derivative
~e
par par Re
has s,
Re a
are e q u a l
a meromorphic
continuation
> max(1,fk
the
I
to
2s-1
I)
-par(s)
two
-
~
to
sides
I
2.5
with non-integer
the
equation
s,a
6 ~
in
Lemma
with 2.4.21
E
exponents
principal
appearing
values
here are defined
by
which are real on
.)
THE RESOLVENT
The evaluation carried
all
par
means of the corresponding D n ]0,~[
For
~par(a)
2a-]
par
(The powers
of
C
TRACE
FORMULA
of the right-hand
side of the equation
out in the three preceding
cial case of the Selberg which bears
Trace Formula,
all information
zeta-function
resp.
sections,
we need
its associated
yields
the resolvent
in Theorem
2.1.2,
an important
spe-
trace
for our elaboration entire
function
E
formula
on the Selberg to be defined
106
in c h a p t e r
3. In c h a p t e r
valent
to the g e n e r a l
2.5.1
Theorem
Suppose
that
multiplier bolic s,a
(Resolvent
system
X
with
linear operator r n 6 i-]O,~[
z
(
I
I
domain
6 IR.
Let
~(F)
F
F .
Assume
of
Ikl-s'
, Hk
U [0,~[
Ikl-a
If
E
tr X(P)
{P }r
(cf.
(n a 0),
(a-2) 1!2 +r n2)
(s-½) 2+r2" n
2s-I
2k
6 p(-Ak ) .
Dk
is e v e n e q u i -
group with a unitary
~ ~o
denote
dxd-
the h y p e r -
further
that
' so t h a t e s p e c i a l l y
1o,11,12 ....
(counted with multiplicities)
-Ak:
case
Formula),
of w e i g h t
~ := a(1-a)
special
Formula.
is a c o f i n i t e
Re s, Re a > I,
of the e i g e n v a l u e s
n>-O
Trace
Trace
a r e a of a f u n d a m e n t a l
6 ~
show that this
Selberg
F < SL(2,IR)
I := s(1-s),
with
4 we
is the
sequence
of the s e l f - a d j o i n t
Notation
1.4.6)
and
2
In = ~+r n
then
= - a~~{-F -) < ~ < s + k )
N(P)
• log N ( P o )
+ ~(s-k) )
-s
I-N (P)
-I
tr P > 2 + ---/--] 2s-I
Z {R}F
tr x(R)
iei2ke 2~(R)2sin
@
O<8<~
v(R)-] K l=O
I / ~-~
e i 8 ( 2 1 + I ) ~ [ ,s+k+l, --~)
- e -i@ (21+1)
(s-k+l,
d
(--d~log
2-
log
N
N
sin ~ j p
j=1 p = m .+1 3
j:1
I
I
+ 4-~ -mr
(S-½) 2+t2
¼ 1
~ - (½+it) dt
(the same e x p r e s s i o n w i t h s r e p l a c e d by a).
107
denotes
the
s u m o n the
right-hand
of h y p e r b o l i c arbitrary
logarithmic
derivative
side
elements
of
representative
of the gamma
extends
over
F
positive
with
of
{P}F
the
which
function.
F-conjugacy trace.
The
first
classes
{P)E
Whenever
is c o n j u g a t e
in
P
is an
SL(2,1R)
to
l a dilatation
_½
with
N(P)
> I
N(P) hyperbolic
element
Proposition {R] F
to
The
elements
R 6 F
k sin O
cos
the o r d e r
of the
T
is the n u m b e r
mj
6 {O,1 ..... d} , Bjp 6
of
The
singularity 1.5.8,
infinite
verge
of
F-equivalence [O,1 [
of X and
sums
@
Remark
that
centralizer
are
is t h e
2.2.3,
F-conjugacy
classes
conjugate
SL(2,1R)
Moreover,
in
F . In a d d i t i o n ,
of c u s p s
of
~(R)
r , the numbers
are defined
T T~ : E mj j :1
T~×T~-rowed
in
]O,n[
R
(p = I ..... d)
and Notation
the
0 6
of
classes
is the p r i m i t i v e o
over
some
d Sj = Z Bjp (j = I ..... T) . p=mj +I
and
tation
(cf.
elliptic
with
half
1.5.5,
P
sum ranges
rotation
denotes
to
second
of t h o s e
a
corresponding
2.2.5).
P '
in Convention
is t h e
scattering
degree
matrix
of No-
~ = det ~ .
and
the
integral
occurring t o t h e t r a c e
formula
con-
absolutely.
Proof.
For
Re s, Re a > m a x ( 1 , 1 k I)
diately
from Theorem
2.1.2
In the c a s e
of
yields
the a s s e r t i o n
that
]k I > I
the
and
the
trace
Propositions
identity
is s t i l l
theorem
valid
for
formula 2.2.5,
follows i m m e 2.3.4
for h o l o m o r p h i c Re
s, Re a > I,
and
2.4.21.
functions Ikl-s, []
Ikl-a ¢ =o By
means
tions
of
introduced
following
way:
Corollaries there,
2.2.6,
Theorem
2.3.5,
2.5.1
may
2.4.22 also be
a n d use of the n o t a formulated
in t h e
108
2.5.2
Theorem
Under
(Resolvent Trace Formula).
the a s s u m p t i o n s
n->O
(s-½) 2+r2 n
of T h e o r e m
2.5.1,
(a-½) 2+r2 n
= - a--~-~- 9 ( s + k ) + 9 ( s - k ) - ~ ( a + k ) - ~ ( a - k )
+
I (-hYP(a) 2a-I Ehyp
-ell ~ -ell
Here
of the
+ :par(a))
-ell
Lz~
and
par
+ ~ell(a)
apar
also
stand
for the m e r o m o r p h i c
continuations
par -ell
functions
and
_par par
-ell rollary
parIs )
_hyP(s ) + = "-hyp -ell
defined
in C o r o l l a r y
2 3.5 and Co-
2.4.22.
The s e r i e s
(2.5.1)
(2s_1) Z ( n~O
{½ ± i r n :
sets of
~ ~ {½ ± i r n :
morphic
function
1
n A O)
(a-~) ~ 2+r2 n
fixed)
n A O}
poles with residues
where
d n 6 IN
converges
with respect
on t h i s d o m a i n .
simple
The
1)
(s-½)2+r2 n
dn
uniformly to
s , hence
A t the p o i n t s if
rn % 0
are the m u l t i p l i c i t i e s
on c o m p a c t
it is a h o l o -
½+ir n , ½-ir n
resp.
2d n
sub-
if
of the e i g e n v a l u e s
it has
rn = 0 , In = ¼+r~ .
function
2.5.2)
s
,-
(2s-I)
+-hyP(s)
-hyp is h o l o m o r p h i c
on
d~4~
<9(s+k)+~(s-k))
+ -ell (s) + - p a r (s)
-ell
{s 6 ~: Re s > I,
-par Ikl-s 6 N o }
.
For
Ikl > I
it
109
has
simple
poles
(O -< 1 <
Ikl
or
- I,
removable
1 6 ]N O )
a~(F) (2k-21-1) ~ - ~ +
singularities with
P =E 9 1
at t h e
non-negative
[
÷
½d(1 -
residues
I ~. (i) ~- 3 3
) 3
Ikll
points
]+
T
½dY -
E Bj - T* 9= I if
(2.5.3)
°~(F) + 9=I 7~ (-2k-21-I) d -~---
I ~.(i) 3 ) - --~. 3 3
½d(1 -
-
~ T + Z pj ~d j=1 if
If o n e
of
O ~ 1 <
the
Ikl
sufficient (note
- I for
that
(2.5.3)
numbers
(2.5.3
is p o s i t i v e
(the c o n d i t i o n s this),
then
r n £ i.]O,~[
is e q u a l
to
the
g A I
Ikl
U
- 1 =
[O,~[
)
integer
is a n e i g e n v a l u e
For
Ikl
right-hand
rem
2.5.1,
-Ak
has
no negative
A.
Selberg
remarked
of
the
may
the
Theorem
spaces
be drawn
2.5.3 real
of
in
the
Definition. weight
2k
A classical
to
F
and the
X
eigenvalue
[Se
on
p.
X
the
of
the
the
84,
1 6 ]lqo
- 1 - I > ~
for
some
operator
Resolvent on case
that
(cf.
formulae
automorphic
forms
, 2~
-~k
[E2],
for
case
"
Trace Re
are
residue
In t h i s
{s 6 C:
.
n 6 ]N O
corresponding 2.5.2).
k <-I
Formula
(Thee-
s > I} , h e n c e p. IO1,
the
Satz 6.1.).
dimensions
of weight
2k > 2
formula.
be
a unitary
dxd-multiplier
system
of
T
entire
automorphic
is a h o l o m o r p h i c
following
of
in t h a t
entire
trace
Let
2],
Ikl
(Theorem
is h o l o m o r p h i c
classical
from
a)
both
2.5.2)
side
or
some
½ - ir n
and
dn
(IkI-l) ( 1 - 1 k I + l )
~ I
for
k>l
conditions:
form
function
of weight f:
IH
2k , {d
with
respect
complying
with
110 i)
(cz+d)-2kf(Mz)
ii)
=
x(M) f(z)
~
=
there on
IR
The
space
b)
Let
which
exists
g > O
such
that
is a c u s p
of
F , then
(yz+6) -2k f(A-Iz)
is b o u n d e d
x [~,~[
of t h e s e
functions
Q(F,-2k, x )
satisfy
f
is d e n o t e d
be the
by
set of t h o s e
if(z) J2 y 2 k d ~ ( z )
< ~
{F,-2k,x}
elements
for
F
.
f E
{F,-2k,x}
a fundamental
domain
F
of
F .
(Cf.
[Ro I ], p.
Note
that
the
297,
Def.
integral
1.2.,
in b)
[E2],
p.
104.)
is i n d e p e n d e n t
of the
choice
of
fundamen-
tal d o m a i n .
The m a p
Q(F,-2k, X) f
I
) {g
6 Dk:
,
ykf
is a c a n o n i c a l
isomorphism.
2.5.4
(dimension
Theorem
multiplier
system
Hence
the
formula).
of w e i g h t
d i m Q ( F , - 2 k , X) =
-~k g : k ( 1 - k ) g }
2k
on
(2k-I)
result
Suppose F •
stated
that
above
k > I
yields:
and
X
is a dxd-
Then
d~ (F) ~ +
P [ ½d(1
Z j=1
+ ½dT
-
_I)_I
aj (O)
~j T Y
-
Bj - T*
j=1 = (2k-1)d(g-1)
(Note
equation
(1.2.2)
+ kd
T +
for the
[P (I . I) . j:1 ~j
hyperbolic
~. j=1 j area
~j. (o)
~(F) !)
z Bj j=1
I-*
•
]
111
2.5.5
Corollary_
Under
d i m { F , - 2 k , X} =
the
(2k-I)
same
o--~
assumptions
we h a v e
+
j:1
=
(2k-1)d(g-1)
Proof.
It
+kd
is a w e l l - k n o w n
automorphic
forms
and a s p a c e
spanned
Usually
these
whereas
in the
(cf.
[Pa I],
T = O,
proof
p.
[He 2],
the
sion of
2.5.6
chap.
Both
In the c a s e
number
follows
mentioned 1 = ±I,
Now
The
are p r o v e d
given
here
10,
Theorem
by m e a n s
Trace
([He
(2.5.3)
k > I
1 = O
from
induction
p.
are
series.
from
a more
even
for formula
expects
general
ver-
433).
for all
the a s s e r t i o n
for the
the
used
7.
the t r a c e
Hejhal
O
Theorem,
are
integers
the R e s o l v e n t yields
entire
section
sense
Theorem.
I],
methods 4,
Bj
8j
Riemann-Roch
chap.
In t h i s
the n u m b e r s
and
j:l
Q ( F , - 2 k , X)
Eisenstein
real-analytic
be d e r i v e d
Formula
s u m of
of the
[He I],
X j:1
of c l a s s i c a l
independent
4.).
can
theory
is the d i r e c t
only
section
Z v-- aj (O) j:1 J
-
of the
p = T = O ,
j=IX --vj
drop
sum of b o t h
Therefore
)
linearly
%~
z
Trace
claim
Formula
for
the
1 6 Z .
first
as a l r e a d y
first
n u m b e r and
±3 .... , since
2~(F)4~
we m a y
by
immediately
above.
±2,
result
T
]+ ½ d T -
3
]
for the R i e m a n n - R o c h
the S e l b e r g
Remark.
Z (Ij=1
{F,-2k, X}
94 for
Riemann-Roch
Proof.
Hence
that
formulae
is an a l t e r n a t i v e that
(~+
] sj(O)
~jp(1)
- ajp(l+l)
the r e s t r i c t i o n
the n u m b e r s
the a s s e r t i o n
k > 1
(2.5.3)
is a l s o
(p = 1,...,d)
by v i r t u e
is e a s i l y
valid
6Z
for the
seen
of R e m a r k to be an
second
number.
1.3.5.
inteoer. []
112 Necessary
conditions
system
weight
of
another
consequence 2.5.6
2.5.7
Proposition.
a)
system
If
If
existence
in
the
case
of
the
of
T = O
Resolvent
Trace
a unitary (cf.
dxd-multiplier
Proposition
Formula.
In
1.3.6)
are
particular,
implies:
of
Assume
~ = 0 ,
2k
weiaht
T = p = O , k 6 1 d
b)
the
2k
Remark
plier
for
.
that on
there
exists
a unitary
dxd-multi-
F .
then
2 ~ e(F)
p -> I
_
and
] d(29-2) ~1,...,Vp
are
I l . c . m . [ ~ I .....
"Up]
then k 6 I_ . 2~ d ~(9)
chosen
as
in
section
2.3,
3,
THE ENTIRE FUNCTIOr~ z
ASSOCIATED WITH
THE SELBERG ZETA-FUNCTION
3,1
DEFINITION
AND
FUNCTIONAL
EQUATION
OF
~ •
THE SELBERG ZETA-FUNCTION
For
fixed
a 6 ~ \ {½+irn:
s I
n
~ O}
~ (2s-I)Z
the
( n~O
function
1
1
(s-½) 2 + r 2
(a-~) 2 + r 2
)
n
is t h e
logarithmic
gularities residues mula
derivative
½+it n , ½-Jr n
being
(Theorem
positive 2.5.2)
(n ~ O)
integers.
suggests
s F--~ - ( 2 s - I )
also
as i h e
logarithmic
o n an a p p r o p r i a t e
3.1.1
are The
function,
simple
poles,
the
G-function
derivative
of a h o l o m o r p h i c
The
function
G:
{
3.].2
is t h e E u l e r - M a s c h e r o n i
Remark.
ly the p o i n t s functional
Trace
function
~ { , explained
2 •Z•1 ••••Z := (2n) ~z e •!Z
(cf.
corresponding For-
function
2
defined
[WW],
G ]-n
p.
function
the order
equation
G(z+I)
,
n
is c a l l e d
the Barnes
264).
is a n e n t i r e with
constant,
by
2 Z -Z + - (I + --~_)n e 2n
oo
n:1 T
its s i n -
~(s+k)+~(s-k)
i
where
the
all
f o r m of t h e R e s o l v e n t
to r e p r e s e n t
~
since
domain.
Definition.
G(Z+])
n
o f an e n t i r e
= F(z).G(z)
n
the
zeros
(n { ~ ) .
of w h i c h G
are
satisfies
exactthe
114
Moreover,
G(1)
=
G'(z+1) G(z+1)
Proof.
Now
See
an e a s y
1
and
= ½1og(2~)
[WW],
p.
[]
264.
calculation
shows
that
(s-½) ( ~ ( s + k ) + ~ ( s - k ) )
-
(-z ¢ IN).
+ ½ - z + zg(z)
= -log(2~)
+ 2s-I
G' + -~-(s+k+1)
(½+k)@(s+k)-(½-k)~(s-k)
G' + --G-(S-k+1)
(s 6 ¢,
This
k-s,
-k-s
~ ]N O )
implies:
3.1.3
Remark.
HI
The
¢
:
Hi(s)
-
function
]-~,lkl]
:= exp
, ¢
[d ~(F)( ~ s
, log(2n)
+ (½-k)logr(s-k)
+s(1-s)+(½+k)logF(s+k)
- log G (s+k+l)- log G (s-k+1))] J
is h o l o m o r p h i c ,
zero
free
Hi
--(s) = -(2s-I)
and
satisfies
d e(F)--~- ( ~ ( s + k ) + @ ( s - k ) )
HI
Hence to
Note are
the
logarithmic
{s 6 ~:
I.
log
k-s,
F
holomorphic
derivative
-k-s
and on
-I %--I
has
a holomorphic
continuation
~ ]N O } .
log G ~ ~
are
]-~,O]
those and
logarithms
real-valued
of on
F
and
]O,~[
G .
which
115 Note 2.
The notation
:ell ' :par )
:I
(by analogy with the notations
is to remind that the expression
in the trace formula arises (I : s(1-s))
3.1.4
from those terms of the series
that correspond
Definition.
For
:hyp '
oI l( ) -d --~-- ~(s+k)+~(s-k)
to
±I
(cf.
for
Gkl
(2.1.2)).
s £ ¢ , Re s > I , s 6 ]1,1k I ]
(if
Ikl > I)
put E(s)
:= :I(S)
Ehyp(S)
= exp [de(F) L -~-
• :ell(S)
• :par(S)
( s log(2n)+s(1-s)+(½+k)logF(s+k)+(½-k)logF(s-k) - log G(s+k+1)
(id (
U U det {Po }F m:O
P
- log G(s-k+1) )]
d (1--~) s
~ [vj j:1
3
-½d(1-~) (r(s+k) F(s-k)) 1
vj~l
(s-½)
We use the symbol to
s+k+l
H p=m. +I J
Pm_½ ] • n f 1
~
1
3
½tr[I T*-4) (½) ] " gl-s
m=U1
J
~, ( l )
F(s_k+l}~ j
j__HI \ ~
.
sin)
)N (
)]
vj % (i)
(sin :6jp)
(s-k) )I:* F(s)r r(s+~)
qn_m----~ ]
exp(-½
also for the holomorphic
)
continuation
{s 6 ¢: Re s < ] , Ikl - s { ~qo } " Then the mesolvent
mula has the form
of
--~
Trace For-
116 (3.1.1) Z
(
1
n>-O
We
fix
a and
morphic the
above,
Hence the We
Z zeros
dn
is
of
which s
Observing
that
the
E(s)
a consequence
(3.1.4)
The
if
to
rn of
(3.1.1)
and
ir n
2d n
if
also
subtract
functional
has
½ +
½ - ir n ,
~ IN o ) •
a mero-
singularities
eigenvalue
function,
Ikl-a
~--
The
resp.
points
a
that
½ - ir n , # O
= (a)
Ikr-s,
I,
plane.
the
entire
get
>
(3.1.1)
points
the
in
=
of
for
Z
equation
(n a O)
,
rn = O .
¼ + r n2
of
-~ k "
denoted
by
E ,
½ +
the
of
ir n
(n Z O).
equation for
=~
ob•
by
-ell
~par
-I
-ell
-par
¢ )
has
½
is
+ irn:
even
we
n
_> 0 } )
deduce
the
integration:
are
2.5.6
integers.
zeta-function
:=-~hyp
at
{½
(s E ¢)
£I
continuation
E
[(1-s)
Remark
on
(s 6 ¢ x
order
of
Z
the
an
a
complex
- ~--(1-S)
equation
Selberg
whole
to
Re
~'
2a-I
'~t
=
(meromorphic
s,
1
-~- (S)
2s-I
formula
dn
1-s
(3.1.1)
"Z'!
the
the
'~' -
Re
exactly
by
from
1
multiplicity
are
:~(s)
As
the
extended
(3.1.2)
(3.1.3)
from
exactly
the
be
functional
6 ¢,
residues
may
thus
(s,a
to
are
with
substitute
tained
n
continuation
poles
)
(a-½ i 2 + r 2 n
conclude
continuation
simple As
1
(S - ~1) 2 + r 2
:= =
to
"
the
all
the
residues
Hence whole
plane
_-1
--1
=-1
Sell
~par
zeros
the
a meromorphic
:m
non-trivial
of
of
is
defined
multiplicities
function
continuation
of
by
dn
at
the
117
points
½ -
finitely At
the
(i.e.
many
due
entire
tion
zeros
be
added.
defined
function
analogy
Selberg
lies
on
the
analogue
of
which
not
the
do
follows
jointly
3,2
3.2.1 A
with
THE
the
there
and
of
on
the
E ,
exists
f(z)
.
rn
.
with
most
non-trivial
may
the
at
-par
• [O,~[
poles
and ,~-1
-ell
6 i or
zeros =-1
-I
zeros
functional
of
not
i.e.
line
{ ,
coincide,
Riemann
line
Z
smaller almost
{s
6 ¢:
are
functional
is
every Re
is
s =
the
zeta-func-
~(s)
product
factors. ¼
(cf.
½} All in
In
JR,
~(1-s)
[E2],
more
every
Satz
zero
6.1.),
of
sense
zeros
Z , an
of
precisely,
.
representation
of
E
in
Def.
3.1.4
(3.1.3).
OF
m x A O
Let , ¢
~
be
is
said
, such
that
= O(exp(IzlX))
an to
as
unbounded be
Izl
of
subset
finite
, ~
of
order
, z 6 m
¢ . of
.
.
the
almost
this
the
= of
As
non-trivial
valid.
contained
equation
the
than
max(1,1kl)]
product
equation
elementary
Hypothesis
this
the
Definition. f:
_-1
~(s)
certain
[min(O,1-1kl), from
with
the
is
Riemann
lie
Trivial
associated
definition
" c r i't i c a l "
GROWTH
function
, ¢
.
factor
trivial
~
2 ¼ + rn
zero
interval
This
the
In =
every
and
the
ir n
satisfies
zeta-function
almost
½ +
> O
n
by
also
with
eigenvalue
do
r
s :: ½ s ( s - l ) n - ~ r ( ~ )
~(s)
By
due
E )
~:
with
n
½ - ir n ,
to
is
ir
are
function
~
This
½ +
poles
to
have
,
n
points
orders
An
ir
growth
if
in
118 In this
case
inf { x
is c a l l e d
3.2.2
the n u m b e r
6 ]19>O: f(z)
the o r d e r
= O (exp(IzlX))
of
f .
Definition.
The
function
I ,
if
nN=IN<1
P(s)
+
as
P: {
Iz
, {
) ~,
z 6 ~}
is e x p l a i n e d
by
T* : 0
q S i ~ ) e x p < ~ < ~ >s-~ 2>1
:=
exp
U n_>N+1
-n-~
%-I
+
)
\~\ q--~-]
if
The product hence
P
converges
uniformly
is an e n t i r e
on c o m p a c t
function.
Lemma
sets
2.4.]6
in
,
\%-1 /
C
implies
T* A I
(cf. Cor. 2.4.22), the
functional
,
if
equation
(3.2.1) f P(1-s)
= P(s) "
Our a i m in this the o r d e r
2
that
has
plicit
E.P
half-plane Using
the
assertion
if
2s-I gl
section
exactly.
product
according
1 ,
the o r d e r
M Z m=1
I
is to p r o v e We a c h i e v e 4
representation
to D e f i n i t i o n
"c* : O
3.1.4
-
qo(s)
T*
this as follows.
function At
(E'P) l{s 6 C: Re s > I} ~
t h a t the r e s t r i c t i o n Ikl)}
H.P
]I, Ik ] ] to the
has the o r d e r
(3.1.3)
for the h a l f - p l a n e
ik := {s 6 ~: Re s ~ m i n ( - 1 , - I k l ) } ~-P
(3.2.1)
has
of the ex-
functional e q u a t i o n s
s t r i p the o r d e r of
and
+
of
E.P
first we p r o v e
T h e n we s h o w by m e a n s
Rk := {s 6 ~: Re .s > m a x ( 2 , 1
In the r e m a i n i n g
_> I
1-S-Pm
that the e n t i r e
at most. of
S-Pm
2 .
we g e t the same
is t h e n e s t i m a t e d
by m e a n s
119
of the
Phragm@n-Lindel6f
the o r d e r and
P
2
exactly.
have
the o r d e r
3.2.3
Lemma.
larger
than
Proof.
The
The
principle. From
this
at m o s t
order
It w i l l
result
we
turn
out
finally
that
E-P
conclude
has
that
2 .
of the
entire
1-2 n
converges
E:
function
¢
~ {
is not
4 .
series
E n_>O
by T h e o r e m
1.6.5.
Because
of
n the
2 In = ¼ + rn
representation
equivalent
to the
convergence
(for a d e f i n i t i o n irn,
-it n
see
[TI],
and
represented
El(S)
:: (s-½)
with
of
p.
rn 6 i
E r -4 n n_>O r %0 n
250,
8.23.)
as a f u n c t i o n
2d¼ H
n_>O r %0 n
of
d¼
-~k E
is the m u l t i p l i c i t y and
r -1-I n
1
is the
of
I - ~
converges.
EI
this
is
product
to the p o i n t s the
form
>
l { s ~ ] j + 1{ s.-½ ~J ) 3\]rn/ 3 \-lrn/
of
is an e n t i r e
has
s-½ I + ir n
the p o s s i b l y element
canonical
s - ½
< <)(
of
smallest
The
U [0,~[
corresponding
• exp( Z1 "j=1 where
]O,~[
occurring
{O,1,2,3}
function
eigenvalue such
the o r d e r
that of w h i c h
n ~ O r %0 n does
not
exceed
E2:
1+1
~
E2(S)
converges function not
~ ~ :=
uniformly which
larger
than
(cf.
for 4.
[T1],
p.
251,
8.25.).
Thus
,
(S-½)
H n ~O r #O n
on c o m p a c t 1 A 2 For
I +
subsets
is e v e n s 6 ~ ~
(S-})2 r2 n
of
equal
exp
rn
¢ . Hence to
{½ ± irn:
it is an e n t i r e
E I . The n ~ O}
order
we h a v e
of
E2
is
120
~2(s)-
: s---] +
(2s-I)
z
-2
n > O r ~O
( -~)2+r2 n
n
n
-~-(S) + (2s-I)
=
d¼" (a_½) 2
n >_O
(a-½)2+r 2 n
r %O
2a-I
-:
n
n
with
a
This
6 { ~
implies
{½,½±irn: that
Consequently, larger
not
n
-
-2
~
E2
~ = E 2 "e f than
2
~ O ]
.
is
with
Hence
arbitrary
but
a constant
a polynomial
the
order
of
fixed
or
a
f
the
~
(cf.
linear
does
(3.1.1)).
polynomial.
degree not
of
which
exceed
4
is
,
either. []
3.2.4
Lemma.
larger
than
The
is
sume
that
that
the
(n =
I ..... N),
s
[,
of
the
nothing
to
be
T*
It
~ I
canonical
P(s+½)
Lemma
3.2.3
which
now
entire
function
P:
~
---~ {
is
not
4 .
There
Proof.
order
has one
is
not
follows
product
½ - gn the
proved
'
from
the
formed
PI
the
4
at
case
with
most.
the of
obtains
that
larger
than
2 . Nevertheless,
is
= O . Thus
of
zeros
the
as-
!%1-4
E n~N+1 ½ - a
-n
function
Similarly
p, T
- P~
T*
convergence
(n A N + I )
~ - qn
order
in
as
in t h e
a polynomial this
the also
proof
of
degree yields
of the
assertion.
In
the
next
Zel I
and
Rk
{s
:=
zero
step ~par
6 ~:
we
examine
' restricted
Re
the to
s ~ max(2,1+Ik
growth
of
the
the
half-plane
)}
where
~k
has
they
functions
are
SI
'
continuous
Ehyp
'
and
free.
3.2.5
Lemma.
Proof.
The
asymptotic
The
function
logarithmic expansion
ZI
derivative
~
the
of
order
the
gamma
2
exactly.
function
has
the
121
~(Z)
(3.2.2)
= log
I
z - ~-~ + O ( z -2)
for
Izl
~ ~ ,
la r g
(cf.
[MOS],
p.
z I -< ~ - 6 <
18).
Hence,
12sll d
aI
=-d-
~(F)
IFI(~ ( s + k ) + ~ ( s - k ) )
"
4n
(s-½)
loG(-~)
+ O
for
We
conclude
by
integration
that
(s-½) -1
Isl
a logarithm
, ~ , s 6 Rk •
of
has
EIIRk
the
expan-
sion
log
H I(s)
: d
• ~\(s-½)
-
2(s-½)21og(s-½)> for
Isl
+ O(log(s-½))
, ~
, s 6 Rk
.
Consequently,
\{d e(F) ~
IEi(s) I = e x p
• Re ((s-½)2-2(s-½)21og(s-½)) for
From To
this
show
in t h e
This
formula
that last
proves
Corollary
the
we
see
order
equation
lemma.
2.2.6
yields
Zhyp(S)
= -
let
that
has
H IRk
is e x a c t l y
and
the
that
r
2 , tend
to
a logarithm
E tr X (P) {P}F
log
~ ~
the
we
log log
Isl
+ Re (O(log(s-½)))> , s 6 Rk
order
at m o s t
substitute
•
2 .
s = ½ + re ±~
infinity.
of
Zhy p
has
the
N(Po )
1
N(P)
I-N (P) -1
form
• N(P)
tr P > 2 (Re s > I)
-S
122 Hence, flog
Hhyp(S) I -< d(1-m(F)
-1)-1
E
N(p)-Re
s = O(m(F)-Res)
{p} tr P > 2 for
where
re(F) := m i n
{N(P)
> I: P 6 F
Isl
* ~ , Re s >_ I+6 > I
hyperbolic}
> I
This proves
3.2.6
Lemma.
The Stirling
B o t h the
formula
functions
for the g a m m a
-:hyplRk
and
ahypiRk_-1 are b o u n d e d .
function
says:
(3.2.3)
(cf.
log
F(z+a)
for
Izl
[MOS],
=
(z+a-½) log z - z + ½ log (2n) + O(z -I)
,~ ,
p.
larg z I ~ ~ - 6 < ]T , a 6 ~
of
(3.2.3)
the h o l o m o r p h i c
F {s 6 ~ ~ {O}: larg z I < ~-6 } and
tations
of
h a v e the o r d e r Hel I (Cor.
3.2.2)
we c o n c l u d e :
3.2.7
Lemma.
(Hpar. P)-1 IR k
The combination
3.2.8
Lemma.
-z-a
~ IN O
13).
As a c o n s e q u e n c e
(O < 6 < ~ )
fixed,
The
2.3.5)
functions
~ {s 6 ~ ~ {O}: larg z 1 < ~-6}
I . F r o m the e x p l i c i t and
Hpa r • PID
i
Hel I R k
have the o r d e r at m o s t
of the l e m m a t a
The f u n c t i o n
functions
3.2.5,
(HP) IRk
product
(Cor.
_- I , ZellIRk ,
represen-
2.4.22,
Def.
(Hpar-P) IRk
and
I .
3.2.6 a n d
3.2.7 yields:
has the o r d e r
2
precisely.
123
Remember
3.2.9
Lemma.
If
Proof. lows
that
ik
:= {s 6 ~:
The
function
T~ = O
immediately
(3.1.3)
and
then
(EP) Iik
E =
from Lemma
(3.2.1)
Re s ~ m i n ( - 1 , - I k l ) ]
entail
EP , a n d by
3.2.8. the
2s-I (~P)(1-s) = (~P)(s) • gl
(Needless
to
For
Re s > I
the
the
Dirichlet
every
By t h e S t i r l i n g
,
From
these the
As
1-s
6 ik
Our
next
For
{s 6 ~: formula
results
tion of
Sk
converges
< F ( s - ½ ) F ( s ) )T* i~(s+k)F(s_k)
s I
and
right-hand ~=~
task
~
Re
is the
s ~ 1
we
A way out
of t h i s
principle
(of.
preliminary
result
p.
245,
telling
Lemma.
(EP) IS k
has
Proof.
For abbreviation
put
mk
:: m i n ( - 1 , Ikl)
the
,
on
to
Rk
on
¢.)
(1.5.3)
is
where
bounded on
on Rk .
conclude Rk
has
that
the
the order
restricat most
is p r o v e d .
is to a p p l y 12.9) EP
order
Mk
(s 6 ~).
.
2. []
of
EP
in the
strip
not considered
representation
Theorem
fol-
equations
• ~(s)
and hence
s < max(2,1+Ikl)}
us t h a t
3.2.10
T• h I
representation
of the g r o w t h
inconvenience
assertion
is h o l o m o r p h i o
3.2.8 we
lemma
estimation
2 .
function
(3.2.4)
the
case
the
, 6 > O , especially
bounded
lack an explicit
[Ru],
the
the
from Lemma
:: {s 6 C: m i n ( - 1 , Ik I) < Re Re
has
(3.2.3)
s 6 Rk ,
side
s ~ I+6}
s i d e of
(3.1.3)
at most
equation
absolutely
is a l s o
the order
M S-Pm I H 1_s_----~m• ~ m=1
right-hand
function
series
half-plane
the
In t h e
functional
(3.2.4)
say t h a t
has
of the
function
yet. EP .
the P h r a g m 6 n - L i n d e l ~ f where
we have
is of
finite
at m o s t
2.
:= m a x ( 2 , 1 + I k I)
order
to u s e
our
of growth.
124
Fix
6 6 ]O,½[ . Obviously
it suffices
to show that the continuous
function f: Sk f(s)
is bounded
'~ ' { 21+6 "s m )2+26h := (-=F)(S) • exp \co--o~[ - k )
(Sk = topological
closure of
S k) •
At first we show that
f {s 6 ¢: Re s = mk}
For
t 6 IR
and
f {s6~: Re s = Mk}
we have / 2 I+6 e X P < c ~ ~6
If(ink+it) I = I (~P)(mk+it) I
= I(EP)(mk+it)I-
exp<-21+6
< CI (6) exp<,mk+it,2+26)
-< C I (6) exp<(2m 2)I+6)
since
The constant on
are bounded.
2+26 " Re[ (it)
])
Itl 2+26)
• exp(-21+6,t,2+26)
.
imk+iti2+26 = (m2+t 2) I+6 -< (2m 2)1+6 + (2t2) I+6
C I (6) > 0
due to Lemma 3.2.9 depends
solely
6 .
There exists
(3.2.5)
T > O , such that
cos ( ( 2 + 2 6 ) a r c t a n
Hence for these
t
we have
~)-<-2
-I-6 cos ~6
(t 6 IR,
Itl _>T) .
125
21+8 ) cos n~ Re [(Mk-mk +it)2+26]
If(Mk+it) I = I (EP)(Mk+it) I exp
= I(EP)(Mk+it)I
-< C 2 ( 6 ) e x p <
/ ~21+8~6 exP\c
IMk-mk+it I2+26 .cos <(2+26)arctan ~ )t )
IMk+itI2+28> e x p < - I M k - m k + i t 1 2 + 2 6 )
by
(3.2.5),
C2(6) The constant
As
f
C2(6)
> O
is continuous,
is due to
Lemma 3.2.8 and depends solely on 6.
f {s 6 ~: Re s : Mk,
IIm s I ~ T}
is also
bounded.
I
is of f Sk u s 6 ]O, 2(Mk_mk) [ there
By virtue of Lemma 3.2.3 and Lemma 3.2.4 the function finite order of growth. exists
Consequently
A > 0 , such that Iff(s) I -< exp < A -
for all Since
for
exp(slIm s[)>
s 6 Sk . f
is bounded on the boundary lines of
Lindel6f principle bounded on
Sk .
(cf.
[Ru], p. 245, Theorem
S k , the Phragm6n12.9) yields that
f
The proof is finished.
[]
We combine Lemma 3.2.8, Lemma 3.2.9 and Lemma 3.2.10 to obtain
3.2.11 Theorem.
The entire function
EF
has the order
Now a theorem of the theory of entire functions
(cf.
8.22.) yields the following improvement of Theorem
3.2.12
Corollary_L
The series
is
2
exactly.
[T]], p. 249,
].6.5.
126
r n- 2 - ~
E n>_O rn#O
for
There
no
convergent. given
at
3.2.13
every
example
It
the
is
end
of
2 .
Proof.
If
= O
Theorem
3.2.11).
already
used
points assertion 3.2.3.
T~
in
conclude
Now ma
3.2.4
is
=' E
the
order
of
shows
of
the
entire
with
a polynomial
PI
E
and
P
P ~ I
and
to
be
Lemma
has
not
E
nothing
a theorem
the
P
larger
[TI],
El
order
is
examples
at most
not
exceed
has
the
than
2
251,
(see
8.25.)
conclude
the
proof
from to the of
polynomial
we
the same
Lemma now
2 .
order
defined (cf.
the
proved
2 . Hence
in
a linear
P1
to
p.
have
corresponding
defined
or
function
is
in o r d e r
product
~2
does
that
(cf.
3.2.3
a constant
one
and
functions
function
way
p, p
entire
canonical
(n ~ O)
obviously
P~
have
for the ~, -2 is Z2
-
order
the
of
true
the
all
-I In
3.3.
the
the
the
¼ + r2n " )
for
be
where
An =
divergent
to
proof
that
now
that
E n~O I ~O n
seen
section
we
Iqnl - 2 - ~
series
the
is
In a s i m i l a r
to
up
apply
-ir n
that
known
again
irn,
Since
(Remember
We
3.2.12
Z
6 > O .
Both
at most
Corollary
and
n_>1
easily
Corollary.
order
~n 1 - 6
n _>0 ln#O
converge
is
z
,
at most in
[T1],
the p.
2 . proof
251,
of
Lem-
8.25.) f
is
a constant f
of
or
degree
a
linear at most
polynomial. 2 .
Hence
P =
PIe []
127
3,3
THE DISTRIBUTION OF THE EIGENVALUES OF
-~k
THE WEYL-SELBERG ASYMPTOTIC FORMULA
3.3.1
Notation.
For
zeros
of
E
which
are
contained
these
points
In
the
Weyl's (n
case
(counted
lie
of
of
the
let
interval
line
denote
on
hyperbolic
with
the
(Needless
s =
of
the
parts
to
say
of
that
½} . )
distribution
T
number
imaginary
a compact
Laplacian
~(F) Z I ~ a-~-n_>O i _
Re
the
the
]O,T[.
{s 6 ~:
group
formula
N(T)
multiplicities)
the
a Fuchsian
the
(3.3.1)
or
> 0
with in
on
asymptotic
A O)
T
fundamental of
the
domain
F
eiaenvalues
In
says:
as
T
, ~
,
as
T
~ ~
.
equivalently,
(3.3.2)
This
formula
unitary ing
~ d e~( F )
N(T)
is
generalized
dxd-multiplier
formula
due
(3.3.3)
T2
to
for
system
an of
arbitrary real
cofinite
weight
2k
group
by
the
with
a
follow-
Selberg.
N(T)
- ~
1 ~ -w~' -~(½+It)dt '
N de(F)--~ T2
as
T
, ~
,
-T where
~ = det
O
%~ = O
.
[He
p.
The of
134,
(See Theorem
proof the
of
(cf. 2],
IO.3.,
(3.3.3)
following
Notation p.
414,
[Ve
and
an
2],
1.5.8)
if
Theorem
6.3.
(3),
(4),
estimation
considerations
(see
T• ~ I , (i),
[Ve
3],
resp.
(ii), p.
of
the
error
also
[Ve
2]).
78,
term
~
(iii),
m I
if
[Ve
I],
Theorem
are
the
4.4.1.).
aim
128
3.3.2
Proposition.
logarithm {S
E ~:
valued
of Re
ZP
s 2
on
Let which
½,
s
]max(l,
arg(EP) is
~
denote
continuous
[½, m a x ( l ,
Ikl),~[
,
the
imaginary
part
of
the
on
Ik[) ],
s
~
, ½ ± irn if
T~
(n ~ O) }
and
~ I , resp.
~
real-
~ I
if
T~ = O . T h e n q~
1
N(T)
~
4~
1
~p'
# ~-(½+it)dt -T
: - arg(_~P)(½+iT) ~ for
(Cf.
[Ve
2 ], L e m m a
Proof.
We
{Re
qn:
n
the
zeros
of
T
]O,~[
\
6
of
the
of
Zp
is ~P
mentioned
in
bounded.
Hence
lie
<{rn:
n
rectangle
in
the
~ O}
x
section
---~ ~ , T
2.4
there
strip
U {Im
[I-A,A]
%
~I
{rn:
+ O(I)
n ~ 0].
{s
qn:
n
[-T,T]
(p.
exists E <:
A
I-A
~ N+I})
=:
96)
RT
the
set
> I , such
< Re
such
does
that
s < A]
that
not
.
all
Choose
boundary
the
contain
that
any
zeros
. With
N(T)
(remember
that
zeros
~P
ment
• -- l o g
18.)
already ~ I}
T
I
+T
of
that
principle
=
Z I nAN+1 Yn < T
Im
lie
qn ) in
and IR
n
denotino
(counted
with
the
number
of
those
multiplicities)
the
argu-
yields
I 2~i
(3.3.4)
Yn
::
[ ~RT
(-~P) ' (s)ds (ZP)
:
Z
ord o
s
(EP)
=
2N(T)
+
2N(T)
+
n
.
s6R T
(Remember gate
The {s
that
the
orders
complex
agree.
rectangle
RT
is
½]
. Let
E ~:
(with
Re
s =
positive
sense)
of
any
non-real
symmetrically ~RT, 1 that
with
denote
leads
from
zero
of
respect
that ½+iT
part
ZP
and
its
to
the
line
of
the
boundary
through
the
left
conju-
~R T half-
129
plane
{s
6 ¢:
through
the
(3.2.4)
we
s
right
~ ½}
to
half-plane
½-iT, to
9~,r
½+iT.
the
By
the
part
leading
functional
½-iT
from
equation
have
I
2~i
Re
(EP)' (s)d s _
f
(_:~)
~RT'I
I
2~i
[(EP)'(s)
f
~RT'r
+ 2 log
[(_-]TZ-P)
~I
M
ds
+ m =X ] (s_~Ipm + 1 - l _ p m ) + ~ ' ( s ) ]
I
(_:p) '
--(s)ds
+ T •
l o g .~ +
(½+it)dt + 2N(T) + O(I)
~RT, r for
This
relation
jointly
with
(3.3.4)
• ~
log
31
+
, ~
•
implies
T
!
~R
(T)
(EP)' (EP) (s)ds
: T
T
(½+it)dt
+ O(I)
r
+ 2 arg(EP)(½+iT)
+ O(I)
-T for
The
proof
is
finished.
Our
next
task
is t h e
we
consider
Therefor arg
Zell(½+iT)
well-defined
and for
estimation the arg
T 6 IR
of
terms
a r g ( Z P ) (½+iT)
arg
- hyp(½+iT). n
~
({±r
n
with
for
arg(ZparP)
function U {O})
arg
, since
.
The
continuous
argument on
---~ ~ .
T
, ~ • (½+iT)
, is
Hhyp (½+iT)
the meromorphic
¢ ~ ]-~,max(1, Ikl )] and its only zeros
{s 6 ¢: Re s ~ I} ~ [~,max(1 1,1kl)] are the p o i n ~
r n 6 IR>o
logarithms
The
: n ~ O}
Selberg zeta-function is even holomorphic on in the set
Zi(½+iT),
T
functions
are
the
½±ir n
imaginary
(n k O) parts
of
130
{s
6 ¢:
Re
s
_> ½, s
valued
on
3.3.3
Lemma.
We
The
tive
~
of
(3.3.5)
(cf. We
(n _> O),
s ~
[½, m a x ( l ,
and
Ikl)]}
real-
] m a x ( l , Ikl) ,~[ .
1 arg
Proof.
# ~+ir n
EI (½+iT)
= d ~e(F)
following
asymptotic
the
function
~(z)
[MOS],
have
gamma
= log
p.
z
I 2z
T 2 + O(I)
expanslon is m o r e
for
of
the
precise
I + O ( z -3) 12z 2
T
~ ~ .
logarithmic
than
formula
for
Izl
larg
z I ~ ~-6
deriva(3.2.2).
, ~
, <
18).
have
~--(s) -I
Hence valued
= -(s-½)
the
logarithm
on
] Ikl,~[
of
d ~O(F)
ZI
9(s+k)+~(s-k)
holomorphic
on
,
C ~
(+k-s
~ INo)
]-~, Ikl]
and
real-
satisfies
log Zi(s) = o~o(~)[, - ~ [ [ s - 2 ) 1,2 -2(s-½)21og(s-½)
for
I ~ ] + O(I) +(2k 2 - ~)log(s-~)
Isl
, ~,
larg(s-½)
I _< ~-6<~.
In p a r t i c u l a r , I arg
E I (½+iT)
= ~I Im
log
E I (½+iT)
= a~-(-F~)
T 2 + O(I) for
3.3.4
Lemma.
We
~ ~
.
have
a r g ( EparP) (~+iT) i . + T
T
z • (T~-log \
= -T ~ • T log
~I - d T
log
T
2 - log
T H 9=I
d sin ~ B j p > + 0(I) U p=mj+] for T ......~..
[]
131
Proof.
Corollary
2.4.22
and
a r g ( E p a r P ) (½+iT)
+
I j=1
(½d-~) J
Definition
3.2.2
= Im { - ( ½ + i T ) d T
(log F (½+iT+k)
yield
log
for
T > O:
2
- log F (½+iT-k)) d -(½+iT)
• log
sin
U
~Bjp]
p=mj+1
+ ½tr[I[,-~(½)
log ~1
] log(iT)-(½+iT)
+ T*(log r (½+iT-k) - log r (½+iT) - log r (1+iT))
+
The
1 + ~ Pm- 2
I log m: 1
Stirling
formula
(3.2.3)
Im log F (½+iT±k)
Noting the
that
by a s h o r t
3.3.5
Lemma.
Proof.
Take
function the
the
~ell:
Stirling
z : s-½
and : iT
thus
and
v,-1 3 E i=0 since
the
(p = 1,...,d)
F
note
~.(1) ]
T-I)
T > O:
+ O(I)
by
for
(1.5.7),
T ---~ ~
one
.
verifies []
Eell(½+iT)
of the
(3.2.3)
" --~ . 3
integers
arg
]-~, Ikl]
log
z = iT,
computation.
logarithm C ~
for
is r e a l - v a l u e d
function
formula
: iT
z : ~ I (s-~) 3 arising
The
implies
= T(log
½tr[IT,-~(½)]
assertion
.
, ~
product (cf.
defining Corollary
to the t e r m s
(--~j) ,s+k+l Consider
is b o u n d e d
the
for
......,. ~
Then
apply
log F (s-k)
with
log F(~-)s-k+l with the
imaginary
part
of the
expression
that
v,-1 ] = E ~j(1) i=0 ~jp(1),
whenever
1
~jp(1) does.
(j = 1 ..... p)
: ½dmj(mj-1)
run
through
It w i l l
turn
.
holomorphic
2.3.5).
log F (s+k),
T
the
set
out
that
,
{O .... ,mj-1}
132
arg
~ell(½+iT)
T
]0,~[
C
As
we
the We
: Im
Eell(½+iT)
is a b o u n d e d
function
do
not
maintain more
have
an e x p l i c i t
of
the using
arg
the
representation
Zhyp(½+iT)
notations
mk
of
demands
:: m i n ( - ] ,
Phragm&n-Lindel~f
IHhyp(~+iT)
I = exp(O(T))
for
Ikl),
some Mk
principle
for
Zhy p
ITI
Lemma.
We
Proof.
IT I
Formula
I = d ~~( F\) { 2 T
s ~ I ,
efforts.
:= m a x ( 2 , 1 + I k l ) .
we
, ~
shall
show
that
, T E IR,
in
d {
2 l°g ITI - T 2 + 2~(~-½) IT1)
~ ~ , T { IR,
(3.2.2)
~(a+iT±k)
Re
[ m k , M k]
have
logl_~i(d+iT) for
for
more
uniformly
3.3.6
of
.
estimation
Once
log
uniformly
in
d E
+ O(l°gl TI )
[ m k , M k]
.
yields
= log
T + i~
~±k-½ + - iT
+ O ( T -2)
~
T log
T + 2~(d-½)
for
0 < T
---* ~ .
Hence, ~w
Re
i~--~I(d+iT)
log
I-~i (~+iT)
"~(F) I : u~-
(2T 2 log T
-T 2
ZI(~+iT)
= ~i(d-iT)
, the
assertion
,
+2~(o-½)T) + 0(log T) for
As
+ O ( T -I)
is p r o v e d .
0 < T
~ ~ .
[]
133
3.3.7
Lemma.
log
We h a v e
I Z e l l ( ~ + i T ) I : O ( l o g l T I)
for
IT1
~~ ,
uniformly
Proof.
J u s t as in the p r o o f
product
defining
to the t e r m s
of L e m m a
Hell(~+iT)
log F (~±k+iT)
3.3.5
take
in
~ 6 [mk,M k] •
the l o g a r i t h m
and a p p l y the S t i r l i n g with
z = iT
and
formula
3.3.8
w a y the
following
estimation
(3.2.3)
obtained
with thus. []
has to be proved.
Lemma.
logl (EparP) (o+iT)
= T*~T + O ( l o g i T !)
for
uniformly
Since
of the
log F { \ ~ + i ~T ± k + l ]~
z = i - T . N o w c o n s i d e r the real p a r t of the e x p r e s s i o n 3 A s h o r t c o m p u t a t i o n w i l l p r o v e the lemma.
In a s i m i l a r
T E IR,
Zhyp'
~-I -hyp
are b o u n d e d
{s 6 C: Re s ~ I+6} , 6 > O ensues
from Lemma
3.3.9
Lemma.
I (Zhyp
3.3.7
° Zell
on e v e r y
(cf. L e m m a
and Lemma
T ---~ = , T 6 IR,
in
~ E [mk,M k]
.
half-plane 3.2.6),
the
following
lemma
3.3.8.
-~par " P) (Mk+iT) I
=exp( ~*'-~2 " ITi + 0(logLTi) )
for
ITI
~ ~ , T [ IR.
3.3. IO Lemma.
I (Hhy p " Hel I " Hpa r
_< exp
(-~ +2(Mk-mk))~
P) (mk+iT) 1
• bTI + O(l~ITl)
for
ITI - - - ~ ,
T 6 JR.
Proof.
The functional equation (3.2.4) implies that
It follows from Lemma 3.3.6
that
By the representation (1.5.3) with an absolutely convergent Dirichlet series,
Iq)(Mk-iT)I
is a bounded function of
T E IR.
Hence Lemma
3.3.9 yields the assertion.
3.3.11
Proposition.
~ W ( C - / T ( )for C
0
The function --,a,
T
E
l ~ ~ ~ ~ ( c 1 ~ +is i Tdominated ) by
IR, uniformly in
o E [rnk,Mk], where
is a positive constant.
Proof.
As a consequence of Lemma 3.3.9 and Lemma 3.3.10 there exists
a constant
c > 0 , such that the continuous function
which has a holomorphic restriction half-strip tion
HS
,
to the interior of the f/& is bounded on the boundary of HS . Since the func-
EP has a finite order of growth (cf. section 3.2, here it is
irrelevant that the order is
2
exactly), we conclude by means of
135
Lemma
3.3.6
Lindel~f modify
h:
~
, ~
Observe
of
[Ru], assume For
that p.
f
is b o u n d e d
245,
that
every
We use
Theorem
12.9
]f(s) ] ~ I £ > O
on
the
HS . T h e r e f o r
as
follows.
for all
we d e f i n e
Phragm6n-
the
we
Without
s 6 ~(HS)
.
function
by
:= e x p ( - ~ - e x p ( - i S s ) )
.
that
lh
=
6
(s) [ = e x p ( - ~ . R e
:= c o s ( B - M k) > O .
Im s ~ t o
[mk,Mk]
exists
x [1,t]
f
that
we
is b o u n d e d
~ > O .
such
[mk,Mk] this
. Letting
HS . F i n a l l y
Applying
Lemma
f
is of
finite
I f ( s ) - h ~ (s) I ~ I for e v e r y
s
(t ~ t o )
if
lying
in the w h o l e
t
infinity
we
tend the
to
limit
for
~
find
on the
, 0
that
for
bound-
modulus
rectangle If.h
and o b t a i n
[ ~ I Ifl ~ ].
P) (o+iT) [ T
is a l s o
and Lemma
3.3.8
IEhyp(O+iT) I ~ e x p ( O ( [ T [ ) )
,~ ,
valid
uniformly
for
T
we c o n c l u d e
for
IT[
uniformly
in
, -=
~ 6
[mk,Mk];
.
that
; ~ , T 6 IR, in
~ 6
of
s 6 HS ,
The m a x i m u m
is v a l i d
order
we h a v e
estimation
3.3.7
As
(s 6 HS)
inequality
consider
_< e x p ( O ( I T [ ) )
this
that
x [1,t]
I (Ehy p " Eel I " Zpa r
by s y m m e t r y
~ e x p ( - 6 8 e ~ Ires)
If(s)-h~(s) I ~ I
rectangle yields
Fix
to _ > I
Hence,
ary of the principle
exp(-iBs))
e x p ( _ E . e B Ires c o s ( B R e s ) )
growth , there
Since
is of finite order.
to s h o w
.
h 8 (S)
in
also
generality,
B 6 ]O,2~u[
where
f
principle
the p r o o f
loss of Fix
that
[mk,M k]
136
3.3.tl
Now proposition T
enables
us to e s t i m a t e
arg
Zhyp(½+iT)
for
)~.
3.3.12
Proposition t
arg
(Cf.
[Ve 2],
Proof.
For
Ehyp(½+iT)
Lemma
T ---~ ~,
T ~ {rn:
= O(T)
in the d e v e l o p m e n t
llog hyp S)i = exists
a > I
such
Re
Now
T h a+1
a zero
any
of
~ -hyp
g:
l-flog
ahyp- (a+it)
The
{z E ¢:
_> ½
with
3.2.6
that
½
for all
Isl
that
flog
In p a r t i c u l a r ,
of L e m m a
s) for
l ~ h y p ( a + i t ) - 11
select
we have
21.)
It is s h o w n
Hence, t h e r e
n A O}
Zhyp(a+it)
I
Zhyp(a+it)l
for all
T { {rn:
t 6 ~ •
t 6 IR.
n ~ O} ,
i.e.
½+iT
is not
function
IIm z I < T}
--~ ¢
,
g(z) :: ~ (Zhyp(Z+iT) + ~hyp(Z-iT)) is h o l o m o r p h i c . disk
{z E ~:
Let Iz-al
n(r)
< r} .
denote
the
number
• log
~a
_<
S
theorem
([B],
of
g
on the
a
~(r__~) dr r
a-½ By J e n s e n ' s
zeros
Then a
n (a-½)
of
p.
2,
1.2.)
_< f n(r) dr r O the
latter
integral
equals
137
1
2~
2-~
f l°glg(a+aei~)Ida° O log
<-~ where
the c o n s t a n t
and Lemma
3.2.6
t ~ ] . Note
Thus
0
the
s > O
that
g(a)
= Re : h y p ( O + ~ T )
into
In the
interior
of e a c h of t h e s e
Re a -hyp(O+iT)
> O
(intermediate
value
, arg
IT+I
o
theorem).
Shy p (o+iT)
for
Thus
and all
A T-a ~ I .
+ log 2 , c o n s e q u e n t l y
o 6 [½,a]
The
by the
iT
zeros of
there
is e i t h e r
subintervals or
a ~ mk
3.3.11
+ log 2)
subintervals
for e v e r y
d~0 + log 2 ,
for all
s(T+a)
(a(T+a)
)]
to P r o p o s i t i o n
I±T + a sin ~I
does not e x c e e d
-I <
sin ~)
according
IEhyp(a+it) I S e at
is split
I
sin ~0) + e ~ ( T - a
is c h o s e n
[½,a]
a
l°glg(a) l
IIm(a+ae I~ ±iT) I =
integral
n(a-½)
Observe
½ eS(T+a
so that
that
last
[(
-
Re Shy p-
(a+iT)
the v a r i a t i o n
< O
of
in e a c h of the s u b i n t e r v a l s
interval gl [½,a].
for e v e r y
the
a
function
is not l a r g e r than ~.
As
Im n h y p ( a + ± T ) larg Ehy p(a+iT) I
(remember
that
we c o n c l u d e
Ehy p(a)
: arctan
6 IR
and
Re E h y p ( a + i T )
Re E h y p ( a + i t )
<
~ ½
for all
t 6 IR),
that
larg Ehyp (½+iT) I < ~~ +
<(log ~ )a
<
(IT+I)~
(a(T+a)+log
_ <
( n (a- ~ )i
+ ~s) ~
)
2) + ~ ~ = O(T)
for
)
co
.
[]
We c o m b i n e 3.3.5, of this
the a s s e r t i o n s
Lemma
3.3.6
section.
of P r o p o s i t i o n
and P r o p o s i t i o n
3.3.]2
3.3.2,
Lemma
to o b t a i n
3.3.4,
the m a i n
Lemma result
138
3.3.13
Theorem
(Weyl-Selberq
asymptotic
formula).
For
T > O
we
have 4 ~I
N(T)
~ -~--(½+it)dt
= dO~(F) ~
T 2 + R T)
-T where R(T)
I
-
T+'T
with
The
known
up
estimate
log
: O(T)
= O(T)
T U j:1
2 - log
as
can
T
be
d U
sin
n~jp)
+ S(T)
p=m.+l J
, ~
.
improved.
The
best
result
to now,
is a c h i e v e d 468,
S(T)
S(T)
O
p.
T
I / • ~ ~+-dT
+ T
Note.
log
T
by a method
Theorem
2.29.,
due [Ve
to B. 2],
Randol
Lem/~a 28.,
([Ra],
cf.
Theorem
also
2.,
[Ve
[He 2], 3],
Lemma
5.2.11.).
3.3.14
Corollary_:
N(T)
We
have
= O ( T 2)
for
4n
the
estimates
T
, ~
,
~-(½+it)dt
= O ( T 2)
for
T
---~ ~
for
T > O .
Corollary
.
-T
Proof.
I
4~
N(T)
k 0
~ ~~'- - ( ½ + i t ) d t
every
is t h e
sum of
a bounded
and
2.4.17
yields
a non-negative
that func-
-T tion
of
T .
This
proves
the
assertion.
D
139
3.3.15
Corollary.
N(T)
:=
The
function
X
1
n->N+ I Yn < T defined
in the p r o o f
N(T)
(See also Theorem
Proof.
of P r o p o s i t i o n
= O ( T 2)
[He
2],
p.
for
437,
T
3.3.2
,~
also
satisfies
.
Proposition
2.13,
[Ve 2],
Lemma
12.,
[Ve
3],
4.4.2.)
Select
any
upper
bound
~
of the
set
{qn:
n A N+I}
and
fix
I c >
I
(qn : Re qn - ½)
arctan For
T _> 1 ,
N(T)
X n _>N+I 1~Yn
= N(1)
+
= N(1)
c X + 2- n - > N + l
I -< N(1)
+ c • X arctan n _> N+I 1-<¥n
T ~n
T __~n dx ] 2+ 2 - T r]n x
I-N+1 -2T 1 _
-< N(1)
c -< - 8
2T , { ~(½+it)dt
qn dt 2 2 nn+(Yn-t)
+ O(T)
for
T
,
T
-
(Corollary
: O ( T 2)
The T last
following ,~
even
theorem
for
specifying
increases
corollaries.
T ---~ ~,
the
the p r e c i s i o n
by C o r o l l a r y
growth of the
of
N(T)
2.4.17)
3.3.14.
+ N(T)
assertions
for
of b o t h
the
140
3.3.16
Theorem.
For
all
~ > 0 ,
I
--T I+~ -
I
4-~
(I+6)T
f
~-
(½+it) dt
+
O(T)
-<
-< N(T)
1 4~
f
~ - (½+it)
at
+
O(T)
-(I+~)T
I I+6
- --T
as
T
,
> ~
.
Hence,
N(T)
in d e t a i l ,
+ N(T)
for all
= d0~ ~ (F) T 2
~ > O
I de(F) (i+~) 2 ~
-< N(T)
First
Fix
for
T ----+ ~
,
have
I
T2
I • -- ~ *
- I+6
• T
log
T
+
O(T)
N(T)
+
_< ( I + ~ ) 2 d ~(~)~
Proof.
we
+ o ( m 2)
6 > 0
T2 -
and
(I+~)
let
as
. ~I m* • T log T + O(T)
T
---+ o~ .
T > 0 .
inequality:
I 1+~ T 1 4~
q)' ~ (½+itldt = ~
S --
I 1+~
1 - - T 1+~
I n~N+1
S O
+(t_~n)2 + ~+(t+,n)2 ) at + O(T) (Corollary
I --T 1+S
1
I --T I+~
S
2nn
z
: 2--{
n~N+1
I
~ + (t-Yn) 2 dt + ~-~
O
J 0
Tn < T
2~ Z n>-N+1 Tn _>T
1
-l +-eT 0
2~ n I n~N+1
dt ~+(t+Yn
)2
+ O (T)
2 2 r~+ (t-yn )
dt
2.4.17)
141 I
--T 1+g 2 nn - ~ d x - ~ qn+X
I Y -< 2-~ n>N+1 Yn
I +~-~
f n->N+1
O
2 ~n dt 2 ~ 2 2 ~n + ( ~ ) Tn
I --T I+~ I
+~-~
S
0
= ~(T)
Second
2 ~n 2 2 hn+Yn
z
n->N+1
+ O(T)
- -
dt
T
, ~
for
+
O(T)
inequality: (I+~)T
(I+~)T
I
<0'
- 4-~ ~ -(I+~)T
7
I
(½+it)dt
f
- 2n
0
2~ m2_%+1( 2+(t_Yn)2 +
2nn ~+(t+Yn)2/ dt + O(T)
(Corollary I
E
(I+~)T jr
2rln
O
Dn2+(t-Yn )2
-> 2--~ n>_N+1 Yn < T
_ ___I E 2~ n->N+1 Tn < T
(2 arctan
_> 1
arctan
~
I
(I+~)T-y n + 2 arctan
~n
E n_>N+1 Yn
(with
dt + O(T)
~~T + a r c t a n
defined
(~
~
of C o r o l l a r y
C ) + o(m)
n_>N+1 Yn < T
= ~(T)
- ~-7 "
N(T) T
~ )_ I + O(T) ~ n_>N+1% Yn < T
+
0 (T)
Yn ) + O(T) ~--
in the proof
I + ~
-[In -
3.3.15)
2.4.17)
t42
= ~(T)
-
= ~(T)
since
The of
~ ~
~(T) T
"
+ O(T)
N(T)
for
O ( T 2)
=
other
assertions
Theorem
3.3.13.
3.3. 17 c I (g),
T
• ~ I dN(t) O ~
---~ ~
for
of
,
T
~ ~
the
+ O(T)
,
by
theorem
Corollary
are
now
an
3.3.15.
immediate
-> O ,
For
every
such
a~(~) 2---/~-
~ > O
there
exist
constants
that
log
T
r-2n+ X
X n>_O Rer
i c 1 (~ )
(l+s)
Z n > 0 Yn* O
Proof.
rn
{
The
~-I n
diverges
{Re
for
qn:
n
finitely
r-2n +
X
n->O Re r n < T rn
[qn I-2
log
T + c2(g)
for
all
(cf.
converges
p.
T > 0
36 ).
k I]
is
many
n E ~
z
I~t
bounded,
-2 =
n>1 Im qn < T
-4 X Yn n_>N+ I
converges,
Hence,
o
Z r -2 + Z --y2 n >_ O n n~N+1 n O
+ O(I)
Yn < T
* O
=ST ~-~I
d (~(t)+~(t))
+ 0(I)
0 _
.
n~l
set
only
x
if
lqn j-2
n_>1 Ima < T ~n
_< (l+s) 2 o - ~
Thus
consequence []
Corollary_ m c2(~)
~ - ~
N(T)+~(T) T2
+
T 2 f N(t)+N(t) O
t 3
dt
+ O(J)
for
T
----+ ~
.
and
143
Now
the
assertion
For
a lot
of
follows
special
from
cases
Theorem
of
F
D
3.3.16.
the
growth
of
4~
(½+it)
dt
-T is k n o w n totic
be of
formula
3.3.18 lar F
to
lower
yields
Example. (i.e.
T ~ = O)
= d0~(F) --~
with
3.3.19
[He
S(T)
on
2],
508,
following)
with
the
T2 + T
have
~(S)
: V~
Riemann
system
log
for
the
, ~
trivial
{
T U j=1
(cf.
T* : also
T
2s-I - ~
It
follows
+ S (T)
multiplier
system
I , and I],
F(I-s) F(s)
p.
= O(T
by Theorem
according 162
and
to the
~(2-2S) ~(2s)
for
we
Itl
(1+2it) + 2 --~(1-2it)
see
that
~
~
,
!
immediately
F,
Hence,
(3.11.9)
: O(logltl)
f ~(½+it)dt -T
on
J ) sin ~B~p
thus, -
is a r e g u -
2 k 6 IR
d ]] p=1
[Ma
log ~ - 9(½+it) - ~(½-it) + 2
53,
X
.
Here
(2.4)
zeta-function
p.
asymp-
....., ~ .
group,
scalar-valued
SL(2,~).
{(2S-I) ~(2s)
[T2],
T
of weight
2 - log
T
F(s-½) F(s)
~0 (½+it)
N(T)
Weyl-Selberg
F . Then
for
(2.3),
of
the
is a c o f i n i t e
I (-dT
•
group
= -2
and
so t h a t
order
F
of
Consider
we
(3.2.2)
that
domain
(2.2),
<0'<0(½+it)
From
precise
2
d×d-multiplier
the modular
p.
than
= O (lo--~T)
Exampl e .
X ~ I
the
Assume
is a f u n d a m e n t a l
N(T)
order
l o g T)
3.3.13
for
that
T ---~ ~
.
.
144
N(T)
A more
(F) T 2 + O ( T 4~
-
precise
N(T)
=
is s t a t e d Theorem
version
T2
4~
in
[He
9.4.
(see a l s o
[He 2],
The
group
theta
8:=
and
the
log T)
for
T
for
T ~
assertion,
T
511,
+ T'~(21 +log ~
(2.12).
T - ~(I 1 + log ~)
p.
{(~
the
510,
~)6
congruence
~)
T + O(io~)
Moreover,
by
[T2],
p.
~
,
181,
• T + O(log
for
T)
T
~
[]
(2.10)).
SL(2,~):
ab ~ cd ~ O mod
2}
subgroups
equipped
with
examples
for g r o u p s
4~
p.
+ O(T
we h a v e
= ~I T log
N(T)
of this
- ~2 T log
2],
= ~2 T2
log T)
trivial such
multiplier
system
X ~ I
are
further
that
-~(½+it)dt
= O(T
p.
528,
538,
F
we do not
log T),
N(T)
: O(T
log T)
for
T ---~
-T (cf.
For
[He 2],
generic
resp.
I
- 4-~
p.
~ ~~'- ( ½ + i t ) d t
[Hu]).
know
the a c t u a l
A strong
order
version
of g r o w t h
of the
of
N(T)
so-called
-T Selberg that
conjecture
claims
that
there
might
exist a constant
6 > O
such
145
I 4~
The
validity
infinitely
many
series
Sarnak
proved F
values
in
the
the
-~k
Selberg
becoming
[PS
that
authors
details
see
3,4
WEIERSTRASS
[PS
off
make
I], [PS 2],
FACTORIZATION
~0
counted (n A O) a zero
3.2.11,
with
and
the p o i n t s
of
~
tation
i
2
I n = ~+r n
of o r d e r
2di 1-qn
contained
a polynomial
is a n e n t i r e
multiplicities,
where
Hadamard's
ZP
1
n
= ¼
ENTIRE
half-plane theorem
of d e g r e e
Re
for
true. of
eigenzeros
of
s = ½} , t h u s matrix.
For
[He 5].
S •
CONSTANT
the
of o r d e r points
of
1-qn , {s 6 ¢:
at m o s t
are
many
FUNCTION
function
([TI],
is f a l s e
many
scattering
[Ve 3],
and
deformations
{s E ¢:
p.
2 , such
1-qn
2 . Its
½+irn,
-Ak
is a n e i g e n v a l u e
(n = I ..... N),
in t h e
conjecture
of
divergence
Phillips
sufficiently
of the
exactly
the
conjectures"
is an e i g e n v a l u e
if
factorization Q
are
However,
line
AN ANALOGUE OF THE EULER-HASCHERONI
By T h e o r e m
even
sufficiently
[DIPS],
OF THE
.
the e x i s t e n c e
quasiconformal
the
determinant
and
1.6).
"standard
that
, ~
yield
Selberg's
Fo(q)
move
T
-~k
Equivalently,
zeta-function
more
2] t h a t
subgroups
of the
of
section
show
for
immediately
certain
disappear.
poles
would
(cf.
I],
provided
= O ( T 2-6)
eigenvalues
I-I n
[PS
congruence of
formula
distinct E n_>O I #O n
In p a r t i c u l a r , certain
~' ( ½ + i t ) d t
of t h i s
of the
generic
] -T
½-ir n
(note t h a t
½
is
of m u l t i p l i c i t y (n A N + I ) ,
Re
s < ½} .
250,
8.24.),
that
zeros,
EP
t h e poles
By virtue there
has
the
d¼)
of
exists represen-
146
(3.4.])
(EP) (s) = eQ(S) (s-½)
2dI ~ •
H
(1 + ( s@- ½ )r2 ) e2 x p
n>O r %0 n
n
s
H n=1
I+
exp
-
qn
2
s-½ h ~ / e x p
+
n
+ - 2~ n2
qn
R (I + s-½ nkN+1 ~n)(1
'
(s-½)2)r2
-(-~) ~ 2nn (s
2 2 ~-~n (2.2,2) rln+Y n )
(s-½) 2
~
+ ~+Yn
(s 6 ¢ ,
The
right-hand
sidered to the
zeros
of the
3.3.17).
(3.4.2)
P ~ I , the as
We m a y
determine
]R 9 s
tor
pleteness, finition
we c o n s i d e r the
and
The
al,
on the
e Q(s)
product
corresponding
(cf.
is,
Corollary
con-
3.2.12,
form
+ aO
a° £ ¢ .
right-hand
the
aI
If
T* = 0 , i.e.
side
substitution
by c o m p a r i n g
logarithmic
coefficient
function Trace
Z
is a l s o
a°
is o n l y
Formula.
coefficient
equation
coefficients
in the
qn ) •
factor
I ..,.. (EP) (s) Q(s)
under
the
canonical
a2,
product
of the
Resolvent
3.1.4
from
Yn =Im
of
(3.4.1)
s [
, 1-s .
as
P ~ I •
~ +~
the
- ½ '
= a2(s-½) 2 + a] (s-½)
a2
entire
by the
s-½
coefficients
if
(3.4.1).
the
s -!2 , the
is i n v a r i a n t
aI : 0
apart
represent
canonical
E(s)
equation
of
We
the
Hence,
since
(3.4.1)
function
Q(s)
and determine
for
of
as a f u n c t i o n
Corollary
Well
side
n n = Re qn
ao
of a s y m p t o t i c
for
of b o t h
is n o t of v e r y determined
corresponding
only
asymptotic
derivatives
Nevertheless,
to be c o m p u t e d
(3.4.1)
the
to
in t h i s real
high
expansions sides
of
interest,
up to a c o n s t a n t for the Z
sake
explained
section.
For
s > m a x ( 1 , I k I)
expansions of c o n t i n u o u s
fac-
of c o m in De-
this
purpose
and
compare
logarithms
of
147
both
sides
in
(3.4
•
I)
when
d
:=
s -!
6 IR
2
tends
to
+~
We
•
have
(3.4.3)
log(EF)
(~+½)
=
log
+
The
growth
The
precision
in
the
the
of
of
proof
O(log
Stirling
each
of
these
and
formula
has
to
be
too.
Barnes
+
log
Zell(O+½)
(~ > m a x ( ½ ,
is
expansion
O(I) - terms, the
Ehyp(d+1)
logarithms
3.2.5
for
log
(u+½)
asymptotic
Lemma
d)-
+
log(ZparF)
of
the
Ei(d+1)
to
be
of
specified
log
now.
Ei(d+1)
increased
as
A useful
help
G-function
Ikl-1))
(see
we
as
now
is
want
the
[Vi],
stated to
get
following
p.
238):
(3.4.4)
log
where
G (x+a+1)
A
is
=
lim n ~
- x +2a
the
n
and
a
Apply and
is
an
Remark formula
3.4.1
Lemma.
log
n ~
12 +
Kinkelin
constant.
2
.22
3.1.3,
the
(3.4.4)
to
For
d
: a~ (~F )
A
+
I 12
+ ax
3 2 4 x - ax
log
x
+ O(
1,28242713
...
for
)
x
)j-oo
. . .. . n n
n I + 2 + I~
arbitrary
Z I (o+I)
- log
+
11 A
log(2[)
n • e
complex
:
2
(cf.
number.
formula
(3.2.3)
for
the
gamma
deduce
<_ +
we
,
4
Stirling
~
[K])
have
2 log
~ +
I log(2~)
la 2 + + ~
(k 2 - ~ 2 ) l o g +
2 log
A
+ O(
)
func~on
,
148
Taking one
notice
also
of
Corollary
establishes
2.3.5
both
the
resp.
Corollary
followina
2.4.22
assertions
by
and some
(3.2.3), computa-
tions.
3.4.2
Lemma.
For
~
- , ~
we
have I),--I
l o g ~~(o+½)ei$ =
E
½d(1 -
)(1-v~) +
j=l
E j=1
- ~
3
(I -
E i=I
v. 3
) log(2~)
+ ½d(vj-1)
l(a4(1)
+ ~j(1))
- loqo
log(2~v4)3
3 %) .--I
I 2 log v. ]
3.4.3
Lemma.
3 v 3
For
log(ZparP)
(~+½)
~
=
l(j
E 1=I
, ~
-T * o l o g
T +
[ -dT
log
we
2 - log
(1))]+ 0(})
+ h
have
a
d
j=IH p=m.+IR
sin
~Bjp
+
T*
- log
.ql] " o
3 +
k d'r -
E
- ½tr
~(½)
- T*k
+ M
- log
j=1 - ½dT
log
2 - ½1og
U j=1
d U p=m
sin +I
j ~6jp
T• - --2 loq(2~)_
- ½log
B1
3 M E log (Pm-½) m=I
-
+
0(~)
Since of
log
Lemma
hand
3.2.6),
side
lemmata
Ehyp(J+½)
in
the
equation
stated
above.
: O ( m ( F ) -u) asymptotic (3.4.])
for
o
behaviour for
~ =
s-½
, +~ of
(cf.
the ~
~
the
logarithm is
development of
the
described
by
left the
149
Notation.
Let
L 6
be the n u m b e r
of
n
6
IN
(~=~X
with
r
O
O
n
6 i]O,~[
< ¼). n
a = S-½ > max(½, [ki-½)
For
(3.4.1)
is g i v e n
a logarithm
of the r i g h t - h a n d
side
in
by
(3.4.5)
a2a
2
+ ala
n E-> O
+ a°
a2 I + ~ n
log
ir < O n E
+ 2d~log
[log<1+a
n
2 ] + iN - ar~ n
2
o_ o r
a
E1)
(::
Z2)
a2
n
>O
(:=
n
N
2
+
E n:1
[log
+
~n2N+l
log
2
(:= E 3 )
2qna
] +-2-~+-T-)r/n+¥ n qn+Yn
q2n_ ¥2 (:= Y4 )
2 2 + , 2 2,2 0n+T n tqn+¥n)
Obviously,
EI
=
i~L
+
[log
E
>- O ir
n
=
ilL
+
2L
--~ -r
-
I
n
log
a
n
z
-
)
2
[2 log
n >- O irn<
r
0
a n
for E2
is d o m i n a t e d
by a m u l t i p l e
of the c o n v e r g e n t
+~
d
series
a
4
E n _> 0 r >0 n
N
Z 3 = N log a -
Z n=1
N
log qn -
N
X __I a + Z n=1 q n n=l
12 a2 + O(ID) 2q n
for
a
-4 r
n
150
~-4
[
x n->N+1
:
log
I+ ~ Yn
2 +
X n>-N+1
-
-
x
2
n>N+1
2
nn+T n
2
2 22 (qn+Tn)
(~
+ Tn
2 -
X
log
1
+
n->N+l
Fix
d > O .
dominated second
first
sum c o n v e r g e s
the
of
sum
last
sums
log (1 +
of the
Yn-4
and
converge
2 q n a + ~ 2n
--n
Corollary
It is o b v i o u s
2.4.17
~+Yn
expression
convergent 2.4.17,
converges
series the
X n2N+1
third
the b o u n d e d n e s s
since
of
T
-4
,
it is the
n
one
again
{qn:
n
by the
~ N+I}
because
)=
O ( ~~nC ) a +T n
2+y2
and
X
nkN+l
in this
by C o r o l l a r y
X nAN+I
+
Tn
by a m u l t i p l e
convergence Both
The
2 h n a + ~ 2n
--~
implies
the
for
convergence
of
n
)
X n2N+1
qn 2 Yn
that
2 qna+ 2
X
log(l+
2qna
- - ) :
n>-N+1
a2+y2n
2 q n ~ + ~ 2n
X n->N+1
2n
+ a2+y2n
2
X
22
n->N+1
a +Yn
2~na + o(1 )
n2N+l
a
n_>N+l a
+T n
+Yn for
Hence,
expression
(3.4.5)
has
the
expansion
C~
~ co
.
t51
(3.4.6) 2 aO
+
i,L
-
2
X n >- O ir < 0
log
[rnl
-
X n=1
log
qn -
X n_>N+1
log
I +
n
2"n O"
+ (2d~+2L+N)-logo+
z
2 2 O +T n
n_>N+1 N +
-
al
I
x
n:1
2Nn x
qn
n>N+1 -
2
2/
"
°
qn+Yn 2
+
a2
N
I
X n _> O ir < O n
r2 + n
2
i
1])
X n=1
~--~ + X 2q n n>-N+1
-
+
2 2,2 (qn~Yn;
2
+ ~ Tn
"a
2 +
X -> O >0
n r
log
I + rn
X n_>N+1
r2
Yn
Yn
n
+ O(I)
3_m.4 . 5
o
, +~
Proposition.
~=p
where A
for
o)
For
:= lira ( T ~
R1(t)
continuous
=
X n >-O O
N(t)
+ N(t)
logarithm
of
o > 0
there
I --
x
2 rn
+
(~) d e- ~
I
2 Tn
n>_N+1 Yn < T
- d O~( F )
exists
t2
(cf.
(EP) (o+½)
has
log
Theorem
the
T -2
T
R I (t)
S O
t(t2+~ 2)
3.3.16)
asymptotic
.
expansion
(3.4.7) 2
N a° +
i~L
-
2
X n > 0 ir < O n
loglr n
-
E n:1
log
2 rlnO +
(2d I + 2 L + N )
• log
4
/ + k ~al
N
-
X n=1
o +
X n>_N+1
2
2 o +Tn
I
~]n
X n>N+1
2 2 qn+Yn
• o
qn
-
Z n_>N+1
log
I + -~ ¥n
dt ]
152 2
+
- ~EP(~)
- n O> X _ ir < O n
Ir2 + NZ n:1 n
2n12 ~n
X n>N+1
2
[ qn-Yn I] 2 2 2 + ~ (Qn+Yn) Yn
+ de(F) --~-
_
de(F)
Proof.
02 log
For
~ + o(I)
o > O
E log n ->O O
for
fixed
I +
+
0
~ +~
we have
Z log n>N+1 Yn < T
n
I +
-
= \/d ~4~-e(F) T 2 + RI(T)
) • <~2
O
: ] log O
I+
d(N+N) (t)
t (t2+o 2 )
) log (I + ~02) + d~(F) ~
o2 O~ t2+o t 2 dt
+ 2 o 2 T R I (t) dt 0 t (t2+~ 2 )
~0~(F) o2 ~0(F) 02 : a ~ + d ~
(log T - log c) + 2o 2~
~ R1(t) dt 0 t (t2+o 2 )
+ o(I)
Note
that
R] (t) = o(t 2)
N o w the a s s e r t i o n s
The e s t i m a t e
for
follow
T)
~~
by m e a n s
RI (t) : o(t 2)
T R](t) 2 S dt = o(log O t(t2+o 2 )
t
by T h e o r e m
of the e x p a n s i o n
for
t ---~ ~
for
T
~
yields
for
T ~
~
.
3.3.16. (3.4.6).
that
[]
153 3.4.6
Notation.
y~p(d)
:=
For
/ lira [ . T~"
d > O
n
N
I 2
Z n->O Re rn< T r
put
I
Z
rn
n_> Z N+I Im qn < T
n:1 2 (qn- ½)
2 (qn-~) 2
2 (qn-½) 2 )
# O T
R1(t)
)
f
- d~2(~F) log T - 2
dt
0 t(t2+o 2 )
Tcp(o)
+
1
z
N
=
Z
n _> O ir < 0 n
Thus
Lemma The
1
r 2n
asymptotic
expansions
~ (ZP) (~+½)
are
given
Z
2q n
,
n>-N+1
of a c o n t i n u o u s by
3.4.2
and L e m m a
3.4.3
unknown
constants
are
of t h e s e
2
n=1
2 2 nn-Yn
[
(3.4.7) jointly
now determined
"
~qn ~n
Yn
logarithm
of the
on the with
1 ]
2)2 +-~
2+
one
hand
(3.4.3)
and
function
by L e m m a
on the o t h e r
by c o m p a r i n g
the
2q n that
Z nkN+1
- o(I)
a2 = Y E P
(3.4.9)
aI :
N Z n:1
:= lim
o
to o b t a i n
will
,
2 qn
- lim
a
yZp(~)
dT
~ + Z qn n_>N+1
- log
for
+~
U2+y2 n
(3.4.8)
A formula
for
T U j=l
be s t a t e d
2
2 2 Dn+¥ n
d U sin ~ p j p = m .+I 3
(T'log
log
+
I ~ - log
~ + @(Y~p-YEp(o))
later
(see
61
)
(3.4.16)).
We h a v e
(3.4.10)
R I {t) (t I y:p-y:p(a)
= 2 lim
~ ~
2+ 2
I )dt t2+ d' 2
hand.
coefficients
expansions:
We o b s e r v e
3.4.1,
154
where
the i n t e g r a l
exists
As an e a s y c o n s e q u e n c e
(3.4.11)
for all
of
(3.4.9)
(
T* = 2 lim
c log ~
R(t)
for the e r r o r
and
lim ,
O'....*co
Moreover,
~,o'
C
..~.o~
6 IR \ {O}
(3.4.10)
7 R1(t) t
.
we get
(t
I 2j~,2
t21+ 2) dt)
.
0
term
T* ( T d U p=m,+l sin ~ j p > + S(t) = ---~ t log t + t • -r~l T * - d r log 2 - log j=l
3 of T h e o r e m
3.3.13
(3.4.12)
2~ 7 R(t) O t
- log
the f o l l o w i n g
.
I t2+a2
formula
is valid:
dt = -T* log o + T* - dT log 2
T d N H sin + 2~ f S(t) dt j=1 p = m +1 ~gJP u t ( t 2 + ~ 2) 3
As a l r e a d y m e n t i o n e d
in s e c t i o n
for
the last t e r m on the r i g h t - h a n d
is
t
, ~ , hence ~
equivalent
(3.4.13)
for
~
it is k n o w n t h a t
, ~ . By these
results,
N I 2~n E -- + E 2 2 n=1 ~n n_>N+1 nn+y n
log ~I
S(t) side of
equation
= O
(3.4.9)
is
to
aI =
+ lis O~ ~ Note
3.3,
.
I
lis 2o S ~'~ ~ \ O
R(t)-R I (t) t
t2 (i +02
i )dt) t2+o '2
that
R ( t ) - R I (t) = - 4-~
~-- ( ~ + i x ) d x - N(t) -t
In p a r t i c u l a r , P~I.
equation
(3.4.13)
ensures
once more that
aI = 0
if
1
155
Dividing under
equation
the s u b s t i t u t i o n
factorization
3.4.7
(3.4.1)
of
Theorem
by
P
s
, 1-s ,
(WeierstraB
is e q u a l
we get the
Z
is i n v a r i a n t
following WeierstraB
factorization
of
• (s-~)1
I +---7 r n
U n->O r#O n
s 6 ¢ . The coefficient
The c o e f f i c i e n t 3.4.7
that
E .
~(s) : exp (yEp • (s-½)2+a o)
for e v e r y
and remembering
0
of the l i n e a r
a
o
E ).
exp
is s t a t e d
in
-
2 r n
(3.4.16).
t e r m in the e x p - e x p r e s s i o n
in T h e o r e m
to
N a I - n=1 I
I ~n
2Dn I n~N+1
2 2 ~n+Yn
Hence,
aI =
(3.4.14)
N I I + Z n=1 ~n n~+1
2N n 2 2 ~n+Yn
~' = -½ ~-- (½) - log ~I +
(cf. C o r o l l a r y
(3.4.14)
(3.4.15)
lim lim O ~ co ~' ~ co
(3.4.7)
(3.4.14), 3.4.3
with (
we r e p l a c e and we r e g a r d
to o b t a i n
E m=1
I
Pm -½
2.4.17).
Comparing
In
M
(3.4.13)
2o
we get
7 R(t)-RI (t) ( O
a2,a I
t
I
I ,[)dt t2+o
t2+ 2
by the e x p r e s s i o n s
(3.4.10),
Lemma
3.4.1,
given
Lemma
in
):
log %1
(3.4.8),
3.4.2 and L e m m a
156
(3.4.16)
a° 6 d ~
+ Z
½1og(2~) + ]-~ + 2 log A
-½d(1-
2 Z log n_> O ir < O n
) lc~j(2n)+~d(v~-l)lcx.j(2m0j)---~log
J=]
- ½d~
+
3
log
~),
u, 3
2 - ~log
T d j=IU p =+mI N
sin
- i~L
Z l(c~_(1) +
3
%* 2
~Bop~
Irnl
(1))
i=I
log(2,)
- ½1oq_ ~I
3 M
Z m=1
D2 + --~ ) Yn
N
log(pm-~)
+
Z n=1
log
qn +
Z n_>N+1
log
(I
X),-- 1
+ lira d ~ O~" -
k 2 - ~)
+
-
+ k(dT
-
/ + (-dT
log
Z j=1
- T*O
log
- ½tr
2 - log
T U j=]
o - 202
lim
O'~
Z n->N+1
A remarkable R I (t)
= N(t)+N(t)
parameters
2DnO } 2 2 + 2~ o +Yn
relation
of
F
between
~(½)
d U p:m
- T*k
sin 3
f
.
the
error
and
follows
the
- 2d$
nBjp
t
i2[
t2 ,
1=I
- 2L-N+M
+ T*
)
• log
- log
~I )
2+2
?
dt
'
2
)
term
zeros
from
t2+~
of
this.
P
resp.
and
o
• o
+I
~Rl(t) co O
- d e(F) X
~. 3
3
2Bj)
k
-
%(I))
j=1
the
157
(3.4.17)
I~
lim
[ 2a 2 l,~ im 0 ~
+ logI ~
g-~ co"
~I RI I(t)t (t 2 +- 2 0
Z 2qn 2 2 n_>N+1 o +Yn
+o
I 2 ) dt t2+o
+ dT log 2 + log
T d U U sin ~Bjp j:1 p=m.+1 3
- T~ + log ~I)] I),--I
]
+ k (d'[ -
Y E
2B~) 3
j=l For
the
ing
theorem:
3.4.8 tion s-i
sake
of
Theorem HP
- itr
clearness
we
(canonical
admits
the
¢(i)
- T*k
- 2d~
summarize
our
factorization
following
- 2L - N + M
main
of
canonical
HP
results
).
in t h e
The
factorization
.
entire
in t e r m s
: 2d¼ (ZP) (s)
U nkO
= exp
I +
|/\ a 2 ( s - i ) 2 + a I ( s - i ) + a O \/|
(-~)
exp
-
2
r 2 n
r
n
(s-i) U n=1
U n>N+1
I +
exp
-
~
I +
2
- + qn -~ 2(qn-i)
I +
exp
s i q n -i
+
s i 2 (-2)
I (
I 2 +
2 (qn-i)
where
• (s-i)
2 ) 2 -- i (qn-~)
(s 6
¢)
follow-
funcof
158
a 2 = yEp
: lim o ~ ~
lim T ~ ~
(
E n ~ O Re r
N E n=1
I r2 n
1 2.=.(qn-~)
n >_N + 1 Im qn
2 (qn-~) I 2" +
T
-d
c°(F)
log
N X n=1
=
~ I q n -½
+
By
is
virtue
the
of
growth
cial be
explained
smaller
3.3). that
Thus there
Theorem
of
groups
the
than
O
3.3.16,
error
we
2
< 6 <
I
(3.4.19)
for
the the the
a constant
R I (T)
this
N(T)
(3.4.14),
we
term
(cf.
suppose
exist
have
RI(T) order
the
improper
every
also
formula
as
T
, ~
growth
of
of
considerations
6 > O
, such
as
T
3.4.6,
(3.4.8)),
(3.4.9),
(3.4.13)),
(½+it)dt
t
RI(T) .
For
a
RI(T) at
the
= o ( T 2) lot is
end
of
known of
for speto
section
in t h i s
section
as
~
that
~ ~
equivalent
integral
~ > O , and
extimate
final
is
T ~, I _f T ~ _ + ~--~
the
mentioned
: O(T 2-6
O for
see
examples
R I (t) Now
Notation
(3.4.16).
F , however,
(3.4.18)
For
by
)
~)dt
1__w + I qn-~ ~n-½)
E n_>N+1
(cf.
a°
2 (~n-½) 2
0 t(t2+
(cf.
i
R I (t)
T - 2 f
2~
a.
2
.
to
= O ( T 2-6)
T
I - t2+d2
dt
converges
absolutely
159
R 1 (t) (3.4.20)
By
~ O
(3.4.11)
we
I__/___ d t t2+o2
t
obtain
= O ( o -6)
an estimate
even
as
sharper
than
o
,
(3.4.20)
if
0 < 6 < I :
(3.4.21)
S O
Especially
we
(3.4.22)
t
now
~ t2+o
o
o
; oo .
have: / <
= lim T ~
Y :-F
2
Re
N E
I
n=1
2._.(qn-~) 2
r
r
-
- -I r n2
E n _> O
~O
n
Z n >_ N + 1
I
{
I
+
\ 2 ( q n - ~i)
2
-a--~,
los
T
2 (-q n _ ~ )~ 2
Im qn < T
and
in t h e
formulae
(3.4.23)
(cf.
stated
y~p
(3.4.10)).
above
- yEp(o)
For
example,
lim o~ ~
Consider
also
20 S O
(3.4.9),
substitute:
: 2 7 RI (t) O t equation
R(t)-R
(3.4.24)
we may
1 t2+02
(3.4.15)
dt
can
now
be
stated
as
I (t)
dt
: log
31
t (t2+o 2 )
(3.4.11),
(3.4.13),
(3.4.22)
suggests
(3.4.16) and (3.4.17)
under
this
aspect.
The
form
logue
of
of
YEP
the
Euler-Mascheroni
factorizations
in
of
the
entire
constant functions
to
interprete
YEP
y . Compare ~
and
~:
also C
as a n the
ana-
WeierstraB
, C ,
S
~(s) to
2
:= ½ S ( S - 1 ) ~
the
Selberg
F(~)~(S)
zeta-function
, respectively.
By
[T2],
which
in q u i t e
Z = Hhy p (2.12.5),
and
a similar the
(2.12.8)
way
Riemann we
have
are
related
zeta-function
160 S
$(s)
: ½ e x p ((log 2 + ½ 1 o g n
- I -½y)s).
H
(1-~)e p
,
~ , i.e.
just
P
the p r o d u c t trivial
enlarging
zeros
Recently
A.
of
point
[Vo]
of t h e
note
that
zeta-function
zeros
stated
group
p
of
for
:=
version
zeta-function
entails Re
z
another
F . Here
~ ~ I)
which
Z(s,l)
has
Selberg
free cocompact
3.3.13,
all
the n o n -
~ .
Voros
factorization
over
s > I
(In-l) -s
case
of a fixed
a s y m p t o t i c formula
(Theorem
the Minakshisundaram-Pleijel
is d e f i n e d
I 6 {
t
canonical
for t h e
Weyl's
that
of t h e
by
suitable
r
nkO
c a n be e x t e n d e d {s 6 {: The
Re
to a f u n c t i o n
s > ½}
continuation
Voros'
result
b y the M e l l i n has
with
a simple
YH
the
side
denoting
:= I
(3.4.25) regular
is s t i l l
true
multiplier
assumption. formula
that
with
assumption
the
at
Z nkO
s = I
3.4.8
e-
Now,
(In-l) t
(t > 0).
a comparison
of
yields
of
(In-¼) -s =
> 0 ,¼
n
r
n
is an a r b i t r a r y
F
since
Voros'
seem very Z,
the
for the
than Weyl's
method
has
assumption. cofinite also
case,
formula
Equation
group with
applies
to o b t a i n
singular
asymptotic
zeta-function
> 0 ,o
above
difficult
-2s rn
I
if
not
Minakshisundaram-Pleijel
part
under
"right"
stronger
n
finite
P ~ I
system
It d o e s
the
I n
s = I . Note
pole
of
= F P 7 * ( s ' ¼ ) is = I '
Z,(s,¼)
at
o n the h a l f - p l a n e
transform
T h e o r e m s 3.4.7,
(3.4.25)
right-hand
meromorphic
under
a this
a corresponding too.
ensures
a continuation
In
[Vo],
that
an
the
meromorphic
161
on t h e w h o l e
s-plane,
the polynomial
holomorphic
in t h e e x p - t e r m
is e x p r e s s e d
in t e r m s
Finally,
considerations
have
our
shown
function
further
(cf.
pp.
s = O . The
the canonical
of the derivative
parallels 57,
of
at
117).
basing
at t h i s
o n the
between
the
constant
factorization
of
E
point.
Resolvent Selberg
term of
and
Trace the
Formula Riemann
zeta-
4, THE GENERAL SELBERG TRACE FORMULA
The
Selberg
zeta-function
Selberg
Trace
lished
in the
compact
group
tured due
from
second that
the
theorem.
tained
Formula,
chapter.
the
chapter
approach
outlined
any
unitary
dxd-multiplier
4.1.I
in the
Theorem
with
X
a unitary
a fundamental equivalence vention
domain
classes
1.5.5
the
Assume
that
h:
system
6 > O
remarks
Formula).
E
= h(-r)
ii)
lh(r) I = O ( I R e
the
co-
resiob-
explanation group
of
with
2k . The m a i n
of
1.5.8 ~
F
and
for and
that
the
F
be a c o f i n i t e
of w e i g h t T
2k 6 IR
is the
remember
Notation
the d e f i n i t i o n s ~ . Let -~k:
result
for all rl -2-6)
and of the
1.5.1,
F-
Con-
T ~ , mj
, ~jp
I n : ¼+r~
(n ~ O)
de-
, Hk
(rn6i]O,~[
function
+ 6}
,
r as
group
of
Dk
holomorphic
number
satisfies h(r)
free
we h a v e
cofinite
weight
estab-
be r e c a p of
which
of the
been
point
can
a detailed
for any
that
IIm r I < m a x ( ½ , 1 k l - ½ )
i
has
application
Let
system
of the o p e r a t o r
{r 6 {:
Formula
function
of real
r . Suppose
of c u s p s
and
for a f i x e d
Trace
that
which
case
theorem.
p = I ..... d),
eigenvalues
and
special
Formula
us to g i v e
dxd-multiplier of
Trace
entire
enable
Trace
and N o t a t i o n
(j = I,...,T; note
(Selberg
important
Selberg
by H e j h a l ,
following
an
by an a p p r o p r i a t e
on the
the
is s t a t e d
Hejhal
(general)
results
third
from
the R e s o l v e n t
zeta-function
The
in the
arises
IRe r 1
)
OO
,
U [0,~[).
163
If g: IR
:= ~ I
g(u)
is the F o u r i e r
~~ ,
transform
~ g(r)e-iru
of
du
h , t h e n the
following
assertions
are
valid:
The s e r i e s
E h(rn) nhO
d~(F) ~
E h(r n) = n_>O
~(F) +a
converges
~ ~ r h(r) -~
E
2~
absolutely,
(Ikl-l-½)
and
sinh 2~r c o s h 2~r + cos
2~k d r
h(i(IkI-l-½) )
icm o
o-
{P }r
lOg N (Po) tr X (P) N ( p ) ½ N ( p ) _ ~
g ( l o g N(P))
tr P>2
+
• i(2k-I)8 co u(k-½) E tr X (R) le f g(u)e {R}F 4v(R) sin e -~
e u - e i28 c o s h u - cos
O
-dT
log 2 - log
d E I j=1 p:mj+1
+ ½ { \½d~
-
+ ¼ h(O)
tr[IT,-¢(½) ]
T~
+ -~
Bjp]/
~
I -e -ku
0
s l n h (~)
1
f h ( r ) ~ (1+Jr)dr
f g(u)
•
I]
du
co
--oo
+~ I
~ --co
h(r)
U j=1
<0' (7i+.i t ) d r ~-
H sin p : m ~+I
3
f g(u) O
g!3jp
sinh(ku) du sinh (~)
28
du
164
I Note.
With
h(r)
I
--
2+r2
2+r2
(s-½) Ikl-s,
Ikl-a { ~ o
as s t a t e d Now
4.1.1
4.1.1
Theorem
in T h e o r e m
Theorem
4.1.2
)
Assume
for the
B 6 ]max(½,1kl-½),
m a x ( ½ , Ikl-½)
4.1.3
series
Lemma.
The
yields
in s e v e r a l
following + 6[
E nkO
h ( r n)
=
h(r
By assumption
lh(r) I ~ br -2-6
ii)
for all
there r ~ T
)
steps.
that
converges
absolutely,
exist
.
constants
, b > 0 , such
T
o
By v i r t u e
of Corollary
that
N(T)
there
exist
Hence,
absolute
T S r-3-6 0
S CT 2
for all
at m o s t
for e v e r y
convergence
3.3.14
T -2 -6 d N (r) -< b f r O
N(r)dr
: 0 I)
N 6 IN
there
of
E nZO
exists
distinct there
n 6 IN
exiSts
at
for
Corollary
T
,
3.3.14).
h(rn).
a constant
T A 0 , in p a r t i c u l a r ,
C.N 2
that
Hence,
o
(cf. the
and
~-- ( ½ + B + i r ) d r -
lh(r) IdN(r)
= b N ( T ) T - 2 - 6 + b(2+6)
yields
Formula
~!
T E [h(rn) I = S n >- O T o T
This
Trace
n
S h(r-iB) -~
n>O
Proof.
Resolvent
is fixed.
oo
E
the
2.5.1.
is to be p r o v e d
Notation.
(Re s, Re a > I,
(a-½)
o
with
C > 0 , such
for e v e r y r
n
6
N 6 ]IN
[N-I,N]
least
one
T 6 [N-],N]
for
all
n ~ O .
. satis-
fying
(4.1.1)
I
IT-rnl
2(CN2+I)
For
each
T
with
this
2C (T+I) 2+2
property
let
WT
be the
boundary
(with p o s i t i v e
165
sense)
of the rectangle
function
the argument
i B ,½+B] [a-
principle
½
I _ 2~i
•
S h(i(½-s))
_
is an even
(h(r n) + h(-rn))
-
I ½+B-iT S
½+B+iT S
+
½-B-iT
(Note assumption
h
-~-(s)ds
WT
I 2~i
Since
yields:
E h(r n) = ½ E n ->0 n ZO Re rn
× [-T,T]
I
=' ~-(s)ds ~"
h(i(½-s))
.
½-B-iT
i) of Theorem
4.1.1
and the functional
equation
(3.1.2)!) TO prove
the second assertion
by the stronger
ii')
Claim.
of this lemma we temporarily
replace
assumption
lh(r) l : o(Ime rl -4-6)
Provided
that
ii')
IRe rl
as
........
is valid we have
½+B-iT lh(i(½-s))
~(s)Ids
O(T -6)
:
for
T ----~ ~ ,
l - B - 1 'T T
Proof of the claim.
2B
Q
The integral
sup
X6[-B,B] It follows
from
(3.1.1)
satisfying
does not exceed
~(x+~-iT) l - o (T-4-6) L
-
that
for
(4.1.1).
ii)
166
LI : iT) <--IxiT + 21x-iTI
X
1
1
nkO
x2-T2+~2-2ixT n
a2+r 2 n
for
where
a 6 ¢
Obviously side
of
Therefor
with
Re a k m a x ( ½ , 1 k l - ½ )
it s u f f i c e s this we
to
inequality
show is
X
split
that
the
O ( T 4)
I'''1
is
into
n k 0
the
bounded
Now
first for
T
observe
2Ix-iT'
one
of
these
three
sums
is
[-B,B]
on
T
, ~ ,
X
I''" I '
the
T satisfying
X
n
<2T
n X> 0 ( T + ~ n O
n
finite,
it is e a s i l y
+ T---lrn + 2 a2+r 2 B- i T
<2T
)
n
(4.1.1)
1
•
~ + 2C(T+I
)2
_<
N(2T)
<
4CT 2 •
~ + 2C(T+1) 2 + 2 +
=
O(T 4 )
for
T
+ 2 +
(B+T)
,
and
21x-iTI
Z
J-..[
]a [ 2 + B 2 + T 2 + 2 B T
-< 2(B+T) X_O n > r _>2T n
n >O rn>2T
< 2 (B+T)
X n_>O r >2T n
ix 2 + ~ r 2 + i 2 x T
I ia 2 + r 2 I
IaI2+B2+½rnT+2BT s
2
2
2,
~r n a +rnl
=
(4.1.1).
]... [
and
n ~ 0 O
that
<
,
right-hand
, ~ .
n >X 0 ' ' ' ' ' O
term
n k 0 i r n ~0
X 1"..I nkO r k 2T n As
x 6
fixed.
second
for
all
O ( T 2)
for
T ---+
seen
to be
167
since
-3 ~ rn n >_ O rn %0
(Corollary 3.2.12).
is c o n v e r g e n t
This p r o v e s our claim.
F r o m the e x p l i c i t r e p r e s e n t a t i o n of Definition
3.1.4,
Remark 3.1.3,
~-(½+B+ir)
(r 6 JR) , given by
C o r o l l a r y 2.2.6,
C o r l l a r y 2.3.5, Co-
rollary 2.4.22 and P r o p o s i t i o n 2.4.21, we c o n c l u d e by t a k i n g notice of
(3.2.2) that
=L'(½+B+ir) I =
= O(Irl)
T h e r e f o r e the integral ii')
Irl
for
h(r-iB)
,~
~--(½+B+ir) dr
is r e p l a c e d by the w e a k e r a s s u m p t i o n
is p r o v e d under the a s s u m p t i o n
ii').
t i s f y i) and ii), we a p p r o x i m a t e
h : Ir 6 {: h
(r)
I
1+~r
2
#
w h i c h satisfy both i) and ii'). the c o n t i n u i t y of
h
If
E h n ~ O
(rn)
h
Thus the lemma
by the h o l o m o r p h i c
}
(max(½, Ikl-½) + 6)
functions
,¢, -2
> ~ ~0.
The d o m i n a t e d c o n v e r g e n c e t h e o r e m and in
By the f o l l o w i n g lemmata the integral
rein 4.1.1.
even if
is m e r e l y s u p p o s e d to sa-
~ : O
finally y i e l d the second
a s s e r t i o n of the lemma under the a s s u m p t i o n s
will be t r a n s f o r m e d
exists,
ii) again.
IIm r I < max(½, Ikl-½) + 6
:= h(r)
, r 6 ~..
~
(i) and
S h(r-iB)
(ii), too.
[]
~-(½+B+ir) dr
into the r i g h t - h a n d side of the e q u a t i o n in Theo-
4.1.4
1 -
2i-I
I
Lemma. z
1
-1 h(r-iB) T(B+B+ir)dr
-1
=
d-o ( f ) 4n
m
sinh 2nr dr h(r) cosh 2nr + cos 2nk
_I_
Proof.
By the residue theorem, this term equals
Since
h
is an even function, the first term of this expression is
equal to m
- d---
4n
i
I
-m
r h (r)
2 2n
[$(:+k+ir)
- $(&-k-ir) + $(B-k+ir)
- $(?+k-kc)]
dr
An application of addition theorems yields
]
4 [cotn (B+k+ir) + cotn (4-k+ir)
This proves the lemma.
=
sinh 2nr -i cash 2nr + cos 2nk
.
169
4. I. 5
Lemma.
I 2~
? h(r-iB) -~ Z
~hyp -hyp
(½+B+ir)dr
log N(Po) i _i g ( l o g N(P))
tr X (P)
m(p)2
{P)r
_ m (p)
2
tr P>2
Proof.
By C o r o l l a r y
2.2.6
and the d o m i n a t e d
convergence
theorem
we
have
I ~-{
f h(r-iB) -~
Z {P}F
-hyp -hyp
tr X (P)
(½+B+ir)dr
log N(Po) I-N (P)-1
_l I ? e-i(r-iB) log N(P) dr N(P) 2 2-~ h(r-iB) -~
tr P>2
Cauchy's
integral
theorem
entails
2--~I S h(r-iB) e -i(r-iB) log N(P) dr = ~I -co
that
7 h(r)e-ir
[]
log N(P) = g(iog N(P))
-oo
4.1.6
Lemma. I 2~
Z
f h(r-iB) -co
tr X (R)
-ell (½+B+ir)dr 9-----ell
iei (2k-I)8 4v(R) s i n @
7
g(u) eu (k-½)
{R} r
e u - e i2@ du cosh u - cos 28
.
0<@<~ Proof. have
As a c o n s e q u e n c e
of C o r o l l a r y
2.3.5
and P r o p o s i t i o n
2.3.4 we
170 I 2~
-ell
S h(r-iB
- -
F----
tr X (R)
(v (R)-I ei0 (21+I)
iei2k0
2~(R)2sin 0\
{ R} r
O<0
(½+B+ir)dr
~ell
-~
I co ~;,(½+B+k+l+irh 2--~ S h(r-iB) ~\ 9(R) / dr
i=O
-~o
v (R) -I -
E
e
-i8 (21+I)
I ~ /½+B-k+l+irh ) 2--~ f h(r-iB) 9\ ~(~ / dr .
i=O
has the
integral
-co
representation -t
(4.1
2)
~(z)
: -y +
•
e O
-zt - e dt --t I - e
where
y
is the E u l e r - M a s c h e r o n i
Since
8
is an i n t e g e r m u l t i p l e
~(R)-I E i=0
e
+i8 (21+I)
(Re z > O )
constant
of
~ (R)
= O , and substituting
~' -ell -ell
I h(r-iB) 2~ - ~
(½+B+ir)dr
([MOS],
p.
16).
we h a v e t =: u " ~(R)
we aet
i2k8 E tr X (R) 2~(R) ie sin@ {R} F
=
O
1 7 h(r-iB) f -~ O
~ (R)-I e-iS (21+I) - u(½+B-k+l+ir) _ el8 (21+I) -u(½+B+k+l+ir) du dr E i=O I - e -u" ~(R)
• i2k8 Z tr X (R) le {R} 2~(R) sin@ O<8
Interchanging
the o r d e r of i n t e g r a t i o n
gral of the m o d u l u s
exists.
Hence
e
i6 -u (½+B+k+ir) e dudr 1_ei2ee-U
is a l l o w e d
as the i t e r a t e d
the last e x p r e s s i o n
equals
inte-
171
i2k8 ie 2v(R)sin8
tr X (R)
E
{R}r
/e-iSe-U (½-k)
ei8 e-U (½+k)
O<e<~
• -!2~ 7 h ( r - i B ) e - i U ( r - i B )
Using Cauchy's denominators
integral
theorem
by the conjugate
and multiplying
complex
numbers
dr du} .
the numerators
and
of the denominators
we
obtain: o~
{...} = e -i8
I -ei28e -u
f g(u).e u(k-½)
O -i8
du 1+e-2U-2e-Ucos2 @
=
- e
e-U(k-½)
e-U(1-e-i2ee-U).e i28
f g(u)
du
O The only thing left to be done
is
second
integral
and to combine
the two integrals.
4. I. 7
Lemma.
I 2~
f h(r-iB) -~
= g(O) (-dT
+ ½
to substitute
-par par
log 2 - log
T E
½d~ -
+ ¼h(O)
I
I
d E
~jp
1_e-kU
du
sinh(2)
7 h(r) ~ (1+ir)dr
7 h(r)
qOl
-6"
~ -u
in the [
T d \ U H sin ~Bjp)/ j=1 p=m ++I 3
tr[IT,-¢(½) ]
T~ ~ + --f f g(u) 0
u [
(½+B+ir)dr
j=1 p=m 3 + I
+~
.
1+e-2U-2e-Ucos2e
(½+ir) dr
)
f g(u) O
sinh(ku) du sinh (2)
172
Proof.
Proposition
1 7
h(r-iB)
2.4.21
~par par
-~
2.4.22
yield
(½+B+ir)dr
I h(r-iB)dr 2~ -~ ,,,,,
and Corollary
-dT log 2 - log
U U sin j=l p=m +1
Bjp ]
3
,,t,
:: 11
+
/ (½d~ \
T d H X ~jp) " ~ 1_ ~ 7 h(r-iB)(9(½+B+k+ir)j=1 p=m .+I ]
-
•
9(½+B-k+ir))dr
,y-
=: ~2 1
oo
+ tr[IT,-~(½) ] ~
dr
1
f h(r-iB)
2 (B+ir)
-co
:: f3
I
- ~* ~
1
+ ~
~ h(r-iB)
17
(9(½+B-k+ir
- ~#(½+B+ir)) dr
(:: f4)
h(r-iB)9(1+B+ir)dr
oo
1
f h(r-iB)
oo [
~-~
-co
~
1
(=: -fs) +
B+i(r+t)
1
B+i(r-t)
--co
• --
+ c • 2-~I
7 h(r-iB)
• 2(B+ir)dr
(½+it)dt
(=: f7 )
(=: f6)
dr
with some constant
-co
It follows
immediately
from Cauchy's
integral
theorem
that
oo
~I = g(O),
f3 : ¼h(O),
I S h{r)~ (1+ir)dr, ~5 = 2--~ -co
Formula
(4.1.2)
implies
that
$7 = O
•
c 6 C.
173
]'2
_
I 2~
co 7 e - U (½ +B+ir) S h (r-iB) --co
-ku -e 1-e -u
= 7 g(u) O
ku
-~ 1 ~ 2 " 2-~ ~ h(r-iB)
"e
sinh(ku) sinh(~)
-e
-ku du dr
I -e -u
O
ku =Te 0
e
du
(Cauchy's
e
-iu(r-iB)
integral
(Fubini)
dr du
theorem).
Moreover,
I co 14 = I* 2-~ I h(r-iB) -co
I -e -ku = T* f - O I -e -u
1-e =
T~
f
•
" e
u -2
7 e - U (½+B+ir)
-ku -1-e - d-uu I -e
0
• ~
I
~ ~ h(r-iB) -~
e
dr
-iu (r-iB)
(by (4.1.2))
dr du
(Fubini)
-ku u
g (U) du
(Cauchy's
integral
theorem).
O 2 s l n h (5)
By i n t e r c h a n g i n g the top of the
4--~ -~ ~ -
the o r d e r of i n t e g r a t i o n
following
(½+it)
page)
~
f6
h(r-iB)
(for a j u s t i f i c a t i o n
is t r a n s f o r m e d
B+i(r+t)
look at
into
B+i(r-t)
--09
N o w we a p p l y C a u c h y ' s inner
integral
equals
integral h(t)
.
theorem
resp.
formula
to see that the
174 Justification
I
of interchanging
1
I
B+i(r+t)
the order of integration:
2 (B+ir)
+ B+i(r-t)
(1 ~
1)
+
1¼-B2+r2-2iBrl = 2[B+irl"
IB+i(Irl+Itl) I-IB+i(Irl-ltl)l-(¼+t 6
2 B+ir
6 B+it -~ •
2)
I ¼-B2+r2-2iBr I IB+i(Irl-ltl) I.(~+t 2)
6
I
<
• O ( , t , - 2 - 2 ) • O (It
12+~)
for
Itl
.....,..~ ,
Is+i(Irl-ltl
Hence,
I
?
2~
I
lh (r_iB)
IB+i (r+t)
--co
+
I B+i (r-t)
_2 - 6_
-< ? ~(irl).O(it I _2 ) dr IB+i(IrJ-ltl)
-~
where
~:
[O,~[
J
, [0,~[
is a continuous
function satisfying
r a r
with some constants
_8
~(r)
N c r 2 o
for
For
Itl > r °
_2__6 O (ltl
o
r o > I , co > 0 .
the last integral does not exceed
Itl-1
2 ) " 2[ ~ o
ltl+l ~(r) dr + Itl-r Itl-1
7 Cro
8 2
~(r) dr + dr B Itl +I r_it I
]
6 _< 0
• 2
[
c I logltl+g-2c
I +
co
,ti
I
I+--
Itl
cI > 0
being a constant,
8 6 = O (It1-2-~ loglt I ) = O (,t, - 2 - ' )
for
,t,
~-1 dx
],
175
C o r o l l a r y 3.3.14 and C o r o l l a r y 2.4.17 imply
4-~
~-- (½+it)
dt = O(T 2)
for
T
)
CO
,
-T
Hence one can show the e x i s t e n c e of
oi~ S
-~
(½+it)
--co
17
~
~ ~r~
~
B+i r+t) + B+i(r-t)
--~o
- 2 (B+ir)
(~
½+it
½
~ t ) ~r ~
by partial i n t e g r a t i o n and then apply F u b i n i ' s theorem.
This proves the lemma.
The a s s e r t i o n s of the p r e c e d i n g lemmata together yield the proof of T h e o r e m 4.1.1.
[]
INDEX
page
Barnes
G-function
page
113
norm of {P}F
canonical factorization
157
order
classical entire automorphic form
109
(of growth)
118
Phragm6n-Lindel6f principle
cofinite group
15
cusp
16
cusp form
39
degree of singularity
28
dimension formula
47
IIO
123, 125, 135
Poincar~ series
45
primitive elliptic element
61
primitive hyperbolic element
50
principal branch of logarithm
17
regular
28
eigenpacket
36
eigenvalue problem
25
resolvent equation
Eisenstein series
29
resolvent kernel
26
resolvent set
25
resolvent
25 40,44
equivalent modulo F
15
Euler-Mascheroni constant
86
Riemann Hypothesis
117
expansion theorem
37
Riemann zeta-function
117
factor system
18
Roelcke's conjecture
36
scattering matrix
33
finite order of growth Fourier expansion fundamental domain Hadamard's factorization theorem
117 30 15
145
resolvent trace formula
Selberg conjecture
144
Selberg Trace Formula
162
Selberg zeta-function
57,116
singular
28 16
hyperbolic plane
14
stabilizer
hypergeometric function
26
WeierstraB factorization
Jensen's theorem MaaB differential operator multiplier system
136
106,108
weight
155 18,20
24
Weyl's asymptotic formula
127
20
Weyl-Selberg asymptotic formula
138
INDEX OF NOTATIONS
page
¢
14
page R(T)
138
R I (T)
151
14 ~,
IN
14
o
14 14
z
SL(2,m)
S(T)
14 T I , • . . ,T T 32
AI,...,A T
138 32 16
164
B
18
D1 E(z,s;v,A,k,
E
14
3P
E~
]p
X)
18
29
(z,s)
32
V
(z, ½+it)
94
Z(P)
5O
26
z (R)
16,60
F(a,b;c;z)
G(z)
19
113 146
Gkl(Zl,Z 2 )
26
a o , a I ,a 2
61
a. 3P
38
Gklell
51
ax
Gklhyp
48
70
b
38
Gklpar,reg H(Zl,Z 2 )
26
L
149
L 2 (F,~)
Mk N{P)
50
P S L (2, IR)
14
Q(F,-2k,x)
110
o
R I , . . . ,R p
dn
I08
d~
145
fn f(t;s,a) m I , . . . ,my
47,50
Po
R
19
37 78
123 96
N
d
23 96
M
3P
28,32
mk
123
m(F)
122
Pl
' " " " 'PM
60 ql 'q2' " " " 66 rn
96 96 46,106
v
=
,.
.
.
.
.
.
~>~
~W
co .
,
~
~
bJ
~
o~
~
co
Co
~ "~.
co
~
.~
~
~~' ~
.-J
to
0
~d u~ ~0 Co
179
page
page Sjp(1),~jp(1) 6j
67 107
T
16
T•
28
q)
33
6jp
32
Y
86
tPjp, lq
32
96
X
19
Yn
153
X
YEP
s
21 45
153
14
32
18
I=,wI
14
qn
96
.l[S,k]l
19
1
38
<
, >
16,60
I
1
16,66
( ,
n ~) ')I ' " " " '~)p
19,22 22
)
23
~(x)
47
II
P
16
IIGXX (z,) II
35
25
{P}F
47
P (-Z k )
~k
18
{R} r
61
a(z,w)
14
{F,-2k, x }
tl
23
110
180
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