Number Theory in the Spirit of Ramanujan

• • Number Theory in the Spirit of Ramanujan T itles in T his Ser ies 3J. Bruce C . Bernd t , Number thfoxy in the _...

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Number Theory in the Spirit of Ramanujan

T itles in T his Ser ies 3J. Bruce C . Bernd t , Number thfoxy in the _pint

01 Ramanujan. 2006

33 Rekba R. Tbornaa, Uctura in !!«>rnelric combi""lorio<s. 2006 3:! Sheldon Kab, E:nu ...... ative potneuy and atn"ll theory, 2006 31 J o hn MeCIea>:y, A tint co"",," in topoIocy: Continuity and dimension,

"'"

30 Serge Tabaclmikov, Geomecry and billiardA, 200~ 29 K r istopher Tapp, Matrix ~JIII 1'0. unl>f'Ol'Ch. 2005

Robert lIard" Editor, Six them .. on van..ion, 2O().l 25 S. V . D,uhin &nd 8 . D . Ch"bo ...,..",aky , n....tor ...... ion UOUJIII for bqi n nen,2(l(l.l 24 Br...,., M . Landma n a nd A aron Rober-taon, R.am.ey theory on .be intqft1l, 200-1

26

23 S . K . Lando, Lectur.. on ~ ....&ti", lunetiona, 2003 22 ADd ....... Arvanitoyeora:o., An introd""' ion to Lie !!1'>uJIII and lbe !!«,md.ry of bo""".... """"" opo.ceot. 2003 21 W . J . K a.,..or and M . T . Nowak, Probu,ms in ma.hemMkallOIlal,.._ Ill: [nte«ra.ioll, 2003 20 KIa ... H u lek. Eknxntary a.lg~caic ~ry . 2003 19 A . SheD and N . K . Vereah cbagln, Compulable functiona, 2003 18 V. V. Yasche nko, Editor , Cryptovaphy: An imroduction, 2002 17 A . Shen a nu N. K . Ve . es hchacin. BasM: Kt theory , 2002 16 Wo](gang KGhne l, Di",,",nti.al &eornetry: CUN8 - ..... f~ - manlfoLda, ~ edition, 2006 1~ C e rd Fi""h e r , f'Ia..., a.l&ebralo: C~, 2001 14 V. A . Vusil iev, Introduction to topoIuc. 2001 13 Fredericlc J . A hng .... n , Jr" Plateau '. problem: An i""'talion to varilold pomelry, 2001 12 W . J . K acsor and M. T . Nawak. Pn>blems in mathematical [t Continui.y IOIld dilJ_nliatiou. 2001

a.naI,....

For a C'QInpLete list of titles in this series, visit .he AMS Bookstore at. www.ams.org/ bookstore/ .

Copyrighted Materiar

Contents

Preface

Cbapter I.

Introduction

§I.l . Notation and Arithmetical F\lDctiona §1.'2. What are q-Serietl &Dd Theta FullCliona?

6

§l.3. Fundamental Theorems aboul q-Series and Theta 7

Functio~

§1.4. Nole8 Chapter '2. Congnaen<:ell for p(n) lUId T(n) §'2 . I. Hi$t.ori<:a1 Background

" 27 27

§2.2.

Elementary Congruences for .. (n)

28

§2.3.

Ramanujan '. Congruenoe p(Sn + 4) • 0 (modS)

31

§2.4. Ramanujan '. Coogruenoe p{7n + 5) . 0 (mod 7)

39

§2.5. The Parity of p{n)

43

§2.6. NoteI

49

Chllpt.er 3. Sums of Square!! and SUllul of 'ITianguIac Numbers

§3. 1.

Lambert Series

§3.2. Sums of T wo Square!! §3.3. Sum. of FOIl!' Squares COpyrighted Material

" " " 56

D. C. DERNDT §3.4. Sums of Six Squares §3.5. Sums of Eight

Squ~

53.6. SUI1l!l of 1'ri&n&ular Numbenl §3.7.

Note!!

Challter -I. ~4.J.

Ei!!enstein Series

Bernou!li Numbers and Eisenstein Seriftl

§U . Trigonometric Series §4.3.

71

,.n 85 85 87

A Clas- of Series from Ramanujan's Lost Notebook

Exp.-ible in Terms of p. Q. and R §4.4.

67

R.ep.eeentations of Integers by x 2 + 2~ . %2 + 3~ , and :r2+xy+~

§3.8.

63

ProoCa of the Congruences p(5n + 4) • 0 (mod 5) and p(7n + 5) • O(mod 7)

§4.5. N _ Chapter 5.

97

'IOS"

The Connection Between Hypergeomeuk Fuoctioll!l and Theta Fullctioll8 109

S5.1.

Definitions of Hypergeomelric Serie. and Ellipt ic Integrals

1119

§$.2.

T he ""ain T heorem

11.

§5.3.

Principle:e of Duplication and Dimidiation

120

§5.4. A CataJor;ue of Fonnula.s for Theta Fuoctiollll and §S.5.

Eiselllltein Series

I?!

Notftl

128

Chapter 6. Applications of the Primary Theorem of Chapter:; 133 §6.1.

Introduction

133

§6.2.

Sums of Squarl:'!l and Triangular Numbers

134

§6.3. Modular Ecluatiollll §6.4.

Notes

Chapter 7. §7.1.

TIle R.ogen-Ramanujan Continued I'rllCtion

Definition and HisjpficaL BacQ:round COpyngrued Matenal

140

150 1$3 153

SPIRIT OF RAMAN UJAN

§7.2.

The Convergence, Di\'ergence, and Values of R(q)

§7.3.

Thll Rogers-Ramanujan F\mctiOIlll

vii 155 I~

§7.4. Identities for R(q)

161

§7.5.

166

Modular Equations for R {q)

§7.6. Notes

167

Bibliography

m

Index

IS'

Copyrighted Material

Preface

Generally acknowledged ILl! India's greatest mathcm3tician, Srill;""'5&. Ramanujan is IIlOIlt often thought of 88 a number theorist, although he made substantial cont ributions to analysis and several other areas of mathemMi(;8. For ffiO$t number 1hoor;$ts, when Ramanujan 's name is mentioned, the paTlitioll and tau functions immediately comc to mind. His interest in these arithmetic fUllctiollll was inextricably intertwined witb his primary interests of theta [undioM and other qseries. In fact, most of Ramanujan's researeh in number theory 81"O@e out of q-seriesllnd theta functions. Theta fun<;tiOIlS are the fundamental building blocks in the theory of elliptic functions, and Ramanujan independently developed his own theory of elliptic functions, which i~ quite wllike the classical theory. Wc do not fonna!ly define an elli ptic function, but, roughly, elliptic furlCtio!l8 are meromorphic fUllctioltll with two linearly independent per iods over lhe real numbers. The concept of double periodicity ill not used in this book, and, to the best of Ollr knowledge, Ramanujan never utilized thi.
The purpose of this book is to provide an introduction to this large e:rpanse of Ramanujan's work in number theory. Needless to say, we shall be able to cover only a very small fraction of Ramanujail's work on theta. functions and q-serie5 and tbeir oolurections with number theory. Ilowever, after de,-eloping only a few facts about

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,

n. C.

BERN DT

q-sene.

and ~het.a functions, we will be equi pped 10 pr~ many interestiD« theo~lIa. The arithmet ic functions on which we focns ace the partition function p(n), Ramaoujan 's t.u function .. (n ), t he nunr heT of representations of .. poIIit;\"\' int~er n loll a aum of 2k squares dellOted by 1'2.('1), and other ari t hmetic functions cloeely a11~ to "2. (n ). Motl of the material upon .... hich we draw can be found in Ramanujan'. IHlblillhed papers on p(n ) and '1"( 11 ), the later chapters in his second ootebook , his Ioat notebook, and hill handwritten manU5Cript 011 p(n) and T(II) published with hia 1000t notebook. We em· ph!JJlize that Rrunanujan left behi nd few of hia proofs, especially for his claims in hi, not.ebook1.s and l06t DOtebook. T hus, for many of the theorelllll that we d i!ICWIS, we do DOt know Ramanujan'. proofs. This is particularly true for the tbeQrem!l on $UIJl.8 of square! aud 5imilar arithmetic functions that we pW'o'e in ChapleT J.

The requiT1'ments far reading and under'Sland iTl& the material in thill book t.fe relatively modest. A.n undergraduate course in "lemetl-tary number theory i!l 1Iodvi$8.b..... For M)me of the fl,nalytic arguments, a t!IOlid undergrlKluate COUI"!Ie in complex anal)'lli!l i3 _ nt ial. However, thl! occll!lio~ when deoep analytical rigor is needed are few, and 50 ~adera who do not have a strong background in anal)'llis can limply verify formally the needed manipulatiollH. Our intent here is not t o give a rigoroWl COUfM in analyllls but to emphll8ize the IIlO8t important ideas about q-teTiell and theta functions and how they interplay with !lumber theory. Thi$ book should be suitllbkl for junior and !lenior undergraduates and ~nning graduate "uden~. Since many readers may not be famili&l" wit h Ramanuj81I " life,

..-e begin with. short account of h.i!I lik where relloders leam .bout the notebooks .nd bit notebook in ...·hich he r«OI"ded his theorems 0\I"eT leVl'!"aJ yellC8. We prm'ide brief histone., fint of the "ordinary" DOtebooke, Md ~nd of the IosI. notebook. After these biopllphicaJ and historical narratiol\lj, !leven cha ple,.,.

~

provide short lummfl,,;es of the book's

fllunanujan Wf1,8 born on December 22, 1887 in the home of his maternal gr&lld mot her in Erode, located in the lIOut hern Indi an st ate of Thmil Nadu. After a few months, his mother. Koma\alammaJ. returned with her 80n to her home in Kumbakorllun approximately 160 COpyrighted Ma/anal

SPIRIT 01-' RAMANUJAN

mUes IIOUth-.outhwes~ of MOOr38. where her h\l!lband WNI a clerk in t he office of a cloth merchant. At t he age of twel~, Ramanujan borroo»'ed • copy of the -'Ond part of Loney'. PI4~ 1hgonomdry [149] from an oIdeT student and worked all the problems in it. Thi$longtime popullLl tJ:'xtbook in India ha!! much more in it IhM iU titll:' suggests. For exampll:', infinite !!eriel!l and elJ:'mentary funcl.ioDII of a oomplex ..ariable are two of iUl topics. At the llI!le of about fi fteen, he borro¥.'ed from t he Kumbakonam College library a copy of G. S. Carr'. A $ynOp~1-I of Blernenta'1l Re~td~ in Pu.re Ma themlltic. [64], which sen'ed &'I his primary IIOUJ"(:e for learning mathematiC!. CarT ....aII a tutor in London and compiled this! compendium of 44 17 rClult.l ( .... it h very few proo&) to facilitate hia tutoring. At the age of sixteen, Ramanujan entered the Government College in Kumbakonam. By t hat lime, Ramanujan ....... com pletely devoted to mathematies and CODlIequently failed hill aamillAtions at the end of hia first ~1Ll , be<:aUge he would DOt study any otheT rubjoct. He therefore lost. his! achollLlllhip and , becaU!le his! family ....B poor. was foroed to terminate his! formal educatiolL He later t .... ice tried to obtain an education at Pachaiyappa', College in ~Iadr&ll, but each time he failed his! examinations. After lelwing the Ckwernment College in Kumbakonam, Ramanujan devoted all of h i~ time to mathemat iC!!, recording his reiUIM with_ out proo& in notebooks. [t was probably around the age of sixteen that Ramanujan began to record his mathematical d~ries in noteboob, although the entries on magic lIquares in ChapleT I in both his first and IIeOOnd notebooks likely emanate from his! acbooI days. U viIII in poverty with no meaDll of ftnanciallUpport, suffering at times from !IerioIu illnesses. and working in ~ation , Ramanl\.ian devoted all of hill efl"or ... to mathematies and continued to record his discoveries .... ithout proofl in noteboob for the nut. fi~ ~an. In 1909. he married Janald, who ....... only nine or I
B . C. BERNDT

xii

After being supported for abou~ fifteen months. for reNlOnS th8t are uuclear, RallllUlujan rdUlled further financial assistance MId became a clerk in the Madras Port Trust Office. T his t urned out to be a watershed in Ramanujan 's career. Several people, including S. NarayMla lyer. the Chief Acwuntant , and Sir Francis Spring, the Chairman, offered support, and Ra!UanujMl wa.s persuaded to write English mathematicians 8bout his mathematical discoveries. Two of them, H. F. BILker and E. W. 110b80n, evidently did Il(It reply. M. J . M. Hill replied but WI\II not cnoouraging. But on January 16, 1913, RamMlujan wrote G. H. Hardy, who responded immedi8tely and encouragingly, inviting Ramanujan to come to Cambridge to develop his mathematical gifts. RamlUlujan MId his family "'~ro lyengars, a oonservatiw branch in the Brahmin tradition. llavelling to a distant land would make a person unclelUl , IUld so RamMlujMl 'S mother W88 particularly adamant abQut ber son's not accepting Hardy's invitation. After a pilgrimage to Namakal with S. N. Iyer and after Goddeti8 Namagiri appeared in a dream to Komalatammal, Ramanujan received permission to travel. So on March 17, 1914, Ramanujan b08rded a passenger ship for England. At about this time, Ramanujan evidently stopped recording his theorems in notebooks, althougb a few entries in his third notebook were undoubtedly recorded in England. T hat Ramanujan no longer ooncentrated on logging entries in his notebooks is \':','ident from two letters that he wrote to friends in Madra.s during his first year in England. In a letter of November 13, 19 14 to his friend R. Krishna [loo [51 , I)P. 112- 1131, Ramanujan oonfided, "I have changed my plan of publishing my re!ult.s. I am 001 going to publi~h any of the old re!uit8 in my notebooks till the war is over." And in a letter of JanulU"y 7, 1915 to S. M. Subramanial1 [51 . pp. 123- 1251, Ramanuja.n admitted, ~I am doing my work v(!ry slowly. My notebook is sleeping in 11 corner for these four or fi,~ months. I am publishing only my present re8earche'lll.'l l have not yet proved the reiullJi in my notebooks rigorously." Ramanujan soon became fam()us (or the papers he published in England, IlQl11e of them coauthored with Hardy. One of hi~ mOtit important 1"'P
S P IRJT OF RAMAN UJAN

xiii

his f&mOUll tau function T("). d~ in Chapter 2 of this book, and the Ei8e~lein ledes, P, Q, and R, which are introduced hefl! in Chapter 4 and which ".,.,re key players in 80 much of his reliellrch. In their palM'r (109), ]192. pp. 262-275). Kllrdy and Ramanujllr! launched the lie'" field of probabilistic number theory, which became an important branch in number theory. In another paper [H O}, (192, pp. 276309(, in the OOUDle of obtaining an ~mptotic .eries for the partition function p(n), Hardy and Ralllanujan introduced the Circle method, which ~till today is the primary tool for analytically attacking prohIeIll8 in additi~ number theory. The gene!lis of the cirde method can be found in Ramanujlln's notebook$ (193), but unfortunately it is IDOIt freq~ntly called the Hardy-Littlewood ci rcle method today. In the latler part of his stay in EngI.and, Ramanujan wrote hill famous pape .... on oon&ruencn for p(n) (188), [192, pp. 210- 213} and ]190}, [19 2, p. 230], about which much is wriuen in thill book. Altbon&h Ramanujan ne-.w doubled his decision to accept Hardy ', invitation to Cambridge, not all W1III well with Ramanujan. World WIU' I beBan shortly after bill arrival, "nd being a 8t rict veg. etarian. he could not always obtain famitillr food and spice! from India. On March 24, 1915. near the end of hill fil'llt winter in Cam· bridge, Ramanujan wrote hill friend E. Vinayaka. Row in MadrMj51, pp. 116-117}. "I WM not well till the beginning of thill t.erm owing to the weather and consequently I couldn't publish any thing fot about (i mont~ .~ By the end of his third year in England. Ramauujll./l was aitically ill, and, for the next two )'Un, he WM oonfined to aanitari· UDlI

and

IIUTl1illl

homes.

Ramanujan'. health turned slightly upward when ill 1918 he became the !!«ODd Indian to be elect.ed all a f ellow of the Royal Society and the 61'!1t Indian to be cboeen .as a fellow of Trillity College. Af· ter World War I ended, in 1919, Ramanujan returned home. but his health continued to deteriorate. IlIld on Apri l 26, 1920 Rruuanll,jll./l died at the age of 32. Doctol1l 111 both England and India had difficulty diagnosing Ramanuj"n'. i1llless. lIe was treated for tuberculosis, but a se,'CTe vi_ tamin de6ciency. liver calleer, le&
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xiv

other diagnoses. However, D. A. B. Young [52, pp. 65-75J made a careful examinati'm of all exlant records aud recorded symptoms of Ramanujan 's illuess and convincingly ooncluded that Ramanujall suffered from hepatic amoebiasis (a parlll!itic infection of the liver). Not only do all of Ramllllujan's symlltorns suggest this disease, but Ramllllujan's medical history in India also favo!"1l this diagnosis. AmoebiMis is a protozolll infection of the large intestine tha t giV(lS rise to dysentery. In 1906 Ramanujan left home to attend Pachaiyappa'$ College in Madras, where he contracted 11 severe case of d)'llentery and had to return home for three months. Unless adequa~ly treated, the infection is permanent, although the patient may go for long periods without exhibiting any symptoms. Relap:ses oc<:ur when tbe host- parasite relatiollBhip is disturbed, which likely happened when he endured a colder climate and perhaps inadequate nutrition after his arrhlll in England. The illness is difficult to diagnose. Our descri ption of Ramanujall'S life has been neo:essarily brief. For several yelU"8, the standard SOurces about R.&manujan '~ li fe hav~ been t he obituaries of P V. Seshu Aiyar, R. Ramacblllldra Ran, and Haniy, found io Ramanujao 'a Collected. Papttr$ (1 92J and C hapter 1 of Hardy's book [107). By far, the most comprehCllBive biography of Ramanujan has been written by R.. Kanigel [134J. The letters from and to Ramanujan are also a source of both mathematical and personal information about Ramanujan , and ll06t of the extant letters ha'-e been compiled with commentary by R. A. Rankin and the author

[51J. After Ramanujan died, Hardy strongly urged that Hamanujan's notebookB be edited and published. By uediting,~ Hardy meant that each claim made by Ramanujllll in his nOlebookB should be examined and proved, if it cannot be found in the literature. Hamanujllll, in fact, had left hi~ fil"'!l notebook wi th Hardy when he returned to India in 1919, and in 1923 Jillrdy wrote a paper [106), [108 , pp. 505-516J about 11 chapter on hypergoometric series found in the first notebook. In this paper, Hardy pointed out t h at Ramanujan had independently di.:;ooo.'Cred most of the important classical results in the subject while al>iO ditlCOVering several new theoren13 !IS well. For the definition of a hypergoometric series, /lee Chapter 5 of this monograph Copyrighted Material

SPIRI T OF RAM A NUJAN Hardy ~nt the first notebook to t~ Univeraity of Madras where Ramanujan's other notebooks and papers lW'rl! being preserved. Plana were undertaken to publish Ramanujan 's collected papers and, potiIIibly, his notebooks and other manuscripts. Handwritten copies of the notebooks were sent to Hardy along with other manWlCriptJI and pa_ pers in 192"3, but the papers were never returned to the Uni>-ersily of Madrllll. It traIlSpired that Ramanujan'! Collected Paper. (192) were published in 1927, bu t hit! notebooks "nd other m"lluscripts were not publ ished. Sometime in the late 1920s, G. N. Watson and B. M. WiLson began the ta.'5k of editing Ramanujan 's notebooks. The second notebook, being a revised, enlarged edition of the 6n;t, was thei r primary focUll. Wilson WII5 assigned Chapten; 2-14, and Watson waB to examine Chapters 15-21. Wilson devoted his efforts to this task until 1935. when he died from an infection at the early age of 38. Watoon wrote over 30 paper!! inspired by t~ notebooks before his interest evidently waned in the late 1930s. Thus, the project w.u never oompleted.

It was not until 1957 that t be notebooks were made 8.\-a.ilable to thl! public when the Tata Institute of Fundamental Research in Bombay published a photocopy edition [193], but no editing was undertaken. The 6rst notebook was published in vol ume I, and vol ume :2 oompri!le!l t he second and t hird notebooks. The present author undertook the task of editing Ramanujan'! notebooks in 1977. With t he help of se>"tral mathematiciaIlS, t be author com pleted his work with the publication of his fifth volume (3 8) on the notebooks in 1998. [n the spring of 1976, George Andr~ of Pennsylvania State UniveI1lity visited Tri nity College, Cambridge, to examine the papers left by Wat50n. Among Wat$On '$ papers, he found a manuscript containing 138 pages in the handwriting of Ramanujan. In view of the fame of Ramanujan's notebooks (193), it waa natural for And~W$ to call this newly found manuscript " Ramanujan ', lost notebook. H How did thill manuscript reoch Thnity College? Wat.90n died in 1965 at t he age of 79. Shortly thereafter, on sepMate oo;a.siOI1ll, J . M. Whittaker and R.. A. Rankin visited Mrs. Wat8(on. Whittaker was a $On of E. T . Whittaker, who coauthored with Copyrighted Material

D . C. DERN DT

xvi

Wat.iOn probably the m(lllt popular and frequently used text 00 anal· ysis in the 20th century [221[. Rankin hl'd succeeded Wal90n as /I.IllSQn Professor of Mathematics at the Universi ty of Birmingham, where Wa\.80n served for most of hill careoer, but was now Professor of Mathematics at the Uni>-ersity o f GIMgow. Both Whittaker and Rankin went to Watson'~ attic office to examine the papenl left by him, and Whittaker found the aforementioned manUllCript by Ra.-manujan. Rankin suggested to Mrs. Wal90n that he might sort her late husband's papeni and send those worth preserving to Trinity Col. lege Li brary, Cambridge. During the next three years, Rankin sorted through Wst.son's papers :;ending them in batches to Trinity College Li brary, with Ramanujan's manuscript being sent on Deeolmbcr 26, 1968. Not realizing the im portance of R.a.manujan's papeni, neither Rankin nor Whitta.ker mentioned them in their obituaries of Wat-son [195J, [222J. T he next question is: How did Watson come into pQIlSeS/:Iion of this sheaf of 138 pages of Ramanujan's work? We mentioned above that in 1923 the University of Madras had sent a package of Ramanujan's p&peni to Hardy. Most likely, this ~hipment contained the uJost notebook." Of the over 30 papenl that Watson wrote on Ramanujan's ",ur k, two of hill last papers were devoted to Ramanujan's mock theta functions. which Ramanujan dillcovered in the last year of hill li fe, which he des<:ribed in a letter to Hardy only about three months befo re he died [5 1 , pp. 220--223J, IIJId which are also found in the 100t notebook. [n these two papers, Wst-SOn made some conject ures a bout t he existence of certain mock thets funct ions. If he had the 1000t notebook at that time, he would hS'"e seen that his COnjectUTllS "'-ere correct. Thus, probably sometime af· ter Watson 's ioterest in Ramanujan's ,,-ork declined io the late \930s, Hardy passed Ramanujan's papenl to Wat50n. In early 1988, just after tile centenary of R.amanujan's birth, Nar088 Publishi ng House in New Delhi published a photocopy edition of the lost notebook (194J. Included in this publicatioo sre partial manuscripts, lOO6C papers, Slid fragments by Ramanujan, as well as letters from Ramanujan to Hardy written from nlll'8ing homes during the last two years of Ramanujan'a sojourn in England. Copyrighted Material

SP IRIT OF RAMA NUJAN

xvii

The first chapter of this book ill devoted lO basic fael.8 about q-aeries and theta funelionl, including the q-bioomial theorem, the Jaoobi triple product identity, the pentagonal numba' theorem, Ra. manujan 's 1111 summation theorem, and the quintuple product iden.tity, Many of the theorell1ll pro\'-ed in Chapter I can be found in Chapter ]6 of Rrunanujan " second notebook (193J, (34J, Chapter 2 focugell on oongruences fm the partition function P(n) and Ramanujan 's tau function T(n ), Much of this materia] is taken from Ramanujan 's handwriuen manU9Cripl on p(n) and T(n), which W!lll !irst published in ]988 along with Ramanuj8n's IotIt notebook (194J. AddinS detail. to many of Ramanujan'sllroofs and discU$iug RamlUlujan'. theorelll.l in light of the literature .... riuen after Ra. mlUlujlUl ', death, the present 8uthor and K. Ono (&11 published 8n expanded version of ltu. manU9Cript. In his notebooks [193J, RamllDujan rl!ClOf"ded 11 Iur;e number of entries OD Lambert $erie!!, 1'bee identities for Lambert $em were U8ed by RalDal1ujan t.o establish theta function identitiel!l and fann .... llIII for the number of repreaentatiowl of an inteser as 11 mm of a certain numbers of aqus.res or of tri8nguJar numbers. We introduce readers 1.0 Lambert serieli in Chapter 3 8nd establish m8ny identities ie!lding 1.0 formul811 for sums of squares and triallsular numbers. A rnanulI(rillt with no proofs on precisely this subject is another of tboee manUllCripU published with Ramanujan'. lose. notebook [1941. 119, Chapter 181. His second notebook al$) oonlaillll a large number of such lheorell1ll. Eiaenatein series permeate Ramanujan 's ooteboob ]1931 and \o8t notebook 11941. Much of our expotiition on Eilenstein series in Chapter 4, however , is taken from Rarnanujan's epic papeT JI 86], [192, ]36- 162]. Olle of RamlUlujan'. approaehet to eonuuencee for p(n ) is based on EilelUltein series, whkb we delnolUl~rll.te at tbe close of Ch8pter 4. In Ch8pleT 5, we introduce readers to hypergeometric functiollll and elliptic integrals. Our goal in this ch8ptcr is to prO\'e one of tile most fundament81 t heorems of elliptic fUD(:tioll8 relMing hypergoometric fUD(:tioll8 and elliptic integrals to theta functions. This thoorem enablerr \.Ill 10 express theta functiollll and Eisenslein !leTies COpyrighted Material

xviii

B . C. BERNDT

at various arlutnent.l in terms of certain elliptic parameterS. Our expOflitiol'l is derived from Chapter 17 of R.ama.nuja.n'. II«!Ond not,e. book [193J, [34, Chapter 17], where, through a ~ 01 preliminary ieIllIIWJ, Rama.nuja.n leads us to the aiorclllf'ntioncd key theorem. Applications 01 tile aforementioned representations for E~tein seriefl and theta functioll5 form the content of Chapter 6. Fim, _ return to the topic of ~ume of Squares and demOf'llltrale how the for· mulas for EileTIIIl.ein serics lead to short proofs of lOltle of the rerultlI from Chapter J. However, most of Chapter 6 is devoted to modular equatiOf'lll, a topic to which Rarnanujan made more contributions than any other mathematician. ChapteTfl 19-21 in hisll«!Ond ootebook an devoted to modular equatioIl$, and our short introduction to this topic is drawn from our previous lICCOunt of R.amanujan's work in these chapteTfl [34J. One of RamaJIuja.n'. favorite topics ..,.. the Ro&en-Ramanujan oontinued fraction, the focus 01 Chapter 7, Because we wish to share 110 much about this continued fraction with readers and because the length of the chapter would be prohibitive if _ proved all theorems offered in this chspter, we forego some of the proofs. H(J9,-ever, we do prove two key theorellUl .-eIMing t he continued frsetion with ita reciprocal. These theoreme are then used to give an alternative, cleaner proof of an identity of R.a.manujan in Chapter 2, yielding immedi. ately the congruence p(&n + 4} O(modS). T he famous RogemRamanujan functiollll are alao diaeus:sed, and , iD particular, _ pl"O\"e that the Rose ......RamaJIujan continued frlloCtion can be repreeented &Il a quotient 01 the two RQsen-Rauumujan functiollll. Our fe!"V'etlt wish is that our aampUfII 0( the many bt>autiful pl"O\M'niel ... tisfitd by tlm continued frllCtiol'l will moti'1lte readers to turn to original ~ to learn mote about it.

=

Ubiquit.ous in t his book are prodUCUI of the form (I _ a)(1 _ aq)(1 _ aq1) ... (I _ aqn _ l) _: (a; q)n

8Il well IIl3 their in6nite versions

Iql <

I,

which are CAlled q--producta. Although we _ume that readers of thi, booIc are familiar with in6nile serieA, it may wcll be that !IOnte COpyrighted Ma/anal

SP IRIT OF RAMANUJAN

xix

arc not familiar with infinite products. A reader desiring to learn a few basic fact.s about the convergence of infinite products may con· suit a good text on complex analysis, such as that of N. LeviJUlOl1 and R. Redheffer [142, pp. 382- 385), for basic properties of infinite products. In particular, all the infinite prod ucts in the present text converge ab:iolutely and uniformly on compact subsets of Iql < I. In particular, taking logarithms of infinite products and differentiating the resulting series termwise is permitted. At fi rst, you may find that working with the products (a; q )~ and {a; q)"" is somewhat tedio\l5. In order to verify q-product identities or to manipulate q-products, it may be helpful to write out t he first three or four terll1.ll of each q-product. This should provide the needed insight in order to justify a given step. After working with q-products for awhile, you wiU begin to handle them more quickly and adroitly, and no longer need to write out any of thei r terll1.ll longhand . When you reach this stage, you should feel qui te w mfortable in manipulating q-series. It ill assumed throughout the entire book that Iql < I. O,..,r 50 exercises are interspersed within the exposition. The author is grateful for the comment8 of graduate students at the Universities of Illinois and Leca: where this material was taught. Y.-S. Choi, S. Cooper. D. Eichhom , AMS copy editor M. Letourneau, J . Sohn, and K. S. WiUiams provided the author with many helpful suggesliollll and corrections for which he is especilllly t hankful. The a uthor also thanks N. D. Baruah , S. Bhargs ,,.,, Z. Can, 11. H. Chan, W. Chu, AMS Acquisitions Editor E. Dunne, M. D. Hirschhorn, M. SomO!!, A. J . Yee, and the referees for their suggestions and correctiollll.

Copyrighted Material

Chapter 1

Introduction

1.1. Notat ion an d Arit hmetica l Functions In this first section, we introduce notation that will be used throughout the entire monograph. Seoondly, we define the arithmetical functions on which we will {OCUlI for most of this book, and which can be studied by employing the theory of theta functions and q-series. A brief introduction to theta functions and q-series will be given in the next section , to be followed in Sect ion 1.3 by 11 few of the most useful theorellU about these functions. Defi n it ion 1.1.1. Defin e

(I.LI )

(a )a: = (a;q)o := 1, ( l.l.2)

(o) ~ := (a;q )n :=

.-,

n(\n al l,

(Ul ),

n 2: I ,

••• (a)""

:=

(a ;q)"" :=

(1 -

Iqj < I.

.~

We roll q the btm, and if the identifioou<m af the ba.se i.! c/w r, we often omit q from the nolotion. Copyrighted Material

8 . C. BERNDT

2

Replacing (I by ~, where a > 0, and usilll l. 'Haspi!al's ru~ while letting q tend to I, we find that

. (q-;q)~ . _ 1- _ t(" _ 1 -_ If+! c"c--' lom - - _ hm _ · .. _"C-'C
,-I{l - q)"

(1.1.3)

_a(a+I)·· ·(a+ n _ I).

The expression on the right side Il~ is called a rilling or shifted factorial, and 110 we _ t hat (11; q)n may be ool\8idered NI an analogue of the rilling factorial. We next define the q·binomial coefficient, or GaWlllian coefficient, [ ::.], which is an analogue of tbe ordinary bill(>.

mia! coefficient (::'). Definit io n 1.1.2. ut n and m derwte integer.. Then the Ca_iliA r.ot:ffiamt u dejinaf ~

(1.1.4)

[~]

(q )"

,- {

~~) ... (q)" -,,, '

./0 S m S n, otllenu&.Jt.

Exe rclse 1.1.3. UtI"g ( 1.l .3) thru timu, MC" tome ""tII 0

=

I,

$IIow thllt

T hlUl, the q·binomlal coefficient tend!! to the ordinary binomial coeffICient when q -

I.

From tbe &linition (1.1.4) it is not obvioutl that the q-binomial coefficients are polynomials in q.

Exercise 1.1.4 . Unng /he defini.tlon (1.1 .4), rmders MOuld finl prow IM fiNI q-GnGlog!.e G/ ptW;td', /ormuki. 9 ' - k/ovl. Lemmll 1. t. ~. For n (1.1.6)

~

I.

[m"] ~["- ']+q·["-'l 1 m m~

Exercise 1.1 .6 . Stalnd, employing UmIl1G 1.1.6 and Inducllon on n, ~ that I:' ) u a ~~e
SPIRI T OF RAMANUJAN

3

ExerciS<) 1.1.7. Eltab/ish a ucorn/ q-ana/ogue of Pa.srol'~ fonnu/(1, which is gi""'n for n ;:.: 1 by

El<er cise 1.1.8. Pro"", that, for f(1ch ptlir of nonntg(1tj"", jnttgcn

m, n,

t [m+j]",~

;.0

J

[m+"+I] . m+1

We now define four primary arithmetical functions on which we focus throughou t the monograph. Definition 1.1.9. Ifn is (1 positive inttger, let pIn) denote the nurnller of unre.!tricted repre.!cn/atiVJU of n !1.! a &urn of po1iti"", inttgcr&, where "'P"'&entatio!l.f with different order& of the &(1me .rummands are not "'9ardw !1.! di~tinct. \Ve roll p{n) the porlitwn function.

For el<aruple, P( 4) = 5, because there are 5 ways to represent 4 as uum of pOIlitive integer.;;, namely, 4 = 3+1 = 2+ 2 = 2+1+ 1 = 1+ 1+ 1 + I. Readers should check that p(5) '" 7 and p(6) = 11, hut reader.;; should not check (at least by hand) that p(ZOO) = 3, 972,999,029,388. A few moments of refle<::tion convinces one that p{n) grows rapidly. More precisely, G. H. Hardy and Ramanujan [110), [192, pp. 276309) showed that, as n ..... 00, I p(n) ....., 4n.;3

(1.1.6)

el
(

"Vr Fl "3) '

i.e., the ratio of the left and right sides of (1.1.6) tends to 1 as n tend'l ' " 00.

The generating function for p(I1 ), due to Euler, is given by .,.,

( U.7)

1

~p( n)q" = (q; q)"" =

where we define prO)

= 1.

1

l - ql

1

~ql···l _ qk

...•

To see this, observe that the factor

~=f:q'k

cop9rii/fted Mflferlal

,

8 . C. IlERNDT

lIetlerala the number of k'. that a ppear in .. particular partition of n . Each partition of n appears once and only 0I"ICe on the right side of ( 1. 1.7) when inttrpreted in this manner.

The ,eneralill& fuoctioo for the number of partitions of n into distinct parts, ~llOted by p~ (n), is si""'" by ( l.l.8)

f

.".

P4{n)q" - (- q;q)""

= ( I + q){ l + q'l '" (\ + qk) .

because IIny particular integer Can occur at mOllt on«l in any given representation of 'I . We now show that, by elementary product ma. nipulationl, we can deduce the rollowi", f.. mollS theorem of EuleT.

Theore m 1.1 . 10 (Euler). ~ number DJ parttLioR.J 0/ a po.tlt"Fe mttgtr n mID dt,lflnd pGrU equals the numher DJ parhliDru 0/ n ,Il10 odd pdrU. Unolm.y p.(n).

Equaling ooefficienlll of q", n:::': I, on both sidCII of (1.1.9), we corn. plete the proof of Euler's identity. 0 For example, p~(6) • 4, becaU$O! 6 hall 4 rcprellentaliom iDlD distinct paru, namely, 6 - S + 1 = 4 + 2 _ 3 + 2 + I. On the other hand, 1'0(6) - 4, because 6 IwI 4 repre8l'nt.atioN into odd paN. namely. 5+ 1 - 3+3 = 3+ I + 1 +1 _ 1 + 1 + 1+1+1 +1.

Exercise 1.1.11 . Evkr's ulemdy OIIn be Illghtll/ rrjinfti. Prove /lull t~ number 0/ pgr1,IIOR8 0/ n mto an ewn (odd) number 0/ odd part. tqIIQlt tAt nllmber 0/ por1,liolU 0/ n ,nto dl.fhnc/ POTU. u'hen IM number 0/ odd pclrt.J ,$ tw:" (odd). Exercise 1. 1. 12. l'row: tAat the n ..mber 0/ part,lIo'" o/the po~itilif! integer n into pclrt.I thal art net dllli.!jblt b!l3 ' $ tqua.l to tht 'IUmber 0/ ""rill/DIU 0/" '" which "0 port " ppe4r$ more Ihon turice. Exercise 1. 1. 13. J:'-uler', ,dent,ly OIIn be rt/ined '" yel anQ!her 1001/. Ut k and n be po.fltllPe '!liege" ""th k > 2. J>r-rn,e Ihat the nllmber 0/ r..;op'fnghled M81ena/

S P llU T 0 .'

,

RA~IA NU JAN

part,lIoru of n lA """ch Il(I part UI d'tILf1ble b~ k fq1I(lU the nllmkr 0/ paThtlOru 0/ n .r.ch that th.t:n oilre- ,'ne,ly len IMn k COp1U of I!fJch part. Nole thQI ...hen k ~ 2, _ "blain Eukr', ,denhl~. The eau k = 3 UI the p~IIIO'" uerciK. Using elementsry product manipulstions ... In the proof of Theorem 1.1. )0, readers ~ encouraged \.0 l~ the theorems in the foHowing exercises. Exer clse 1. 1.14. ut lI (n } deno~ the numkr 0/ partilion.! of /he muger" m lO pol rll congrueJ\1 102, 5. or 11 modulo 12. ut B{n} deJ\oU the "umkr ofporl.ili~ 0/" m lo dUllmcl part.!: congru eJ\t 10 either 2. >t, or 5 mlNlulo 6. ~ tIult

po~iIi~

A(n)

= B(n).

Exercise 1. 1. U . f>rr1w IMI the numkr 011/ parllhOfl8 of n lA ""nch repealed i6 eqIUIllo the n"mber 0/ porl.lfton.f of n In ,,'Inch Il(I pori. appear. mo~ Uw.n 3 bm«.

o.uv odd porll _~ k

Exercise 1. 1. 16. Prove /hat /he nwnkr 0/ por1llionl 0/ n m which each pori. appuN exactly 2. 3, ar S timu equals th e n llmber o/porl.ilion.! of n m to par1, rongruenl 10 ± 2, ±3, or 6 modulo 12. Exe rcise 1. 1. 11. Prow. Iho t the nllmkr 0/ par1llion.! of 'I III ..... hich no pari. oppur' u(lclly once tqIIol5 the nllmkr of parlitiofU 0/" into parll not OC)J\grtlent to ±I rnodulo 6 . Definition 1.1.1 8 . Ihfine the eM/fictenll T{n). n ~ I, by ~

(1.1.10)

..,

q(q;q)~ - L T(n)q~.

Iql < l.

For example, T(2) _ - 24 and T(15) = 1.2]7, ]60. A glance at a t(lble of T(n) showlJ that IT{n) 1 grOWl quickly. It ha.s long been conject ured, but never pro_-ed. th al T{n) ;. 0 for all n > O. It may seem at fil"$t sight th&.t the definit ion of T(n) iSllrtifici&.l. lIowe>-er, it is one of the mOISt import&.nt &.rit hmetic&.l functions in Ilum~r theory &.nd Ari8es ill ma"y oontexU, in particular. in the theory of modular forms. We study this e2,~~~RR/in Chapter 2.

B. C. BEItNDT

6

DeHnltlon 1.1.19. For pnitive 'nlfSOeY"' n and t , kt r~ (n ) "mote /he numbtr of ~pre8m tatKml 0/ n III 11 81irn 0/ k 'qlUlnl. ~ rep1UentatiDru ""Ill d.ffemtl urden ond dIfferent ngn.r lire aot
dllllnct.. 8, eonvmtwn, r, (O) = I . for example, f'J(2) "" 4, becaUlle 2 _ l' + \' _ I' + (-If "" (_ I)' + \' _ (_ 1)2 + (_I)' , r, (9) '" 4, becaU8e 9 _ 3' + ()2 = 0' + 3' _ (-3)1 + 01 _ 0' + (-3)' ; and T, (7) _ 0, because there &re TJ(lt any ways Wll can write 7 as a sum of 2 squares, DcHnitloll 1. 1.20. For n 2: 0, the tnonguwr num~n lire the "limber, n{n + 1)/2. 1/ n /lnd k are po$itive mltger" Id t.(n ) "mote Ihe number 0/ reprulntatiOJU of n 41 11 6tlm 0/ k trilmgWllr numben, ~ reprumtlll.(m.I ""Ill different order. lUll cOIm/m III distinct. Se! It (O) _ I. For example, 1,(7) = 2, becal1$l! 7 _ t + 6 _ 6+ 1, while t, (16) .. 4, becaU3e 16_1 + 16 _ 15+ 1 =6+ 10 _ 10+6.

1.2. What arc q-Sc ries and Theta FUnctions? Generally, a q·series hu expTeS!lioll5 of the type {a;q)n in the summands. In Section 1.3, we OOIl.'lidcr a few of the moet impOrtant examples. lIowew:r, many q-serie8 do !lOt have 811ch prodllcl.II in their lummanda, but t hey may arise &$ limiting cases of IICriel oontaini~ producl.ll (1.1 .1), or their theory may be intimately intertwined with .ueh eerie!. Theta functioml are such enmplel. Defin ition 1.2. 1. R.lrnanllj an'5 geneTGt thda /line/IOn ! (ft , t.) u de· fined t.y

E ~

(1.2. 1)

! (a,t.):=

Thru lpeclal ClUU are

a" ("'H )!1 b,, (,, -ll/2,

d~fined

RamanllJan " notaltOn ,

by,

In

=

E ,.'. ~

(1.2.2)

r,I>(q) :- / (q, q)

(1.2.3)

~( q) :"" /(q,q3 ) =

labl < I.

~

E qn ("+ I)/~,

Cop~ed'JflJterial

S PIRIT OF RAM A NU JAN

7 ~

I( - q):·f(-q, - i )-

(1.2.4 )

E:o:efciR 1.2.2 . JUJlIfy the ucond (1.2.7) 01 Ummo 1.2.4 belmu.

L

(_W qR(", -I)!1.

equal.,~

01 (1.2.3).

St:t: IIl80

The numbers ,,(3n - 1)/2, 'I :::.: 0, are t he pentagonal nllmbeN, and the numbe ... '1 (3" ± 1)/2, " :::.: 0, aRl the genemli.:w pe1ItQgoIUII numbe,., . Note that ~

.;,t (q) _

(1.2.5)

L It(n)q".

1'he8e cenerallnc functions will be used In the ll«luel to find formulas for "t(") and 1. (,,). for certain poo!Iltive integers k.

Readers may wonder why the fullClions defined above are called tMl a func&ions, tince 110 tMlaappeanl in the n.otlltions. In the cJ.a&.. licaI defini tions. e.c., Bee Whittaker and Wauon'. text j2 21 . p. 464J. the notation () is empl~. We em phlllli:.e thlt Ramanujan's definition i. different from the W1Ual definitions, but the gerleTaJity ilr identical to that in j22 1J, i.e .• all o f RamlUlujan 's theta functions fall under the purview in [22 1J, and oonversely.

Exercise 1. 2.3. Prove the 1m.lowing balle and propertlU 01 thdo lunetWm.

enormoti,!l~ ti,!~JW

Lc:oJnnta 1.2.4 . We hQw.

(1.2.6)

1 (0 , bo) "" I (bo, a).

(1.2.7)

1(1,41) = 2/(a,Qs), 1(- 1,0) =0,

(1.2.8) and,

.f n

(1.2.9)

.., Qn~ IDlegff,

f (a, bo) _

(I,,\ .. +1)/2 b ,,( .. - 1)/2 l (a(atJ)" ,6o{Qb)- R).

1.3 . f\mda lllcnt a l T heore m s abo u t Q-Scrics a nd T het a FUnctio ns Perhaps t he mOlt important theorem In the subject of q-!ll!Ties is tbe q. binomial theorem . Copyrif;lted Material

B. C. BERNDT

8

Theore m 1.3 .1 (q-analogue of ~he binomial theorem). For 1,

Iql. 1:1 <

(1.3.1) Proof. Note that the produd on the right side of (1.3. 1) converges uni formly on compact subsets of 1:::1 < 1 and so represents an analytic function On 1:1 < 1. 11IW!, we may write F {:-):=

(1.3.2)

(az),.,

-I') '"" ""

~

n

1:1<

~ An: ,

I.

n_ O

From t he prod uct representation in (1.3.2) , we can relldily verify that

( 1 - z)F(z)

( 1.3_3)

= (I -

az l F (qz) .

Equating coefficients of zn, n ;:>: 1, on both sides of (1.3.3), we find that

A ,, - A .. _ ! =q" A .. - nq,,- IA .. _ I ,

"'

1 _ a.qn _1

(1.3.4)

An -

I

qn An_ I.

n 2: l.

iterating (1.3.4) and using the va.lue Ao "" 1. wbich is readily apparent from (1.3.2), we deduce t hat

An

( L '5 . )

Using (1.3.5) in (1.3.2),

_(a)n (q) .. '

,,~complete

n

2: O.

the proof of (1.3.1 ).

0

To understand why Theorem 1.3. 1 ill called the q-analogue of the binom ial thw mn, replace /I by q~ in Theorem 1.3.1. Arguing lI-'i we did in (1.1.3), we find tbat lim (q" )~

= a(a +

1)·· · (a+" - I) n! Assuming now ~bat a is a poo;it i\"
(q)n

~a{ a+I )·.·(a +n - l ) L zn ~~O

rr -' 1 =Q - - = (1 -

n! . 0 1- : Copyrighted Mat'e(;al

: )- 0

'

SPUUT OF RAMA NUJ A N

9

which In ealcuhu is often called ~be S"'ner... iud binom ial theorem .

Corollary 1.3.2 (Euler ). Fur Iql < 1. ~"

1

l: -'(q)" - -(.: )"" ,

(1.3.6)

ft .(l

,.,

1'1 < I .

1,1 < 00.

( 1.3.7)

P roof. _ 0.

Equali~y

(1.3.6) follow8 imrnediaU!ly from ( 1.3.1) by Betting

Cl

Replace a by al b and .: by b.: in the q-binomi'" theorem. Thus, for

Ib:1< I ,

f.

(1.3.8)

ft ...o

(a / b).. (b.:)" (q)"

= (Ol ).... {h)""

We let b - O. Now,

t.~(a/b)"bft _ ~ (I - ~) (I - ~) '" (1 - CJq~-I) b" (1.3.9) Eql.ll!.lity ( 1.3.7)

now foliO'WllupoD

!letting a _ I.

o

Alert. ~aden will have immediately 1lOI.ic:ed that \VI! havoe taken the limit ()Jl b under the III.IIllmation qn without justifyiDt!: i~. In the thoory of q-Bene., taking limits under the III.IIllmation sign , all ...., have do ne, is a1 .....ys l!.8!Iumed to be justi6ed. but hardly anyone ever ~ it . No nel hele88. a rifl:orou8 lLI"~nt almost ........ys ruM aIoIlJll the !IIUIIe lincs. In the CI!.I:Ie at hand , clw:>o:wIe a number AI such that Iql:S: M < I. Let { > 0 be given such that 0 < 'h < I - M . With Ibl temporarily fixed, let No be that unique poIIilive integ~r such

:s: (

t hat

O:S: k

< No,

D. C. B ERNDT

iO

Since 2t

< \-

AI,

Thus,

f: (a/b)~ .. -0

(blr

(q)1\

OOIlVttges uniformly for 1111 ::; l by tbe Wt ierst . ... AI -test , and !O leltiIl& '" __ 0 under the $UIIlIJl.a.lion &ign is j Wltified.

ru the q-&JIaIosue of the binomial theorem is per haps the 10051 fundamental result in the thoory of q-seriea, 50 the J &OObi triple product identity in the 11eX! theorem is likely t he mOllI important and useful rl'Sui t in the theory of theta fuuctioTU .

T heore m 1.3.3 (J aoobi Triple Product Identity) . For .: ". 0 and Iq) < I , ~

(1.3.10)

L

. ..""

.:"qn· _ (-zq:ql).,.,(- q/z;q').,.,(q' ;,/l"".

In Ramanujan'. notation ( 1.2. I}, the Jarobi triple product identity taketl the shape

/ (a,6) _ (- o.;ah)""(- II;ab)""(ob;ab),,,,.

(1.3.11)

P roof. In (1.3.7), ~place q by ql 'IInd (_:q;q2 )00 _

(1.3.12)

!

by -z q 1.0 dedu« that

"" L --- ~ --L zn qn· (q2n+1. ql ) (q~ ; q2 )"" ....() • 00

1

J n qn>

n ()( ql; q2 )"

00

_

]

~""'2n+22

(q2; q1 )00 ..!:-oo z q (q

COpyrighted Ma/anal

;q

loo,

SPIRJT 01·' H.AMANUJAN

11

lince (q2"H; '12).,. _ 0 ..h~D n is a ~ative in\.eg~r . Now apply ( 1.3.7) again with 'I replaoed by q'- hut thi3 tim~ with : .. q'-'"H . ThWl, from (1.3. 12).

1

" " . . . .... 'I •

~",,:

- ('12;'11)_( ql:;q2)"" ...

by ( 1.3.6) with : ~Iaoed by -'11: and 'I ~aoed byq2. and tberef~

1'1 /:1 < 1. RearraogiDi~ . we oompleu the 1'1 /:1 < I. HD"-'ever, by analytic OOIItinuation, (1.3.10) holds for all oomplel[ : -F 0, and.tO the proof is oomplete. 0 with the restriction

proof of (1.3.10) for

Coro llary 1.3.4 . RtaJll that &ma""jan', thetaluneti(m.l

{1.3. 14}

0(, 1- (q', q'I~. 2 ('I; '1

(1.3.15)

)""

I {-q) "" ('1;'1 )"".

Proof. The product (1.3. 13) follows from (1. 2.2) and (1.3. 11 ) by settinga .. b _ q. LeHing a .. 'I and b = ql in (1,3.11) and using (1.2.3) and (1.1.9),

we find

tha~

B . C. BERNDT ],Mtly,

!!et "

= - q,b =

_ ql in (1.3.11). Then, from (1.2.4),

f( - q) _ (q,q')oo (ql ,q')<»(qJ; qJ)oo

(1.3.17)

= (q;q )"".

o Comhining (1.2.4) and (1.3.15), we obtain Euler's pentagonal number theorem, ...·hieh, be<:suse of its importance, ""<1 sUIte in a !!eparate corollary. Coroll ary \. 3 .5 (Euler's Pentagonal Number T heorem). ~

(1.3. 18)

L

(_I)~q"(3"- I)/1

~

L

=

(_I)"qnpn+ ll/l

= (q;q)"".

Note that too second repTC8enta tion in (1. 3.l8) arises from the former by replacing" by - " in the fir!lt sum. In the sequel, we ,hall use these representatioll!l inU::rcIlBngeably. for

~iLII

From Corollary 1.3.5, ""<1 values of p{n).

easily derive a recurrence formula

cai~ulati ng

Corollary 1.3.6. For Then (1. 3.19)

brevit~, $et

L

p{n) -

0", ..,

w)

=

j(3j - 1)/2, - 00 < j < 00.

(-I)1+l p (n -

IoIj)'

:s ..

P roof. By Corollary 1.3.5, (1. 3.20) Now equate coefficients of qn,,, ;:: I , on both sides of (1.3.20). The 0 rec urrence relation (1.3.19) then immediately follows.

Exercise 1.3.7. There U onother $im"le recllmmu rdation for v(n) that U d~e to E~ler. Consider the yeMm!my",ncha" ~

(1.3.21)

F(q)

;=

,

L p(11)qn = ~(q..~q)~. '"'

n _O

Logarilhmioollll differentiate both &idu of (1.3.21). Then upond the re$u/ting dena",inatorr 0" the right·hand 'ide inlo grometric $eriu. Copynghted Material

SP IRJT OF RAMANUJAN

13

.-, np(n )

=L

p(j)a(n - j ).

,~

EuJer '. pentB&onat number theorem h8ll interpret&lion gi\~n in t he next corollary.

&

beautiful combinatoriAl

Coroll ll.ry 1.3.8 (Combinatorial Version of Euler '. Pentll80nal Number TheQrem ). D~ ( n ) dtnott the n~lIIber 0/ parl,/i0'l8 0/ n into nn w.:n nUlllbtr 0/ d,$!mc/ parts, and Itt D o(n ) dt note the number 0/ parlitioft$ 0/ n .nlo nn odd n umber 0/ di..!linct parts. Tht n

ut

(1.3.22)

D. (n ) - D..{n ) =

Proof. Let

III ~D

(1.3.23 )

H P ( 0, '

"n _

j{3j ± 1)/2,

,,"~

by clo!lely examining the product

(q:q)""

= (1 -

q)( l - rI )(1 - q') ... ,

.. hich "'"e _ is similar to the generating function (1. 1.8) for p... (n ), that in (1.3.23 ) there arC minus signa in the product. When we multiply t he terms of ( 1.3.23) toget her, we obtai n a plus sign if we multiply Iltl C\"en number of powefll of q , and we obtain a minus sign if we multiply an odd nwnber of powers of q. Thus, partitions of an integer n into an even number o f distinct para are weighted by & plus aip, while parti tions of n int..o an odd number of d istind parts an! weight«! by a m;nUII ai&n. ThUII, from the pent8(ODII.1 number theorem ( 1.3.18). ex~pt

~

1+

L,(D.(n) -

..

L

~

Do(n )} q" - (q;q)... -

T he result now follows.

(_ I y .,.,(!j- Ilfl.

o

For example, 0.(5 ) - D,,(5) _ I , since 5 = 2(3 · 2 - 1)/ 2, i.e. , j _ 2. Indeed , t he parti tions of 5 into distinct parts lUe 5, 4+ 1. and 3+2. Second, D.(6) - D,,(6) = 0, since 6 is not 11 generlllized pentagonal number. Indeed, the parti t ions of 6 into dist inct p&rt.$ are 6, 3+ 2+ 1, 4+ 2, &nd 5tbpyrigIlted Ma/anal

8 . C . BERNDT

14

T he next theorem is fundamental, and we shall

U$e

it eevera.l

l inlel! in the &equel.

Theorem 1.3.9 (Jaoobi '. ldentity ). We .....1Ie

E(

- 1}" (2" + 1)q,,(,,+I){2 =

(1.3.24)

(q; q)~.

.~

Proof. In (1.3.10). replace ~ by :-' q to deduce that

L ~

(1.3.25 )

~2"q"' +" _ ( _.l2q2 ; q2 ) "" (_ 1 /~2 ; q2 ),,,, (q' :q2) ....

( 1.3.27)

,,--""

ThWl, letting : ~ I in ( 1.3.26), empklyin& (1.3.27), and applyin& I.. 'H& pital'a rule, we !lod th8t 1

(1.3.28)

'2

~

L

. --..,

(- 1)" (211

+ l )q"t..+I) _ (q' ;q') !., .

We IlQW divide the sum above into t wo parUl, -00 < n 'S. -1, O !> n < 00. [n the former sum, replace n by - n - I and .implify. [..astlr, ~ace q' by q in (1.3.28 ). We thWl arrlw; at (1.3.2-4 ) to complete the proof.

0

Besidell the functions ..,(q), w(q), and / (-q), Ramanujan defines one further function ( 1.3.29)

which is not . theta fur>etion but which pla~ .. prominent role iD the theory of theta functions. ThC'lle four fUllcliorlll satisfy a myrilld of r~latiolll!l. We offer here tnCWIt of thoee recorded by Ramanujan in Entry 24 of Chapl.er 16 of his IIeOOftd ooubook j1931; see [34. pp. 39401 for proof8. Copyrighted Material

SPIRlT OF RAMANUJAN

15

Theor<)m 1.3.10.

(1.3.30) (1.3. 31 )

/(q) '" \I>(q) / (-q) w(- q)

=

/(q) ( ) = /(_q2)

X q

t(q)

X(-q)

= /(q)

=

J

/(_q2) '" I/.'(- q) ,

.,2(_ q2) '"
(1.3. 32) (1. 3.33)

x(q)xi - q) = X(_q2),

( 1.3.34)

/3( _ q2) '" 'I'2( _q2)"'(q2) .,

2(q).

ExercisEl 1.3. 11. U3e the juIlCtiOt\3' product repre!elllatia .... 1(1 prow 0/ the idenlitil!3 (1/ Thwrem 1.3.10.

$ome

The q-binomial theorem is perhapS the most widely used summation formula in the t heory of q-series. Ramanujan 's ,I/.', summation t heonlm is a bilateral (i.e. , the summation index ru .... from -00 to 00 instead of from a finite integer to 00) analogue and is undoubtedly the most useful bilateral ~ummalion in q-series. In both ( 1.3.1) and (1.3.36) below, the series On the left sides have exactly One q-product in the numeral.:" and ODe q-product in the denominator, and both theonlIIU give product representat ions for the $l'!ries. The prefix 1 On I/.' indicates that there is one q-product in the numerator, and t he suffix I on '" indicates that there is one q-product in the denominator . Before we state and pl"Qve Ramanujan 's ,"', summat ion theonlm, it will be convenient to defi ne an ana logue of (a; q)" for negative integers JI. So, for - 00 < n < 00, define

(a; q}.., ( ) .. '" (aqn;q)",,' a;q

(1.3.3.5)

Observe that (1.3.35) agree!! wi th Definition 1.1.1 when n is a non-

negative integer. Theorem 1.3. 12 (Rrunanujan's ."', Summation). For 1 and Iql < 1. (1.3.36)

~ (0) .. ~n L... (b)" n __ OO

Ib/al < 1:1 <

= (o.~)""(q/(oz))""(q)oo(b/a),,,, .

( z )""(b/{a~),,,,{b)..,(qla)oo Copyrighted Material

8 . C. BERNDT

16

P roof. DefiTl'! ( J.3.37)

In ~be anllullll.lb/al <: 1:1 < 1, /( .. ) is a nalytic, and a

l.au~nt

tj()

therdore has

elrpa!U:ion ~

L

J(z ) ""

(1.3.38)

From tile definition ( \ .3.37),

we

can deduce that

(/I - aqz)J(q:)

(1.3.39)

c"z".

= q( 1 -

:)/ (:).

Obeerve t hat the Laurent upansion of / (q%) is valid in the annulUl, 1"/{oq)1 < 1:1 < I/Iql. and !IQ the La~nt expansions of both 1(:) and I (q:) are valid in t he intenteet.io:m of tbe two annuli, i.e.. for Ib/(aq)1 < 1:1 <: I. Thus, _ ooed le) IWIIIUme that Iql > Ib/"I. a reBlriclion t hat weahalllatu relD(l'\"e by analytic continuation. Heooe, by (1.3.38) and (\.3.39), for 11I/(aq)1 < III < I. ~

(1.3.40)

L

~

(be,. - oc,, _l)q"z" -

L

q(e.. - c,, _I) Z".

Equatinl!! coefficicnt.ll of zn on both sidi!'!l of (1.3.40). we find thllt

(oc.. - aC.. _l Iq'· _ q(c.. - e.. _ I)· Solving {or e.. . _ 600 that 1 - aq,, - I

( 1.3.41)

c" -

I

bq,,_Ic,,_I.

If n > 0, iterale ( 1.3.41 ) to conclude that

( 1.3.42) If n

< 0,

(cl"

c" = (b) ..

CO·

r(lwrite (1.3.41 ) in thi! foml

( 1.3. 43)

Iterate (1.3.43), utile ( 1.3.35). IInd rcpl&Cf! n by ' l + I to ar.hoe ( 1.3.44 )

lit

SPIRIT OF RA MANUJAN

17

Puttill8 (1.3.42) IUld (1.3.44) in (1.3.38) and using lUlalytie continUAtion, we conelude that ( 1.3.4$)

I (' ) -CO

;c~

. --..,

(a)• •

Ib/ o l < 1' 1 < I.

(b) RZ ,

It rema.inll 1.0 eYllluate CO.

Write

(1.3.46) I t:) - CO ..

too ~:~:~" + £<> ~ ~:~: zn-: CO (g(:) + h( :)) .

From the dclini tion ( 1.3.37) of /Cz) , we see that It:) hM a simple pole at : _ I. Since g(:) i$ a pO\..:~:r aeriel about: _ QO con~rging !Or 1: 1 > Ib/ oj, it follows that g(.z;) is an.aIytic at : _ I. lience, by (1.3.46), if we let et:) :_ £<> (1 - z)h(:), (1.3.47)

!~( I -

:)/ (:)

CO ~ ( I

=:

-

:)h(:) _:

~~

e (:).

:s

et:) is analytic on 0 1:1 < 1/191, because 0(:) conin " neighborhood a bout : "" 0, ill simple pole at : ..., 1 is removable, and the next \argegl pole of both I t:) and h(:) is at z _ I / q. T hus, from tile definitio n of h(z) gi~n in (1.3.46) and tile definition of e(:),

The funetion ~rges

0 (:) _ Iim CO

N__

f. R -(I

{ (a) .. _ (a) .. _1 (IJ)" (IJ)R_ I

with the underatandiDf; that (o)_I/(b)_ 1 _ 0 , and

l zn,

1

IKI,

by (1.3.47),

)" _ (a).. - Il N__ eo E{ (O (IJ)" (IJl 1

G( l )- Iim

RooO

R_ 1

. (o l N (0) .... - CO J~"" (b)N - Co (IJl

oo '

(1.3.48)

On the other hand, by ( 1.3.37), (1.3.49)

.

11In( ._ 11 - :)/{:) -

(a)..,(q/a)"" () q .... (b,) a "" .

Combining together (1.3.48) and ( 1.3.49), we conclude that (1.3.50)

(IJ)_(q/II )"" COp~bi~)Jji,l

B . C . BERNDT Putting (1.3. f>O) in (1.3. 4$),

wtl

complete the proof of (1.3.36).

0

Exe rcise 1.3.13. A ! a corollary, con~rt Tlieorem 1.3.12 into Iht followmg $ymmetric form not inllO/",n9 a b,/atem/ 3meo.

Corollary 1.3.14. If lPql

(1.3.51)

1+

< 1:1 < l / Ioql.

thtn

f, (I/o;q')ft(-oq)" z" + f .... ,

(ftq2; q2 )"

"_,

(I/P; q')ft(-Pq)" : _n (oq';q')"

{q'; q')"" (OPq2; q')..,( -q z; q2)oo( -q/ z; q')""

= (oq2 ;q2)..,(pq'; q2)..,(

oqz; ql)",,( Pq/z;q2)",,·

Exe rcise 1.3.15. Pro~ that }acobi'$ triple product identity, ( 1.3.11), can b<: deduudfrom Ramanujan'6 I ¥-I 6ummation theorem. Hint: fir!! let b = 0 and replact a bye. Now let z = - ble and q = ab, and /hen

/etc-O. Exe rcise 1.3. 16. Pro~ tliat tht q .bH1(lmiQ1 theortm, Thwrem 1.3.1, is a $p«iaJ ca.se of Ramanujan'3 ,-.,It, .rummation theortm. We dosoe thi~ chapter with one further major identity, the quintuple product identity, which is rnormously useful in the theory of theta functions. Wc follow our proof of the quintuple produet identity wilb proof!! of two beautiful corollaries that are analogues of J aoobi's identity in (1.3.24). We use the!ie two:> corollaries in Chapt.er 6 to help derive certain modular equatioll!l o:>f degree 3. The quintuple product identity can be formulated in several ways, and "'"e give three of them below. We provide a proof of the identity in the first SOltting. The<>rem 1.3. 17 (The Quintuple Product Identity; Finlt Version). For J", 0,

L ~

(I.3.52)

q-'n'+"(z3"q- 3,. _ z - 3" - l q 3n + l)

SPIRJ T OF RAMANUJAN

19

Theorem 1.3.18 (The Quintuple Product Identity; Second Venion ). For a " 0, ( _aq; q)"" ( - 1/a; q )",,(a 2q; q2)",,{ q/a2; q1)",,(q; q)""

= a - L(a3q; q3)"" {q2 /a3; q3 )",,(q3; q3}oo + (a3l; q3)",,(q/a.3; q3)",,(~; q3)"".

(1.3.aJ)

Theorem 1.3.19 (The Quintuple Product Identity; Third Version). We ha~ 2

3 ,,) f( _ x , - ;"x)/( _ ).,"z 3) (I .. f(x,>.z2)

= / (_ ;..2%3 _>.xG) +'/I_;" ,

"

_ >.2%9)

To deduce (1.3.aJ) from (1.3.52), replace l by q. set .: = -a..;q, and employ the JlIOObi triple product identity (1.3. 10). Setting a = l/z Md a 3 = >./q in (1.3.53) and utilizing the Jaoobi triple product identity again, we deduce ( 1 .3.~). Proof. ~t I(l) denote the right side of (1. 3.52). Then, for 0 < 1:1< we can elCpTell5/(:) 8$ a Lament series

00,

~

(1.3.:>5) From the definition of

1(:)

I,

=

L

a~.:".

we find that

1(1/:) '" _: - 2/{Z), or, from (1.3.55),

Equating constant terms, we deduce that ao = - G:! , and equating coefficients of : _ 1, we find that a, = O. From the definition of I, we a1so find that

ThLl8, from (1.3.55),

B.C. 8ERNDT

20

EqU.lltilll coefficients of :ft on both sides, ..~ fi nd that, for each inteser

•• Excrd!le 1.3.20 . Bl1l1erotion, prove thai, fur every mles-er n, (1.3.56 )

(l3~

q6ft -~Q.3" _3

'"'

_

q-'''' - 2Y>ao.

.•• _

= ... = q6..- 143oo _ 1 _ ... _

( 1.3.57)

IIl.. +I = q6,, -30. 3.. _ 3

q~'" (1\,

( 1.3.58 )

" 3" +1

q3,,'+1 ,,0.,.

By the Jaoobi triple product identity, T heorem 1.3.3,

I (-z' . - 'l' / ,'lf {-'l:, - q/:)

(q. ; q') ... / (I) -

~

L

=

(1.3.59)

( _ I YHq'Jl.i-I)"· 'z'JH,

j"'--",

Equating constant coefficients on both sides of ( 1.3.59), ....... find tlw ~

(q4: q4).. ao

L

"

(_ I y" qSi'-2j

(q4 ;,').."

_

J--"" by tbe pentagonal number t heorem , Corollary 1.3.6. Henoe, ao = 1,

and

~ince

Q, \ . ,

"1 _ -ao .....e also ded uce that

a, _

- I. Recalling aJao that

0, "'~ oondude from ( 1.3.56 )-( 1.3.58 ) that, for each integer n,

P uuilli ,belie value\! in ( 1.3.SS), ,..~ complete the proof of ( 1.3.52), after rep1adna: n by -n- I in tbeseoond sum arising from ( 1.3.:.5). 0 Corollary 1.3 .:U . &coli fh
-

L

(6"

+ I)qh' .... .. cp'(-q')/(-q').

Proo f. Divide both side/! of ( 1.3.52) by I - q/l. By I. 'H&!pital's rule,

S PIRIT OF RAMANUJA N

21

Hen<:e, lett ing z tend to q in (1.3.:>2) and employill& (1.3.13) and (1.3.15), we find that

-...

L

(6n + ' )q~'''' '" (q'; q')~(q' ; q4 )!, _ .jl(-q'l/( - q'),

".-

and this complelell the proof.

0

C .... r .... llary 1.3.22 . 1/ .p(q) and I (-q) are dtJintd by (1.2.3) and (1. 2.4 ), l"e8putlwly, then

-

L

( 1.3.61)

(3 n + I)q~''''''' ... y,(q'l/'(-q).

P roof. We apin \IIIe the quintu ple prod uct identity. Divide both side& of (1.3.52) by 1 - %' and let:: tend to 1. First Il(lte that as % _ t, the left. ~ide of (1.3.:>2) tends to

by (1.2.9). Thus,

we apply L' II OIIpital's rule

1.0 deduce that

't""'00

L..".-oo tf"'+"(Z'''q-3o> _ Z-S.. -If''' +I ) - -~ C~OO 3111l~'-2" + "r;oo(3)1+ 1}tf"·H.. +I ) - ~ Ct-oo 3mq3,..'+'... + ..t-"",(3m + 2)q3..• U ... ) lilll

._1

I

-

.. L

:;:2

,

(3 m + \ )q3 ... H "',

"'.-"'" where, in the antepenultimate line, we !let n - - m in the former sum and n _ - m - \ in the lalter sum. Remembering that ...-e have dividtd both sidel of tf~~RrcfM"8t:,end hll\·e let 1" - I, we find

" f: from

B. C. BERNDT ~be

calculation above, (1.3.1 4), a nd (1.3.1&) that

(3m + 1)q3""+~'"

= (ql; q2)",,(q; ll!.,(q4; q4)!, = ,p(q~)/l( _q),

which oomp\etes the proof.

o

Although we have e8tabli5hed only a few theorems about q-soeries and theta funct.ions , we have developed enough machinery to prove some beautiful, significant theorems about the arithmetical functions introduced at t he beginning of thi s chapter.

1.4. Notes For an introduction to the elementary theory of partitiotlll that is eminently acressible to undergrad uates , read Integer ParlitlOfl.f (231. by G. E. Andr(!w$ and K. Eriksson . The mOllt authoritative account on t he theory of partitions is also by Andrews (1 4J. several portions of wh ich are suitAble for undergraduates. Hardy and Ra.manujan [llOJ, [191 , pp. 276-309) actually pl"
SPIRI T OF RAMA NUJA N

23

circle method, because Hardy and J. E. Littlewood extensively developed the method in a seriee of papers. In 1947, Lehmer (141) wrote, ~ ... it is natural to aU whether 1"(n) = 0 for any n > O.~ Although not stated as a conjecture, since that time the nonvanishing of 1"(n ) ha.'i been known as Lehmer's con· jecture. Theorem 1.3.1, the Q-binomiaJ theorem, is due to A. Canchy (66, p. 4:>J, ""'hile Corollary 1.3.2 was first proved by L. Enler [91 , Chap. 16). T he Jaoobi triple product identity was proved by C. G. J. Jaoobi in his Fimdamenta Nova [131], the paper in which the theory of elliptic functious was founded and one of the mOll! important papers in the history of mathematiC!!. Hov."tlver, the Jaoobi triple product identity was first proved by C. F . Gau~ (91, p. 464]. The proof that we have gh-en here is independently due to AndreWll (12) and P K Menon (155J. F. F'l"anklin devised a beautiful oombinatorial proof of Corollary 1.3.:> (o~ Corollary 1.3.8). See Hardy and Wright'S book [112, pp. 286-287] or Andrews's text [14, pp. IQ111 for F'l"anklin's proof. Jaoobi'" identity, Theorem 1.3.9, has an elegant generalization found by S. Bhargava, C. Adiga , and O. O. Somashekara ]60]. Ramanujan's I>PI sum mation theorem was fil"5t stated by Ramallujan in his ootebooks [193, Chap. 16, Entry 17]. In fact, he stated it in preci$ely the form giwlfi in Corollary 1.3.14. It was found by Hardy, who called it, "a remarkable formula with many parameters" and intimated that it could be established by employing the qbinomial theorem [101, pp. 222- 223] . Howe"er, it was not until 1949 and 1900 that the first published proofs were given by W. Hann ]103) and M. Jacbon (130), respectively. The proof that "re have given here is due to K. Venkatachaliengar snd was presented for the first time in the monograph by Adiga, Berndt, Bhargava, and G. N. Watson [51; see also Berndt's book [3 4 , pp. 32-34J. As with the other thoorems;n this chapter, there are now many proofs of the Ilh th"", rem. Almost all proofs employ the q-binomial theorem at some stage. W. P. JohllSOn (132) has written an interesting semi-expository paper showing how Cauchy could have diso;xr.-ered Ramanujan's 1>P1 summ ation formula (but did not). Rams.nujan left uo clues about his proof Copyrighted Material

8. C. BERNDT

"

or proors. However, it iI quite pOS'Iible that Ram/UlujAll ~ the theory of partial fr8(1.Ions, which i8 a g(>~ization of the method of partial fractions all e&kulus studenu learn la evaluate integrals of rational functiorul. For a proof of Ramanujan 's IVlI lummation theorem employing partial fractions, 8ee a pllper by S. 11 . ChlUl [73]. but this proof al50 utilius the q-binomial theorem. Readert might a)llllider the following

ilUltructi~

e.xereil!e.

E"",rcise 1.4 . 1. COll.!ider the Lau""n! expalUion of

(a~ )oo ( blz) ""

I ( Z) :. (dz)"" (q/ (dz ))",, -:

Proc«dmg like _

~

~

~::oo eR: .

did in the proof of Jhml
th«>l1!m. d«luce thol

( 1.4.1)

(· I /). (')' e.. - (ql /( bd).. b C(J,

- 00 < 0 < 00.

11 fQII()~ that

(\. 4.2)

;c-

(· I d). (") " 1(: ) =C(I ~ (ql /( bd» .. b . ~ . _oo

Applying Rarnanujan', lW, &um ma tion theorem IQ Ihe righl side of

(1.4.2), we find that ( 1.4 .3 )

{
'The q-billOmilll theorem is .. specilll CM!! of a p roduct repre!Jentation of 11 hMic hypergeometric Mri('JIII, where here oo.n c is !lOt meant to he 11 !lynonym for /undamell/cU but instead. refen to the "'ut: Q. Roughly, ill general , a basic hypergeoll\etric series , t/!k has j q.. produclJI in the numerator and k + I q- produCIJI tn the denomina-tor. where one of the q-produclJI in the denominator is (q; Q)", if n ill the ind ex of summation. Clear and more extensive introductions to q-M'rics can he fou~Jp:i~~~ A drews [141 and Andrews,

SPl lUT OF RAMA NUJAN

Ror [18, Chap. 10[.

"

By fM, the most. oompreben~ sj,"t tTelltise on q-llerie!l is by G. Gasper and M. Rahman [96). An int roduet ion to Ramanujan's extell!live contributions to theta fuoctioDll can be found in Berndt'a lecture IlOtf:!l [36]. The interaction of q-:leries with numbl':r t heory is best demonstrated in N. J . Fine's beautiful but .uccinctly written monograph [94]. Some illuminating historical remarka on q-series Me given in Roy '~ review [200].

R. Askey, and R.

T he qu in tuple product identity hM been di!lOOveroo many ti mes in t he pMt century, and an lICCOun t of all proofs known to the author in 1991 can be found in [34, p. 83]. However, a more oomprehensi,-e survey di8cUMIing more t han 25 Imown proofs has been written by S. Cooper [79[. The formulation given in ( 1.3, ~ l is that given by Ramanujan in his lost notebook [194, p . 207]. The proof of the quintuple product identity that ..-e ha\"
book [94, p. 83].

COpyrighted Material

Chapter 2

Congruences for p(n) and T(n)

2. 1. H ist orical Background This chapter is primarily devoted to proving some of Ramallujan 's congruences for pIn) and T(n). We begin wi~h a remark on notation. Throughout the chapter. we shall see congruence.; of the sort

(2.L I)

J(q) = y(q)(modm),

where J(q) = LOnq" and g(q) = 2:bnqn are power series in q. T he congruence (2.1.1) is equivalent te> the wndi tion an ;;; b" (mod m) for every integer n appearing as an index in either pov.-er series. In 1919, Ramanujan [188J, [19 2, pp. 210--213J announced that he had found three simple congruences satisfied by p(n), uamely,

(2.\.2)

p(5" + 4)

(2.1.3)

p(7n

(2.1.4)

== 0 (mod :» ,

+ 1i) == 0 (mod 7),

p(ll n + 6) ;;; o{mod 11).

He gave proolil of (2.1.2) and (2.L3) in [188] and later in a soort one page note [190), (192, p. 230J announced that he had &Iso found a proof of (2.1 .4). He also remark'! i n [l OO] that "It appears that there are no equally 5imple prOpeTties for any moduli involving primes other th&t1 these t hree ." In a posthumously published paper (1911, (192, Copyrighted Material

8 .C. BERNDT

"

pp. 232- 238]. Hardy extracted differen~ proofs of (2. 1.2)-( 2. 1.4) from an unpublished manuacript of Ramanujau on p{n ) and T(n ) [194, pp. Ill-In]. [00).

Let"

In [1881. lUmanujan oif...-l a mon! &enerlll conjecture. = let ,\ be an int~ such ~hat 24>. • I (mod 6). Then

~. ~ 11 C and (2.1.~)

p(n6 + >.) "" 0 (mod d) .

[n his unpublished manusc;ript [194, pp. 133- 171], ]501. Rarnanujan gave" I)roof of (2.1.:1) for arbitrary Q. and b = c = O. lie also began a proof of hill conjecture for arbitrary band 11 _ C ,., 0, but he did not complete it. If he bad com pleted bis proof, he would have noticed that bill conjecture in tbi8 ~ needed Ul be modified. RamMujan had formulated hit COIIjectures after studying 11 table of ,'IIlU('ll of p(n), O S n 200, made by P. M"CMahon. After Ramanujan died , H. Cupta. ateoded MaeMahoo '. table up to n = 300. Upon ex.amining Gupta'. table in 11134, S. Chow!.. ['15J found thllt P(243) is not divisible by~. despite the fact that 24 . 243 == I (mod 1 3 ), To correct R.amJwujll.ll's conjecture. define 6' .. ser' lI c, wh,,", 1I .. b, if" _ 0, 1, 2, ll!ld 11' _ 1(b + 2)/2J, if b > 2. TheD

:s

(2.1.6)

p(nd + >. ) ;: 0 (mod d' ).

In 1938, C . N. WII.L90n {2IS] publjghed 11. proof of (2.1.6) for a ". t '" 0 rind gll.Vl!! 11. more detll.iled \'~rsion of Ramanujan 's proof of (2. 1.6) in the <:age b _ c _ O. It WM not until 1967 that A. O. L. At kin (28J pllMld (2.1.6) for arbitrary c and a "" b _ O. The tau fuoction T(n ) was introduced by Rlunanujan iD his fa. PllpeT (186], 119 2, pp. 136-1621. Althou&h he pTO\'ed titt~ about T(n) in this PllpeT, hf! did formulale'.ome fundamental \XlIIi«lure. about T(n ). ID [l OO], Ramanujan Slated without proof ooogru-ern:e!I for T(n) modulo 5, 7, and 23. In his unpublished manll8Cliptoo P(n) and T(" ) 1194 , pp. 133- 177J , Iso), he proved thet!e oongrWnceII and lIeveral further r(!I:IulUl on T(") .

InOU!I

2.2 . E lem e ntary Congruences for 1"{n) First , ~ show that T leij\.~

ltMenaJ

SPIRIT OF RAMANUJAN Theorem 2.2 .1. The

num~

01 valuu 01 n ::;

:t

lor which 1'(n) is

odd t.quals

(2.2.1 ) when:

[ (:tJ

P roof. orem,

1

+,fi].

denote.! th~ greGtul integer le33 than or equal 10 :to

Obse..,.~

that, for any positive integer j, by the binomial the--

Hence,

== (q! ; qt )_(mod2).

(q;q)!,

Therefore, using the d(lfinition of 1'(n) in (1. 1.10), t~ congruence above. and Jaoobi's identity, T heorem \.3 .9, we find that ~

L 1'(n)q" '" q(q;q)~ == q(q!;q!)!, =

Thus, 1'(n) is odd or

(l~n

E(

-1)"(2n + l )q(2,,+II' (mod 2).

according aJl n is an odd square

Or

not.

E"ercise 2.2.2. AJ an elementary trercL!e, $how that the number 01 odd $quare! leu than or t.qua/ to :t ;3 precisely (2.2.1). So th(l proof of Theorem 2.2.1 is complete.

o

Theorem 2.2 ,3. For ooch nonnegati m: integer n, (2.2.2 )

1'(7n), 1'(7n

+ 3). 1'(7n + 5), r(7n + 6) == 0 (mod 7).

Proof. Applying the binomial theorem, proof, we eaJlily deduce that (q; q);'"

and 80 (2.2.3)

f. ~ .. l

1'(n)q"

8.!!

we did in the previous

== {q'; q7)oo (mod 7),

= q(q; q)~ == q(q; q)!,(ql: q' )!, (mod 7). Copyrighted Material

B.C. BERNDT Since the powel1l in (qT: qT)!., are all mult ipiet 017 , we need ooIy . . . . .tt ~

q(q:q)!., _ ~) - 1 }" ( 2n + l )ql+ ..(.. t-I 1!1,

(2.2.4 )

by Jacobi', identity, Theorem 1.3.9. O~ that 1+ " (n + 1}/2 == 0, I, 2, 4 (mod 7). Moreover, I + n (n + 1)/ 2 • 0 (mod 7) if and only if n E 3 (mod 7) or 2n + I '" 0 (mod. 7). It now follows from (2.2.3) and (2.2.4) that T(7n) • 0 (mod 7). It aJso follows that there lIle 1\(1 po.,,.ers of q on the right !ide of (2.2.4 ) thM lIle OOTIiruent to either J, 5, or 6 modulo 7. The latter three oongruencelll in (2.2.2) thUII foIJow from (2.2.3) and (2.2.4). 0 For example, r (3) ., 252 :: 0 (mod 7), T(S} _ 4830 • O(mod 7), 1'(6) _ - 6048 :: O(mod7) , and '7'(7) _ - 16744 • O{mod7) . ThI! reader will o'-n'e that 3, 5, and 6 are the quadratic nonnsidlll!ti moduk> 7 and that indeed the proof of ~m 2.2.3 de~rar.es this.

Theorem 2.2 .4 . Lt.1 r, 0 :5 r < 23, denote ony quadrot" rucdut moduJo 23. Then. f or /'.IleA pQ.ritive integer n , (2.2.5)

T(23n - r)

E!!:

0 (mod 23).

Pn)Of. By t he bitlOmial theorem , ~

..L,T(n)q" Since the

po'I"erl

q(q: q)~ = q( q;q)O<){qu: q1l )O<) (mod 23) .

in (~ ; qU ) oo are all multiples of 23, we need only

oona~ ~

(2.2.6)

q{q ;q)oo

=

L

(_ I}"qlh()OO+1)/2,

by EUler '. pentagonal number theorem (1.3.18). Ob8en'e that (2.2.7) 1+ n(3'~+ 1) '" (6n+I)2 _ 23,.,(3; + I ) . (6n + I )2 (mod 23). Thus, T(m) will be a multiple of 23 when In is not oongruent WI 8Qll~ llIodulo 23. In other wordSl, if ....'e II(!t on _: 23n + t _ .1,1,0 $

( < 23, for IOme in~~~m~I~~ ( mWlt he .. quadratic

SPIR.I T OF RAMANUJAN nonresidue modulo 23. Recall that r ill a quad~tic residue modulo 23. Since 23 • - 1 (mod 4), - r . r t- 0, ill a quadratic nonresidue modulo 23. It folJowa that (2.2.8)

r (230 - r ) • 0 (mod 23),

0 < r < 23.

o For example, 3 and 6 are qua dratic residues modulo 23, and so r (2O) - - 7109760 • 0 (mod 23}.md .,.(17) .. - 6905934 • 0 (mod 23).

We shaH later examine congruences for .,.(n) modulo S. However , to establillh these congruences, ...~ first need to prove Ramanujan 'a

famoUll COiIIrucnce f.:x- p{n) modulo S.

2.3. Ramanujan's Congrue nce p(5n. + 4) == 0 (mod 5) In tllia mOllOpaph , _ present four proofs of Ramanujan '8 congruence for pen) modulo S. We offer three proofs in thit IleCtion. The fint and the IIKOnd are more elementary than the thi rd, but the third gives more information. Theore m 2 .3 . 1. For each (2.3. 1)

nonnegati~ inttg~r

n.

p(Sn + 4) :; 0 (mod SI·

First P roof of Theore m 2 .3 . 1. Our first proof ilI l.aken from Ramanujan'. paper 1188), [J92 , pp. 2W- 213j and is reproduced in Hardy '. book 1107, pp. 87-881· We begin by .... riting 4 (~: ,h", '" ( 2.3.2) q(q,.q) 00 (q; q)~

q (
'::-0

(q:q)""

By the binomial lhoorern, (2.3.3) (q:q)!o _

(q~; q& )",, (modS)

or

(
' '')~OC . 1 (modS)

COpyrighted Ma/anal q,q ""

8 .C. BERNDT

"

Hence, by (2.3.2) and (2.3.3), ~

q(q ; q) ~ =: (q~;

(2.3. 4)

L p(rn),,,,+1 (modS ).

We now lee from (2.3.4 ) that in otder to ,bow Iha~ p(Sn + 4) :; O{mod 5) we rnll8t show that the ooefficienu of ,f"+$ on t he left. side of (2.3.4) are multip\eo of 5. By the pentagonal number t heorem, Corollary 1.3.5, and Jaoobi'a

identity,

1.3.9,

T~rem

q(q;q)!, ..q(q; qloo(q; q)!, =q

f:

(_ I )'q'(3J +1I/ Z

J--""

.. L

~

(2.3.5)

E <-lt(2.\: + 1 )q~(HI1I2 k .,1)

~

L {- I)-'+i {2k +

l )ql+J(3J+ 1 )/H ~ ( HI )/1

Our objective is 10 determine when the expor>enu on the right aide ~

multipll'S of 5.

O~n~

that

+ (2k + 1)2 .. 8 { I + !i{3j + I ) + ik(k + I J} - IOj2 - 5. + b(3j + I ) + !k(k + I) is a multiple of :; if and only if (2.3.6) 2/j + \ )2 + (2k + 1)' I! 0 (mod 5). It is tasHy checked thllt 2(J + \)2 "" 0,2, or 3 modulo:; and that 2() + 1)2

Thus, I

(2,1; + 1)2 .0, L, or 4 modulo 5. We the~ro", if and only if (2k

1ft

that (2.3.6) ill true

+ 1)2 •

0 (mod 5).

III particular, 2k + I • O{mod5 ), whkb , by ( 2.3. ~), Implies t W I ~ coefficient of ~+5, 11 ~ 0, in q(q; q)!c, is a multiple of~. The coefficient of r/"+$ on the right side of (2.3.4) is therefore &00 • mult iple of ~, i.e., p(~1l + 4) is a multiple of S. 0 We now Ki~ a .geC<Jnd simple proof due to G . E. Andrews [I S] "nd bII8ed on the ~im ple lemma given below. See .I!tO all extclll!ive genc •• Hution of thia lemma by AndrCWII and It. I\oy (l4J. In par' ticular , taking a special CIL'le of their general theorem , Andrews and !toy ftltabHsh t he congruence P{7,. + SI ii! O{mod 7). COpyngtrted Ma(enal

S PIRJT OF RAMAN UJA N

33

Le mma 2 .3.2 . Let {(I~ ) , "
(2.3. 7) 16 divi6ible by S,

P roof. Write (2.3. 7) in t he form

L() q

=

(

)3 2:::-0 a.",qm' _ ( )3 2::. 0 amq"" ( od 5) 00 (q;q)l, = q;q"" (<1;<1)"" m ,

q;q

by the binomial theorem. Using Jaeobi '$ identity, Theorem 1.3.9, we

thus see that it suffices to examine the coefficient of .r~+3 in (2.3.8)

(q; q)~

..f:,

a",q"" "" f ) - IY(2j + 1)q-'(j+1)/l

,.,

..f:,

(lmq"" .

j(j + 1)/2 + m l _ {in + 3, where n 2: 0. It is eILSy to see that thill condition is e<jui.....tent to the congruence We want those ter ms above for which

(2.3.9)

(2j

+ 1)2 + 3m 2 ::

°

(mod S).

Since (2j + 1)2 :: 0, ± I (modS) and 3m 2 :: 0, 2, 3 (mod S), (2.3. 9) hold!! only when (2.3.\0)

WO!

see that

m :: 2j+1 :: O(modS).

The coefficients of q~~+3 in (2.3.8) are then composed of terms of the .'IOrt (- I)j{2j+ 1)(1"" which , by (2.3.10), are aU multiples of 5. 0 Exercise 2 .3.3. Use (1.3. 13) to show Ihat

- ) - (q;q)""

'11 ( q - (_q;q),.,"

(23 .. 1 I)

Second Proo f of Theore m 2.3 . 1. Using (2.3.11),

..,

~ p(k)ql~ = _ _I _ ~

-

1

= _ 1_

(ql;q2)oo - (q;q)",,( q; q)""

= (q;:)1

( 1+2

f (_l)mqm')

eop~tfJl:rMaterial

we

find that

(q;q)""

(q;q)~ (_q;q)""

B.C. BERNDT By IA-mma 2.3.2. the cof'fficienta P(k) on the ~ft aide aoo.'e IU'e muJ. tiples of ~ whene>""er 2k i f ~j + 3 {mod ~). Le., whelle\"eT k _ ~n + 4. Thls then completes our second proof. 0 Our third proof is also due to Ramanujan in [188J, but it is only briefly indica.ted in that paper. In his unpublished IDanWlCript on ptn ) and T(n ), [194J, [:;OJ, fuunanujan gives a more detalled liketeh. In this proof. the congruence p(5n + 4) ;;; o(mod 5) follows from a beautiful identi ty. Following Ramanujan, throughout the proof _let Jk {q), k _ 1, 2. denote lX"''er aeries with integral powe.. and integral coelficient.l, not ~ily the same at each appearance. T he precise identities of J I (q) and J 2 (q) are not important for the proof. Theore m 2.3.4 . We have

f:. pt5" +

(2.3. 12)

R-o

4)q" _

~ (.r:.r!~. (q: q)...

P roof. We begin by writing

(2.3.13)

(

I/~·

I /~)

q(.r ; ~ ) .. 00

= J I{q) _ ql/ a + h

(q)q2 / ! .

To

Re this, UIlC the pentagonal number theorem (1.3.18) in the numerator on the left side of (2.3.13) ./Uld. if n is the index of summation. divide the ltrms Into rtllidue classe!! modulo~. If ,, ;;; O, 2 (mod5). then the p<:I'Nt" are integral, and these. ",hen divided by (~ : q~) .. , aooount for the ltJ'IDlI in J I(q). If n E 3 , 4 ( mod~). then the pmoen are of the form k + 2/:; . where k ia 11 posit ive inte&er, i.e .. "'-e obtain the t.Nma J 2(q),r/a.

Exen:~ 2 .3 .5. !.cutl". when n "" I (mod~). lUe the penlGgolllll number theorem, CoroUa~ \ .3.5 . to -"'aw tha t thue term!" m (ql/l: ql /&) ... are equal to _q l/~ (q'I ; .r )"".

Hence. Wll obtAin the term _ ql/a in (2.3. 13). Now cube botb aidCIL of (2.3.13) to obtain ( 1/5. Ila) 3 q(rr :: ! )!'.. 3JM - ql / ! (3Jf - JM + 3Jlq2'~( 1 + JIJ,)

(J: -

(2.3.14)

- q3/~J;;;;fi!J M!fJ~& ( 1 + Jlh ),

SPIRIT OF RAMANUJAN

35

where for brevity we have deleted t h
( 1/$. (/5)'

q(q<~)~00 = Cl (q) + G2 (q)ql/~ + 5'13/5,

where G,(q) and G 2{q) are power !lerie!! with integral powers and integral coefficients. Hence, equat ing coefficients on the right side!! of (2.3. 14) and (2.3. 15), we find that

From anyone of these equalities, we deduce that (2.3.16) Our next tllSk is t.o use (2.3. 16), (2.3.13), and "'ratiolUl.lization~ to show that (2. 3.17)

(q5;
JI

q'/5

(J: +

+ Jzq~/'

3J~q)

J~

1[3 / '(3J I

+

+ q l/5(J~ + 2J~q) + q2/$(2J~ + Jfq }

J?

l1q + q2J1

+ J~q) + 5q~/' lI q+q2J1

.

We now demolllltrate how to prove the !leWnd equality in (2.3.17). Return to (2. 313) and replace ql/' by wql/&, where w is any fifth root of unity. Tbus , (2.3.18)

B. C. BERNDT

36

Let w ruU through all live fifth root.8 of unity and multiply all five such equa.lities (2.3.18) to obtain

(2.3.19)

I] ""q(qS:~q)"" (

1 /~)

1/ &.

..,

=

1] { J1(q) - wql /~ + h(q)wlql/~ } .

First examine the product on the left aide of (2.3.19). Using the fact t hat the sum of tbe live fifth roots of unity equals 0, we see that if n ill not 11 multiple of S, we obtai n product!! of the form

(2.3.20) (1 _

q~/~)(l

_

However, i f n

wqn/~)( l _

= Srn, then the correspooding terms are

instead of I - qft that

(2.3.21)

..,lqn/5)(I _ w3qB/5)(1 _ ",.qft/5) = 1 - q",

IT..

we obtl:liooo in (2.3.20). Thus, ...-e find that

("'9'{5,,,,q'/5),,,,

(q5 ;qS).,.,

(q:q)!,

= (rr;qS)~ '

We now examine the product on the right side of (2.3. 19). Sin<:e th~ lUe no frl'Ctional powel'll of q on the left side of (2.3.19) by (2.3.21), there are none on the right side 8.8 well. Thus, we only need toe:umine thO$C t/ll"lllB in the product tha.t give rise to integral powers of q. A brief inspection t hen oonvince8 us that (2. 3.22 )

I1 {JI(q) -

r.Jq" ~ + J2(q)w2q2'~}

= J:(q) -

•

where C l is the sum of the

(,.Ll =

q+ J~(q)q~ +C,q+C1Q, 20 terms of the form

-Jlh"lW1WJW~ = r.J1~r.J~, and C~ ill the sum of the (1.~.1) = 30 ten IIll of the form - J?JJW1W~W~ = - r.JIW~r.J~, since J,J1 = _ I. lI ere. for each j, r.JJ is a fifth root of unity, a.nd in each product r.J, '" "'. if j '" k. Fi rst, el
L-

""

......

_1

W."'~

= -"'~Ws- \ = - I ,

where the 8Um ill over the four fifth rooUi of unity "'4, ex~pt %. SillOl! there are five possibilitIes for <Js, ~ (:Onclude that Cl = -5. Second, Gopynghrea Malenal

S PIRIT O F IlAMANUJA N

37

exam ine the lerms -wlo.1wl Now,

L ..........

-

"'1<.1"'~ = (o..>;z + 1J;J)o4..,~ - ~<4 + <40.4,

.... "....

There are five IK*libilit ie$ for '-'3, /,Ond !\O it would lleem that we obtain a oontribution of - 10 to the value of C,_ However, beeaU6E! of the symmetry of o..>;z and '-'3, we have counted each contribution t.o C 2 t",·ice. i.e., in fact, C, = -5. Using our values Cl _ - 5 = C2 in (2.3.12), "''eo deduce that (2.3.23)

I) {J1(q) - ..,qlt ! + J,(q)w'q' fB} = J:(q) - lIq +Ji(q}q'·

In summary, from (2.3.23), _ ha, "", shown that I fL,t1 (J1(q) - ..,q1t. + J, (q)w'q2t!} 'J~,(~q)~_-q:;'''''~+~J'''(q~)q':v.f! = fL (J1(q) - ..,ql/$ + J,(q)w'q't&}

(2.3.24)

F

n ",,.l (Jl(q) - ..,q1/S + J, (q""'q'f B} JHq) IIq+Jl(q)q2 F(q) llq+Jl(q)q"

JHq)-llq+Ji (q)q' (q) - J1(q) ql /B + J,(q)q2/5·

If we mnaider the numerator and denominator .!:Iove 811 polynomials UIIe long divillion , ...'I! find that F(q) indeed ill the numerator OD the right aide of (2.3. 17). Th ...., by (2.3.24). the proof of (2.3. 17) ill complete. Recalling t hat on the left side of (2.3. 17) I/( ql /l: ql /l)"" ill the generating funct ion for pIn), ....... select thOlle term. on both aid611 where the po",en of q (U"f! congruent to 4 /5 modulo I. We then divide both sidell by q4/B to find that in q1 f! and

(2.3.25)

8 .C. BERNDT

38

However, from (2.3.1 9), (2 .3.21 j, and (2.3 _23 ), we al$c) know that (q~;~)~

(2.3.26)

(q; q)t,

1

Jf(q)

=

11q + J~ (q)q~'

Utilizing (2.3.26) in (2.3.25), we complete the prOQf of (2. 3.12).

0

Theorem 2.3.4 clln be utilized to provide a proof of Ramanujan's oongruence for pIn) modulo 25.

Theorem 2.3.6. For

e~ry

nonntgal1t>e inttger n,

p(2,sn + 24) :: O(mod25).

(2.3.27)

Proof. Applying the binomial thoorem OD the right side of (2.3.12),

we find that (2.3.28) ~

LP(5n

('~)'

~

(q;q)oo

....0

+ 4)qn == 5~ = S(q5;q~) !, L

.. _ 0

p(n)q" (mod 25).

From Theorem 2.3.1 we know that the coefficients of q4, q~ , qU " q5 .. H " .. on the far right side of (2. 3.28) are all multiples of 25. It follows that the coefficients of qh .H , n ;:: 0, on the far left side of (2.3.28) are also multiples of 25, i.e., p(2:;n

+ 24) == 0 (mod 25).

o

This completes the proof.

Ramanujan's congruence in Theorem 2.3.1 yields a simple proof of Ramanujan's congruence for t he tau function moduJo 5, as "'"1) next demonstrate. Theorem 2.3.7. For each nOl"megatiue mleger n, (2.3.29)

r(5n)

E

O{mod5).

Proof. By the definition of r{n) and t he binomial theorem, (2.3.30)

L,., T(n)qn :

". 1

q(q; q )~

(.)2~'"

= q (:: q)

00

== q(q~; ~)!,

copyrigh1e'3 Ma/enal

L p{n)q" (modS).

n_ g

SPIRlT 0)<' RAMANUJAN

39

By Theorem 2.3.1, t he coefficient of I/'~, n ~ I, on the far right side of (2.3.30) is a multi ple of 5. Thus, tile coefficient of 1/''' on the far left side of (2.3.30) ia 11 multiple of 5, i.e., T(S,,) :;:;: O(modS). 0

2.4. Ramanujan's Congru en ce p(7n + 5) =: 0 (mod 7) We consider next Rarnanujan '. congruence for p{n) modulo 1.

T heorem 2.4.1. For each

nOtln~gahve jn!~er

p{1n + 5)

(2.4.1 )

E:

n,

0 (mod 1).

Proof. Our proof is again taken from I~amanujan's paper (188) and sketched by Hardy [101, p. 88).

I¥M

First, by the binomial theorem , ql(q7; q7)""

f:..

.. ~ O

p{n)q"

= ql (/ll; q7)"" = l(q; q):" (q7 ; q1~"" (q;q)""

(2.4.2)

(q;q)""

:;:;: l (q; q):" (mod 1).

Hence, if we can show that the coe1licient of q7"+1, n ~ 0, in ql(q; q):' is 11 multiple of 1, it will follow from (2.4.2) that the coefficient of q7n+1 on t he far left side ill a multiple of 1, Le., p(1n + 5) == 0 (mod 1). Applying Jacobi 's identity, Theorem 1.3.9, "..e find that

rl(q; q):" = ql{(q; q)!.,}l ~

=L

(2.4.3)

~

L ( - I)J+k{2j

+ 1)(2k + l )qH j(j +L )/Hk(k+ L)/l.

j.{) h . O

As we saw in the previous paragraph, we want to know when the exponents aoov" are multiples of 7. Now o~rve that (2j

and

$0

(2.4.4)

+ 1)1 + (2k + 1)2 = 8{2 + !iU + I) + ik(k + I )} -

2 + ij(j + I} + lk(k

+ l)

14,

is a multiple of 7 if and only if

(2j+I )2+(2 k+l )1 :;:;:0 (mod 7).

We easily ~ that (2j + 1)1, (2k + 1)1 ::;: 0, 1, 2, 4 (mod 1), and so t he only way (2. 4.4) can hold is if boLh (2j + 1)2, (2 k + 1)2 == O(mod1). (n such cases, we triv~9'p~Jn3~~~cienls on the right Bide of

B. C. BERN DT

40

(2.4.3) are multiples of7. Hence, tile ooefficient of q7 nH,fI;:: 1. on the left side of (2.4.3) is a multiple of 7. As we demonstrated in the foregoing paragra ph, this implies that p{7n + 5) ;;; O(rnod 7). 0

We now consider the analogue of Theorem 2.3.4 for p{7n + 5), which w!IS sta ted without proof by RAmanujan in his paper [188]. In hill unpublished manuscript on p{n) and r(n), he give!l a w:ry brief sketch of its proof [194, pp. \33-171]. [50]. There are now several proofs of T heorem 2.4.2, but t he details of Ra.manujao's proof were worked out only ~ntly by Berndt, A. J. Yee, and J. Yi [541. Because tbe details are cumbl!['/I()me, we provide only the central ideas

and refer readers to [54J for a complete proof. Readers should not aUcmpt to oomplele missing det ails hut merely try to grasp the ideas

behind Ramanujan's proof. Theorem 2.4.2. W( hat'<:

f:

(2 .4.5)

p{ 7n

... &

+ 5}q" = 7 {qT; qT!:'., + 49q (qT; qT"' . {q;q)."

(q;q)."

It is cle3l" that Theorem 2.4.1 is an immediate oorollary of"I1leQ. rem 2.4.2. P r()()f. Usi ng (1.3. 18) in both the numerator and denominator and then separating the indices of summation in the numerator into I"ilSidue classes modulo 7, "'"Cl readily find that (2.4.6)

where J" Jl, and J3 are power series in q with integTaI coefficients, and where the pentagonal number theorem W8.S used to calC11!ate the coefficient of ql/7. Cubing both sides of (2.4.6), we find that

(qIIT; q'/T):'., (q7;qT)!, =

(2.4.7)

(J? + 3Jihq - 6J , h q) + ql 17(3J~ J 2 - 6hJJq + Jfq2)

+ 3q2/1(J,J? - J f + JJq) + qJ/1{Jt - 6J')2 + 3J, J~q) + 3q~f7(J1 - Ji + hJ~q) + 3q~/7( J2 + J~J3 - JJq) + q~f7(6JlhJ3 - I). Copyrigl!/ecj Material

SPIRlT O F RAMANUJAN On the other hand. using Jacobi 's identity, Theorem 1.3.9, and separating tbe indices of summation in the numerator On the left side of (2.4.7) into residue cJ_ modulo 7, we ellllily find that

(2.4.8) where Cl. G 2, and G 3 are power series in q with integral coefficienLs, llTId where Jaoobi's identity, Theorem 1.3.9, WIIIl used to determine the coefficient of q61T , Comparing coefficient.!! in (2.4.7) and {2.4 .8}, """ condude that

Jf + J 3q = o. J, - J? + hJ1q _ 0. JIJ? -

(2,4.9)

+ J P3 - .!Jq = 0, 6J, hJa - l .. -7,

{ J~

Replace ql/7 by wq'/1 in (2.4.6), where w is any seventh root of unity. Therefore, (2,4 .10)

(wq 1/7.,wq 1/7) oo=J+wqI/7 J _ l<J2q2IT+w6q~/TJ .

(qT;q1)""

1

2

3

Taking the products of both sides of (2.4.10) over all seven seventh roots of unity, we find tbat (2.4.1I)

(q; q)!, = II {Jl+l<Jql/ Th _ wq211+wq3/TJJ)' (q7 ;q1 )!, '"

(2.4.12)

n",,,,

We only need to compute the terTIL'l in (J I +wqJ2_wq2 +l<Jq3 J3) where the powers of q ~%MmJw~J!i};tJj w complete the proof. In

8.C. DERNDT

42

order todo this, we need to prove !!e~ral identities using the identities of {2.4.9}. We do not give the detaim. ChOOlling only thoiie terll1$ on each side of (2 .4.12) where the p(l'.vers of q are of t he form 7.! + ~ and using the omitted cakulations, we find that

.-.

n .. S(modT)

-

(2.4.13)

( 49. 49)'

( <9 . • 9)'

" p(7n+S)'" = 7 Q , 9 ""+49q,9 , 9 co. ~ q (q1;q7)!., (qT;q1)~

Replacing q1 by q in (2. 4.]3), we complete the proof of (2.4.:;).

0

Recall that the identity ofThoorem 2.3.4 yielded in Theorem 2.3.6 a congruence for p(.. ) moduLo 52, Similarly, the identity in T boorem 2.4.2 yie lds 11 oongrueJlC
Theorem 2.4.3. For well notmtgative integer.!. (2.4.14 )

p( 49n+ 47) :

o(mod 49).

Proof. Write (2.4.$) in the form

f n sO

7

p(7n + 5)qn = 7(9 ; q');:"(q; q)::" (q:q)~

(2.4.15)

'" 7(q7;q7)!o

-

+ 4!1q (9'; qT)~ (q;q)~

L (-I)"'(2m + l)q",(",+1)/2 (mod 49). m.'

by the bi nomiallheorem and J1IOObi's identity (1.3.24). We now examitle the terms on the right side of (2 .4.1.)) where the 1"J"''eTS of q Ilre of the form 7n + 6. Separating t he summands into residue d!lSSe!l modulo 7, "''e see that the only terlIlll yielding ~uch eJ'I)Qnents are when m =- 3 (mod 7). Bul then 2"1 + 1 == 0 (mod 7). Thus. the <» efficient of the I>O"-'er q7"+6, n 2: 1, on the right side of (2. 4.1$) is 11 multiple of 49. The same mUllt be true, of course, on the left side of (2. 4.13), i.e., the coefficient p( 49n + 47) must be a multiple of 49, Le., (2.4. 14) hllS been establ i s h e d . . 0 Copyrigllloo Matenal

SPIRlT OF RAMANUJAN

43

We shall return to congrueru:es for p{n) after"~ introduoe Eisenstein series in Chapter 4.

2.S. The Parity of p(n) In contrast to Theorem 2.2. \ which provides a criterion for determining wben T(n) is either even Or odd, we know much less abom the parity of p{n). It has long been conjectured that p{n) is even appr(lJcimately half of the time, or. more precisely, #{n S N : p(n)is e'~n} '" !N,

(2.5.1)

a.s N - IXI. T . R. Parkin and O. Shanks [1781 undertook the first extensive computations, providing strong evidem:e that indeed (2.5.1) is IIIOI'lt likely true. Despite t be wnerability of the problem, it was not ewn known that p(n) assumes either e'~n or odd values infinitely often until 1959. when O. Kolberg [1 3 5[ established these facts. Otber proofs of Kolberg's theorem were later found by J. Fabrykow1!ki and M. V. Subb&rao [931 and by M. Newman [1681. In 1983, L. Mirsky [159J establi.,hed the first quantitative result by showing that log log N (2.5.2) #- {n S N: p(n) is even (odd )) > 2log2 . An impfO\~ment was made by J.-L. Nkolas and A. S.!:rkOzy [171 1, who proved tbat

#- {n :S N : p(n) is e'~n (odd)} > (log N )e,

(2.5. 3)

for some poIlitive oonstant c. In the most r
been somewbat di fferent from tbose for odd values of p(n). Greatly improving on previoU!l results, Nicolas, I. Z. Ruzsa, and Silrkmy [I 701 in 1998 proved that

#- { n

(2.5.4)

::; N: p(n) is even } :>.IN

l > 0, #- (n S N : p(n) is odd } :> .JNe- {Ioi~+')~.

.. nd. for each (2.5.5)

(We pa.~wexplain the notation :>. We write F(N) :> C(N) , ifand only if there exists a nositi~ ..cplW1.ant (,such that F(N) <:!: cC(N), t;OpY"lI')fea Ma/eoal '

8 . C. BERNDT for 1lI1 N sufficiently large.) In an append ix to their paper 1170), J.-P, Serre used modular forms to prove that (2.3.6)

.

] N~""

#{n
= 00.

At pre>ent, this is the best known result for even valUe!! of pr,,). The 10'0\'er bound (2.3.3) ha!! been improvW first by S. Ahlgnm 16), who utilized modular forms, and socond by Nioolas [1691, who used mOre elementary methods, to prove that

(2.3. 7)

#{n::=;N:p(n) isodd}>

...IN(I~I~N)K,

for some posith'e number K. Ablgren proved (2.3.7) with K = I) An elegant, elementary proof of (2.5.7) when K = 0 was established by O. Eichhorn [86]. The lower bound (2.5.7) is currently the best known result for odd \"Blues of p(n). Our goal in thill section is to prove the resul ts (2.3.3) and (2.3A) of Nioolas, Ruzsa. /lIld SkkCizy 1170] by relati\'ely simple means. Our proof is a special instance of an argument devised by 6erndt, Yet! , and A. Zahareso;u 155[, woo prm-ed oonsiderably more general theorems that are applicable to a wide VlU"iety of partition functions. Except for one step, when we Ullll!t appeal to a theorem of S. Wiger! 12231 and RIImanujan 1185), our proof is elementary and self·ooutained. Th eorem 2.5. 1. For ooch /ix(d c u.';!h c > 2 log 2 and N I
$uffici(nll~

(2.5.8) Theo rem 2.5.2. For ooch foud COn8!on! c tdth c < 1/.;6, and for N III.fficiently large, (2. 5.9) Before we begin our proofs of Theorems 2.5.1 and 2.5,2. we need \J8Oe language from modern algebra, readers need not know allY theorems from the :;ubje<;t. In fact, some of the information conveyed in the next two para-graph!! will not be u"oo in the sequel. but we think these facts are interesting in the111~yrighted Material to Clltablish ""me terminology. Although we sh all

SPlIUT OF RAMAN UJAN Let A ,= F,llX)) be the ring offormal power series in one varia.ble X over the field with two elemenUl F , = Z/2 Z, i.e., (2.:>.10)

A =

f J( X ) = f. onxn 'an E F~,

1.

0:'0"

< 00).

n_ O

The ring A ill an integral domain; note that an element J(X ) E::".o.ll)

J(X) = X n ,

+ X n , + ... ,

where the Sum may be finite or infinite and 0 :'0 any /(Xl E A , obl!e~ that

nl

< n, < .... For

(2.5.1 2) In other words, if /(X ) is given by (2.5.11), then (2. :>.13)

/'(XJ = X1<> , + X~n, + .

On A there rucis\.9 a natural derivation which sends fIX ) € A to /,(X) = ~ E A, i.e., if ~

then

(2.:>.14) Note thllt for any (2.S.IS)

/ (Xl €

f'(X) =

L na"Xn - l .

.-,

A,

f"{ X ) = O.

We al!lO remllrk that for any J(X) given in the form (2.:>.11), the condition (2.5.1 6)

f'(X) = 0

is equivalent to the condition that an the exponents nj are even num-

"'_.

In our proof of Theorem 2.5.2, we need to know the shape (2.5.1 1) of the series /(X)/(I ~ Xl. For any integers 0 :s a < b, lire see that

in A (2.5.17)

B.C.BERNDT

"

\ \'e put together paiI1l of consecutive tenns X """ +X""' · to obtain the equality

+ (X" 'H' + ... + X ....... - ' ) +. If the sum on the right side of (2.~_11 ) defining fIX) is finite, say fIX) ,.. X '" + X'" + ... + X", , then

I(X) = (Xn, + K"'+!

1- X

+ ... + X"' - ') + .

(2.5.19) if ~ is even, and

I(X) .. (X'''+ ... + X",-I)+-

I - X

~

+(X"·-'+· ·· +X ..·-,-I)+

(2.5.20)

L X",

if ~ is odd.

Before wmmencing our proofs of Theorems 2.5. 1 and 2.5.2, we introduce some standard notation in analytic number theory. We say that I(N) = O(g(N)) , all N lends to 00, if there exis\.9 a posili>'e oon.'ltant A > 0 and a number No > 0 such that If(N)1 Alg(N)1 for aU N ~ No_ To emphasize that this po.sitive constant A above may depend upon another parameter c, we write I{N) = Oc(g(N)) . &8 N tends to 00.

:s

Proof o f Theorem 2.5.1. We begin with the pentagonal number theorem ~

(2.:'>.21 )

~) _1)"q.,13n- l l/2 n _O

~

+ 2:(_1 )"qn (3n+I )/2 '"

(q;q )"".

~_!

By reducing the coefficients mooulo 2 and replacing q by X in (2.5.21 ), we find t hat, if 1/ F~),YMRf~e1Mhe infinite series of (2.5.21 )

SPJRlT OF RAMANU JA N in A, then

(2.;'.22)

1 = F(X)

(1 +~ (X"P"- Ilf1 +X (3n+Ll/')). n

We write F(X) in the form (2.;.23)

F (X) = I +xn,

+ X'" + ···+X", + ... ,

where, of course, ",."2,. lire po;;itive integers. Clearly, from the generating function of the partition function Pt,,) and (2.5.23),

#{l :0:;" :S N: pr,,) ~ odd} '"' #{n, :S N}

(2.;.24) ..d

#{l :S ":0:; N : pr,,) is even} = N - #{", $: N}. We first establish alo,,"er bound for #{"i :0:; N}. Using (2.5.23), write (2. 5.22) in the form

(fxn,) + f (x" ( I

, .. ,

(2.5.26)

=

13.. - I)/1 +

X"(3"+1)/2) )

.... I

f:. --,

(xmI3m -!)/2+

x m (3 m +l)12) .

A$ymptotic1lIly, there are /2N/3 terllUl of the form xm(3m - I)/2 less than XN on the right side of (2.5.26). For a fixed positive integer "i' ~ determine how many of these tcrlIl.'l appear in 11 series of the form

(2.5. 27)

X" ,

(I + ~ (x ..

{3n -' J/2

+ x n (3 n +l)/2) )

•

lU"ising from the left side of (2.5.26). Thus, for fixed " j < N, we esti. mate the number of integral pair! (m, n) of solutions of the equlltion (2.5.28)

nj

+ !,,(3n -

1) _ !m{3m - 1),

which we put in the form

(2.5. 29)

2nj ~

(m - ,,)(3m + 3" - 1).

Bya result of Wigert (223] and Ramanujan (1851, (192, p. 801, the number of divisors of2nj is no mOre than 0 < (N~) for IU"Ly fixed

c> 1082. ThU!! , each 0C))1p'H~~wfeeNaieMfd 3m+3n- 1 e&n _urnI'

8. C. BERNDT

"

at rnO!lt 0 0 (N~) ,,,,,lues. Sinoo the pair (m -

n,3m + 3'1 -

1)

uniquely determiTW)S the pair (rn, .. ), it follows t hat the number of SQlutioll!l to

( 2 .~. Z9) is 0 < (N~), where c is any constant such

that c > 21og2. A similar argument can be made for the terlIlll iD (2.;',26) of the ronn X", (3 ...+ I)/2. Returning form

\.0

{2.5.26} and (2. 5.27), we see that each serif'S of the leema X .. (3 ... -LI/2 up to

(2.".27) h&5 at mO!l!

Oc (Nr;;Cv:)

Xl" that IIppesr on the right side of (2.$.26). It follows that there are

at least.

0< (N! -~ )

numbers n, ~

N that are needed

to match

all the (asympt.otically ,j2N/3) term.s X ", (3"' - 1)/1 up 10 X'" on the right side of (2.5. 26). Again, IUI analogous argument holds for ter!Il$ mpm+!)" We have therefore completed the proof of of the form

x

Theorem 2.5.1.

0

Proof o f Theorem 2.5.2. Next, we provide a lower bound {Of #(n ~ N : p(n) is e>1ln} . Let {mJ, m2 ."' ) be th", oompl",ment of th", tI(lt (O,"b "1,"') in the set of natural numbers {O, I , 2, ... ), and defiM (Z.5 .30)

G(X} '''' X ""

+ X "" + ... E

A.

T,"," (2.5.31 ) G{X ) + F {X } '" I

+ X + X2 + . .. + X~ + ... '" __ ,- X

Sine"" by (2 ..5.2(5), (2 ..5.32)

Hlmj:S: NI =

N - #{nj

we need a ]()W(lI" bound for

#Imj

:s: NI = ('I :s: N, P(n) is even} , :s: N). Using (2.5.31) in (2.5.22),

we find that

1+ G(X)

= =

(1+ ~ (X~{3~- I)f2 + X(Jft+1)/2)) ft

I~ (I+ ..t , X

AJ' +

(X ft (3ft - l){l

+ Xft IJn+l)f2) )

X + X 2 + X~ + X 1 + ... )

- •-r,.;opyrigl!/ecj Matenal

SPIRIT OF RAMA NUJ AN

"

_ 1~X((l+ X) +(Xl+ X5)+ . + ( X (n- 1)(3(,,- 1)+1)/2 + X" (3" -I)/2) +

(2.5.33)

( X ~ (J"+I )/l +

x (n+ I)(3{ .. + I)- I)/l) + . .. ).

By (2.5.IS), we see that the right side of (2.5.33) equal'!

(2.5.34) 1 +(X 2 + X 3 + X ' ) + ... + (x (n- 1)( 3(n_ I)+I)/2 + ... + xn(3 n -l)f2- I)

+ (xn (3n+l )/2 + .. . + x

(n+l )(3(n+l )- I)/2- 1)

+.

x

Obsel""'ffl that the gap bet~n x n(3n- . )/ 2- 1 and n (3 n +l)12 OOlltairtJ> n t
- 1) - 1 - Hn - 1)(3(n - I ) + I ) + 1 = 2n - I

U!rmB tha t do appear ill (2.5.34). So we see that (2.5.34) conuuns lI8ymplOtically

:;--,2'i"~-~'"N " 11 +(211

tenns up lO X N

Now the

I)

~um

2n - 1 N

3n - 1

_ ~N 3

in parentheses on the left side of

(2.5.33) has asymplOt ically 2..j2N/ 3 nonuro terms up \.0 XN. Thus G(X) must haV
2.6. Notes Theorem 2.2. 1 hBB been slightly refilled by M. R.. Murty, V. K. Murty , and T. N. Shorey [165]' using a more sophisticated argumcnt They also ohtain lower hounds for t he \l8.Iuetl of "T {n ) .... hen T(n) is odd. Anothcr proof o( Theorem 2.3.1, rivalling Rarnanujan's first proof in simplicity, has been giycn by J. Drost [84). See M. O. Hirschhom's paper [120) for still anothcr clementary proof. Many referellce'! \.0 further proofs of both T~ffl;m~J M.l~(f.worem 2.3.4 can be found

8 .C. BERNDT

50

in t he latest edi t ion of R.a.ma.lluj!U\'s CollulM Pllper$ (1 92. pp. 372-

375J. T hese pages also oontR.in references to other proofs of Th~ms 2.4.\ and 2.4.2. Rarnanujan himself summarized the congruences he proved and the methods he utilized to ptm-e them in 11. letter to Hardy writum from t he nUf$ing home, F it zroy House. in the summer of HH8. In part icular, he wrote [5 1, pp. 192- 193), "T hus the d ivisibi lity by

SOPll' when a _ 0,1.2,3; b = 0,1,2,3; c

=

0, 1,2 amounting to

4 )( 4 x 3 ~ I or 47 C8SeS of the conject ured theorem are proved." This statement is interesting for several reasons. First, Ramanuj!lll had evidently prO\w special cases of his general conjecture without

leaving us proofs in these CIISe8. Second, he claimed a proof for tbe "''e had noted in t he introduct ion to this chspler that Rama!lujan's conjecture was false in this case. Third. Ramalll.. modulus 7J , but

jan's proof of hill conjoct ure for arbitrary established afU,r this letter Will! written.

JlO"'~1'lI

of & waB obviously

We have not glven a proof of Ramanujau's congruerw:e P{ l lfI+6):E 0 (mod 11). The 010111 elementary proof is d ue to L. Win-quisL [221] and uses Winquist'g Identity. Further proofs of W inqui!;t's identity h&\~ beo:-n found by Hirs<;hhorn [1171 and S.- Y. Kang [1331. Another elementaTY a pproach to proving that p{lln+ 6 ) == 0 (mod II I has beo:-n devised by Bernd1, S. H. Chan, Z.- G. Liu, and H. Yesilyurt 148J, who established a new ident ity for (q; q)~. lli rscllhorn 111 81hllll devised a COmmOn approach to proving all three congruences (2.1.2}-

(2.1.4). Hirschhorn and O. C. Hunt [126J gave an alternatj,~ proof along c!as/jica.l lines to that of Ramanujan aud Watson for Ramanujan's oongruence mod ulo 54. ThOlle relldel'll intrigued by our sketch of RAlnallujan's T heorem 2.4.2 might consult F. G. Carvan·s [951 proof of t he more general congruence modulo .,. Ramauujan appeared to coujccture that the only congruences of t he form p((n + B) == (mod I), where t is a prime, are thOlle wheo ,;:, &,7, Or 11. This WB.8llot proved unt il 2003, wh.m Ahlgrell and M. Boylan 18J pro....~1 that indeed these are the only three such con· gruern:e!l.

o

Copyrigl!/ecj Material

SPIRlT OF R AMANUJAN

"

Suppose, h.,..,.,ver, that _ drop the re!!triction that the modul i of the arithmetic progreseions are the same 8.!1 the moduli of the congruences. Th&t is, are there congruences when the moduli are not the same or are not prime!!? The first theorems estahlishing lots of o:mgruences for p(n ) were found by Atkin [29J. Then K. Ono [175J proved that , given any prime i :-er (219J devised an algorithm based on 0110·8 work and found over 76,000 explicit examples, all with

Define 6,: =

i::; 31.

e""""24. - I

Except for a oouple of sporadic examples, it t urns out that for sll of the congruences p{ An + B ) ;;; 0 (mod i) found by Ramanujan, Ono, Ahlgren, and ot hers, B '!!i - 6, (mod l) . In another breakthrough, Ah.lgren and Ono [10]. proved that this residue class is only olle of (1 + 1)( 2 residue classes where p{n ) possesses many such oongruences. An informative historical description of the quest for congruences for the partition function has been given by Ahlgren and Ono (9). 5ubb&rao [211] first conjectured that in every arithmetic progression n == .. (mod t) there are infinitely many values of n such that p{n) is even and tbat there are infinitely many valUe! of n for which p{n) is odd . The moot extensive results poi nting toward the uuth of this conjecture have been fou nd by Ono [J73 ], [174) and Ahlgren (6], with a !>Il1I1mary of previous results provided in these papers. The best lo_r bounds for t he number of even and odd valu"," of p(n) in arithmetic progressions are (2.5.6) ]170) and (2.5.7) (61, respectively, while the most gelleral theorems of this _ t are foulld in [55] and (56).

Copyrigl!/ecj Material

B. C. BERNDT In hiB unpublished manuscript on p(n) and r(n), Ramanujlln asserted and, in some CIISeS, proved furt her congruences for T("), mOl!\ involving divisor functions. References to most of the many papen written on this subject can be found in Berndt and Ono's ao:colll1t of Ramllllujan'! manuscri pt [50). We now have a complete understand_ ing of congruences for r(n) through the theory of t- adic representa. tions, IInd an lI such th at rf"I" but p"+l I n is called the order of n modulo p and is denoted by ord.n.

Theorem 2.6.1 . For

an~

po8itive integer n,

o (mod 23), if then! eri..u a pnme p ,ueh that

r(n)

iC

(fJ)=-1 and 2 l ordpn, 0 (nlod 23), if then! eri..u 0 prime p ,,,ch that p = 2xl + xV + 310'2 and or
(- w

IT

(I

+ ordpn ) {mod 23),

otherwi.!e,

pw1. ' +ZMHM' ,.n

when!

"

,, .... 1%'+~r+3.· ""d. n _ l

(_ ~ l

As mentioned in the IIistorical Background beginuing this chapteT, Ramanujan introduced his arithmetical function r (n) in 1186], [192, pp. 136-16), which WII8 to beoome one of the most important papeTli in the history of number theory. In this paper, Ramanujan made three important ooujectures about r(n}. (a ) T(") is multiplicative, i.e., r(m)r{n) = T(m"), whene'llr ("','I) = I. Copyrigl!/ecj Material

S PIRIT OF RAM A NUJAN

53

(b) If pi.!! any prime and n is any in!.eger exceOOing I, then

.. (pn+1 ) = ..(p) ..(p") _ p ll .. (p" _ I ), (c) For ell.Ch prime p ,

).. (p»):5 2pll / ~ . The fil'llt t....o C(lnjectures were first prm-ed by L. J. Mordell [1611 in 1917 and then greatly generalized by E. Hecke [113J. beginning one of ~he most important chapters in ~he theory of modutlU" forma. Conjecture (c) was also COll!iiderably generalized and was one of the mOBt famous unproved conjectures in uumoor theory until it was proved by P. Deligne )8 1J in 1974. Readers wallting to learn more about .. (n) might begin by reading two expo!litory papers, one by R. A. Rankin 11971, and the other by M. R. IIf urty )164). See also It. paper by V. K. Murty [166J.

Copyrigl!/ecj Material

Chapter 3

Sums of Squares and Sums of Triangular Numbers

3.1. La mber t Series In this chapter, we shsll derive formulas for rn (n), for le '" 1, 2,3,4, and formulas for tlk(n). for k = 1,2. ObserV<) that rlk(n) and t2k(n) ~ generated by 'f2t(q) =

-"-.

L rlk(n)q"

and

",n(q) =

-

L

"-.

tl.(n)q",

respectively. where ,.,{q) and ,,(q) are defined in (1.2.2 ) and (1.2.3), respect.ively, 8nd where, by convention , we define ru(Oj .. 1 - tn(O). Appearing in our proofs are Lambert $erie:'!. Strictly speaking, a Lambert series is a series of the type

(3.i.I) A generlllized Lambert series allows mOre general exponents in both the Ilumerato,." aud denomi natol'!l of (3. 1.1 ), with the minus sign in the denominator al90 possibly being replaced by a plus sign. Nonetheless, we use the appellation. Lambert series, for all such .'Ieries. Copyrigl!/ecj Material

B. C . BERNDT We also prm"e three further theorems On the reptelentations of numbers by certai n quadrati<: form$. For our purpOSe, a quadratic form is a fun<:tion of k variable6, nI, n2," ., nk, and hlL'! the shape L a; ... n;n), where a;.j are nonnegative integers and ni, n2,"" nk Ille integers. T hus, for example, rk(n) denot.el3 the number of repl"I!Sen· tat ions of n by the quadratic form + ~ + ... + n~. In another example, we exam ine the number of ways n <:an be represented by the quadrati<: form .,2 + xy + y2.

nr

3.2. Sums of Two Squares Theore m 3 .2.1 . For rock pruillw in teger n, (2 (n ) = 4

(3.2.1)

L

( _I )ld-l)/l.

""

.~.

We can state Theorem 3.2.1 in the

a1lernali~"e

formulation

(3.2.2) where dj.k(n) denol<Jll the number of positive divisors

(3.2.3)

cl of n such that d

E

j (modk).

Immediately deducible from (3.2. 2) is the ...-ell.known theorem that every prime p congruent to 1 modulo 4 can be representoo as a sum of two squares [172 , p. M]. E xe rcise 3.2.2. U.lUg (3.2.2), prow that r1(n) > 0 if and only if tWill pnm~ p congruent to 3 mOOulo 4 ;n the canomcal fadon:anon ofn appttlr~ flIith an etlen t:lpOnenl [172. p. [,[,. Theorem 2.1:;1. First P roof of Theore m 3.2. 1. Using the Jacohi triple product identity, Theorem 1.3.3, ",-e 6rsl doouce that 2 (a - l /a)(a q; q)""(a - 1q;q)",,,(q; q)"" = a(a 2q; q)""(a- 2;q),,,,(q; q)".

-

- L

n __ OQ

( _ t )ft,,2 n +l qn (n+l)/2

Copyrigl!/ecj Material

S PIRIT OF RAMAN UJ AN

f. + f: )(_lr

= (

n w_ OO

.. w_oo

" ."'''

.. odd

-

L

=

57 a2n+l q,,( n+l)/2

-

L

a· ,,+l q,,(l ,,+I) -

a h - 1 q,,(1,, - I )

= a( _a 4q': q' )",, ( - a - "q; q' )",, (q4; q" )"" (3.2.4 )

,

_ .!. (-a'q : q')",,( _ a- ' q3; q' )",, (q' : q')"".

where we applied the Jacobi triple product identity , T heorem 13.3. t wo add itional limcs. We next use logarit hm ic differentiation to differentiate bot h sides of (3.2.4) with respect to a and then set a = I . Not.e that on the far left side of (3.2.4) the differentiation of the infi nite products ill unnece!5ary, be<;aWle when a _ I , t he factor a - l / a = O. We therefore obtain (3.2.5)

2(q;q)!o = 2(- q): q' )",, ( -q;q' ).",(q' ; q4)00

+ (- l ; q' )",, ( _ q; q' )",, (q4; q4)""

- ( 4 -1 q' ..

X " " £.-

'"'

I + /f,, - l

-

4q4" - 3 l +q, .. - 3

-

4q',,- 3 4q, .. - I ) l+q4.. 3 + l+q'''- l

--C'-:.=

"" 2( _ q3; q' )",, ( _ q;q4 )",,(q' : q4 )""

Now di,·ide bot h sides of (3.2.5) by 2 and by ( _ q:q);,(q;q)"" = (_ q;q).,.,(q1: q2)00

= (_q; q1)",,( _ q2: q2)",, (q2: q2)"" _ ( _ q; q2)",,(q' : q') ""

= (- q\ q' )",,{ - q; q' )",, (q' : q')"" to deduce that (3.2.6)

(q;q )!., 00 ( _ q. q )1 , OQ(;opyrig/1f!e(J

=1-: 4L

0,

q''' -I )

q,,,- a +;t,,- a- t + q' n-l· ale al

O. C. BERNDT

"

By ( L3.IS) and EuleT's identity (1.1.9),

(3 .2.7)

(q;q2)!,(q2;q~)oo = {~~~~i,.,"

Using (3.2. 7) in (3.2.6) IUld replacing q by - q, we conclude that

.,1(q) "" I + 4 =

I

+4

L- ('H I q q4~-3 - I ''"_' q' Q_ l ) --,

f: (f:

", .. I

= I+4

(3.2.8)

f(

"_I

f q(~m-Ll') L 1),0.

q (t m _ 3 )r -

r_1

L

, .. I

1-

~In

dIn

ol(_ i )

dIl3(m<>
Lastly, equate ooefficients of q'" n :::: I, on both sides of (3. 2.8) to deduce thM

r 2(n) '" 4 (dl.4(n) - du(n» , which, as we baWl seen, ill an alternative formulation of (3. 2.1).

0

We IlQW give a second proof of Theorem 3.2. 1 that ia quite diff~r ent and somewhat sborter than the proof above. It requirell, however, a deeper theorem, namely, Ramanujan's lW, ~ummation theorem.

Second P roof of Theorem 3.2. 1. In R.&manujan 'sl"" summation theorem, Theorem 1.3.12, set <> = {J = - \ and z = I. After simplifying the product on the right-hand side, ,'re find that (3.2.9) where in t he next t(liast step we used (l. 1.9), Md in the last 5tep u5ed (3.2.7).

On the other hand ,

f

1 :':1" =

,,_ I

f. f.

(_ I)mq"+2",.. = t

,,_ 1 ... .. 0

(-I )'" t

m _O

fo 00

(3.2. 10)

=

Copyrigl!/ecj Material

q(lm+1) ..

",01 (

1)"' q(2"'+1)

I

q1m+1

SPIIUT OF RAMANUJAN

"

Usil\i (3.2.10) in (3.2. 9), we see that we have readied the first equality of (3.2.8). The remainder of the proof then follows as before. 0

3.3 . S ums of Four Squares ThOOn'lm 3.3. 1. For Ndi positive inttgel" n ,

(3.3.1 )

Observe that the swn in (3.3.1) is clearly never equal to O. Thus, as a corollary of Theorem 3.3.1, we deduce Lagral\ie's thoorem that every positive integer n can be represented as & sum of 4 squares (112, p. 317).

F irst Proof of Theorem 3 .3 .1. We begin with Jaoohi's identity in lhe form ( \.3.28) (3.3.2)

(q;q)!, =

,

~

z".L-00(- 1)" (211 +

l )q~ (ML )12.

Squaril\i both sides of (3.3.2), we easily find that

(q;q)!, =

= (3.3.3)

i of

(_ I )m+n (2m + I)(ZII + l )q(""+"' +m+,,)/2

i (. .,f _oo f-'"') + ....

(_ I )",+n (2m+ 1)(2n + I )

", +n .... m+ " odd )( q(m'+n' +"'+" J12.

In the first sum on the right side of (3.3.3), set

m=r+8,II=r-& or r=t(m+n),8=!<m - n). and in the second on the right side of (3.3.3), set

m = r H, n = &- r cJptt'l9t)t~JJm(;ar

- I),

&

= H m+ n + 1).

.,

8 . C. BERNDT

In both CMeII, U m and n run ~ their resp
(q; q)!. ..

~ C.~""(2r + 2~ + 1)( 2r - 2~ + l)q" +f'H ~.,.,(2.- + 2. + 1)(2. -

- , ..

.. ~ of

({2r

~ -2"l( Lq" • __ ...

..

2. - I)' '' ..•... )

+ I)' - (ZI)2 )q,· H>.. r

(

1 + 4q -

d) L~ q>'+'

dq

' .-coo

~ ( (-q:q' )!o (q' ; q' )... (I + 4q~) 2( - q'; q' )!a(q': ,')"" - 2( - q': q' );",(q': q1 )"" )(

4q~ ( _ 9': q2);"(l

: q2 )"" ) ,

wlM're.....e used the product repre.entation ( 1.3. 13) foe .,,(q), and where applied the Jacobi triple product identity (1.3.\0) 1.0

"'''e

~

L

qr'+r = /( 1,q2) .. (_ I;q') ... ( _ q' ;q' )",,(q'; q2 )..,

--~

.. :l( _ q'; q2);"(q' ;",2).." Now in (3.3.4), "-e logarithmically dlffetf!ntiatll 10 deduce that

(q:q)!. .. (- 4; q2 );'(,l:Q')!.( - q' ;q' )!, X

oo 2nq:l» Loo 2n q2ft ) -- . l +qlR l __ ( 1 + 8 ftL_I q 2ft .. _ I

_ (- q': l l!.\q': q')'

(-q' q' ):!.,

copyr@l/edMGffJfldl

SP IRJT OF RAMANUJAN

"

(3.3.5) Now di vide both sides of (3.3.5) by

t
(3. 3.6)

(q; q)~ ~ {(2n _ 1)q2"-1 2nqln} (_q;q):., = 1 - 8 l+qln- I - I + ql'"

t=i

Thus, substitu ting (3.2.7) inUl (3.3.6) and tben replacing q by - q, we find that ' ( ) = 1+8~{ (2n - l )ql" - 1 + 2nq~n} ~ I q'Pol I+qln

'l'q

--.

--

, ' +SL;L; "'m _ IoI .. '

Copyrigtmd Material

B. C. BERNDT ~

(3.3.7)

~'+8L L dq" · .. _1 41..

"

where in the penultimate step we merely replaced n by d, and whe~ in the llIllt ~tep we IlIlt n ; dm and wHocted all coefficients of qft. Now eq uate coefficientll of q", n ;;:= I, on the extremal Bides of (3.3.7)

to deduce (3.3.1).

0

We now ghoe a 9OCOnd proof of Theorem 3.3.1 t hat is based 011 Our work in the second proof of Theorem 3.2.1.

Second P roof of Theorem 3 .3 .1. In Ramanujan 's Ilbl summatioa theorem, Theorem 1.3.1 2, first replace q by q2, and then set z '"' qt", a _ - I, and b = where 6 is real.

-e,

Exe rcise 3.3.2. U&;ng (1.3.35), protlf! that jor every integer n, (3.3.8) Using (3.3.8) and replacing n by - n for each negath'e integer n,

we find from (1.3.36) that

;1+ 4 f q"COS(nO) .. _ I I +q2n (3.3.9)

( - qe"; q1)",,( - qe- j , ; q2)",,(q1; q1)~ = (qe;';q2)""(qe-"; q' ),,, ( q2 ;q2)~ .

Now repl8.C
SPIRIT OF RAMANUJAN

63

by (3.2.1). Now lntegnu.e both sides of (3. 3. 10) with mlpect. to 9 0\1'1" [-., ..J. Uling theorthogonality of (COIl(n9», I S n <: 00, on [- ",. 1,

• find thlt (3.3. 11 )

2• ..,4( -q')

~ = 2.,. + IS. L.

(- I)"q'"

".1 (I + q''' )'

.

Rep1acint! - q' by q in (3.3.l l ), we deduce thl t (3.3.12)

..,4(q) - I+8~ ( I +rq)")2'

N~,

-

---.

- L (-I)-- 'm Lc- I)-C-,)-(3.3.13)

-

1: +7:)",' --. 1

Substitute (3.3. 13) in (3.3.12) Md observe that we have !lOW reached the first Hne of (3.3.7). ThU'l, the remainder of t he proof is identical to that given above. 0

3.4. Sums of Six Squares Of Jacobi.', formulu for ",,('1). 1 S k S 4. the mott. difficult to pl"O\'e is perhapll t hat for r.(n). The proof of S. H. Chan 17l] that we &i~-e ill poesibly !.he si mplest. Ari thmetic proofa of the formula for rs (n) an! &i~ in [167] and [IM].

Theorem 3 .4. 1.

(3.4.1)

'1-

d')

d .. I(n>t>dOj

+16( L <11"

} _ l ( _ 4j

COpyrighted Ma/anal

d' -

L ",). "1"

t . ' 1_4j

D.C. BERNDT

"

proor. Hep1acilll n, b, and ~ by r, n , and :t, respectiwly, in Ramanlljan's I ~I summation {1.3.36), ...-e obtain the Wieful corollary

(3.4.2)

Exere i.&e 3.4 .2. Oburu! th4t the right-hond 'ide of (3.4.2) i& Igm. mdric In x and 1/. He"CJe.

(3.4.3)

GII.oe 4 direct proof 0/ (3.4.3),

'nd~enf of (3.4.2) .

Differentiating both sides of (3.4.2) with plyinl! both sidea by;r, we find that

re!lpect

to r and mult~

(3. 4.4)

Intercltanp z and 11 in (3.4.4) and subtract the raultifll identity from (3.4.4) to deduce that

(3. 4.5)

f:

ru:"

f

~

"", _... 1- IIq" - n__ O<> l _rqn

=

(XII;q).., (q/(ZII) ;q)..,(q;q)!.. (y; q)"",(q!lI;q)""(r ;q)oo{q!:r:;q)",,

SPIRlT OF RA.M ANUJAN where Iq l < I!E I, 11/1 < I. Letting 1/ ...-e find t hat (3.4.6)

L(,),~

f (--"'-"- _"(-X)")

"._""

1 + .:rq"

Upon sim plifying,

Wf!

1 - .:rq"

"

- .:r in (3.4.5) and simplifying,

(_ .:r1;q)",,( -q/z~; q)",,(q; q):'" (.:r2; ql ).,.,(q? / .:rl; q2)""

see that

(3.4.7)

R(q) = (_ !E2 ;q)""(_q/ .:r 2 ;q).,,, (q;q ):'" (.:rl ; q2)"" (q2 / .:r2; q2)"" =2x (_ !E2; q)",,( _q/ x 2; q)..,(q;q):.c. (Xlq; q2 )",, (q/ X2; q2)",, (q2; q2):'" (!El; q2 ).",{q2 / !E2; q2 loo (!E2; q2loo( q2 / x2; q2)"", (q; ql)!, 2 2 = h (1 + x2 ) (_x q; q)",, ( _ q/ x ;q)""( x 2 q;q2),,,, (q/ x 2 ; q2).",(ql ;q2)~ , (x2; q2)!,(q2/!E2; ql )~ where in the second equlllity we applied (3 .4,2), with x, 1/, IUId q replaced by q, !El, and q2 , respectively. Simplifying the left- hand side of (3.4.6) yie lds

=

L-

.. _ I

( n!E .. - nzn+lq" - - - n(- z ).. 1 + xq n

nz-ft-Iq" l+qn / x

-."'

+

+-"""(-CX~):·O'~"'-" 1

!Eq"

n (_x)-n -l q" ) I qn / x

= :L ="(I -(- I j"j

+

-

L ( _ 1)m+1nz"+m+l q,,,n +n + ( _ I ),, +l n!En+... +lqmn+n) ( _1)m+l n!E - n- "'-lqmn+n

+ (- I )n+lnx-n-... - 1q.....+.. )

Copyrighted Material

B. C . BERNDT

66

". , ~

L

+

«_ l )m _ (_ l n n (l:.. +m

+", ~n - m) qmn .

Not ing that the !ummands disappear whenever m and n ll1e of t be f>ame parit y, we deduce t hM ~

L(q) = 2 L (2n - 1)",' ,, -1

".,

(3.4 .8) We nOW multiply both L(q) and R(q) by 2/( %(1 + x'}) and let x -i. Fi rst, from (3.4.7) we fi nd thlll

r

(q ; q )~(

2R(q)

z'!!: x( 1 + x')

(3.4.9)

q; q');,(q7; q1 ):'"

( q2; ql )!,

=

(q';q4):;"(q'; q')!, (q2; q7)!.,

(ql ; ql)~

=(

q'; ql)g" ,

wheTe we have appealed to Euler's identity (1. 1.9) in the last step. 5e<:ond, from (3 .4.!!) we deduce t hat

2£( )

lim ( q 1 z_. xl + :.: j

= 1+ 4

co

' " (2n - I - 2m)(_ I )'H m ~

n,m . L

t~eia~;;all)lm

SPIRIT OF RAMANUJAN _ 1+4

L

61

~

I )~

( 2., _

_

4m~) (_ I )"( _q~ )(~"_I)m

f:( f + 16 f: m f (_ I)"+L ( _q~)(~"_l)'"

- I+4

- 1)" (2,.. - 1)2

.... ,

{_q.2)("ho-l )'"

... _ 1

2

..... 1

" .1

~

L

- 1 +4 L (-q2) t t_ 1 ~

L

+ 16 L {-q~)t

(3.4.10)

{_ I )"(2n_l)2

(2
t_ 1

(_i)"+L m 2 .

(2,,-1 ) ... _ .

Now IlUbstitute (3.4.9) and (3.4.10) into (3.4.6) to deduce that (22)'

q :q OG ( , ' .','Ill ...

00 _ 1+4"' (_q2) ~

£.-

t _1

'" (- 1 ) " (2"' -1)~ £.-

( 2
~

L

+ 16 L (-q2). t_ 1

(_I) .. +l m 2.

(2
Squating the ooefficienUL of (_ q2 )" on both aidee above and using (3.2.7), ...~ complete the proof 01 Jacobi 'a six Wjuares t~m. 0

3.S. Sums of E ight Squares Our proofof Jacobi', formula for r, {n ) will focUl on the function ( ) 3..5.1

F{ ) F( ' ) ' " ( I + z)( -:q:q)_(-ql z:q)oo(q;q)!o , q,:. (1 :)(zq;q)oo{q/:; q)",,{ q; q)~ .

Lemma 3 •.5. 1. (3..5.2)

For Iql < 1:1 < 1/1ql "nd :;' F (:)

= .!...±.! + 2 1 -:

f

".1

I,

q"(, - " - z") . I+q"

Proof. In Ramanujan's ,tJoI summation formula, Corollary 1.3.14 , Jet 01 _ (j "" - I, replace ~,byrbl~ *lltmn>laoe z by .jij z. We then

8 . C.BERNDT

68 readily find that , for I < I

+zE

q"; ~

~ .I I

+q"

+zf:

1'1< ' /l.ql. ~

.. _ I I + q"

_

(q;q)!o(-;q;q)...(- I/:; q)oo

( 9;9 ):'.,(: Q;q)""( I/z; q)",,

_ _ (I + 2)(q; q) ~( -zq;q)... ( -'l/': .,)"" (1 - :)( ":,,)!o(z,,;,,)... (,,I :;,)...

(H 3) . .

_

- F (:)

.

< I: J.

8UI, for I

(3.5.4)

Putting (3.5.4) in (3.S.3), multiplying both sides by - 1, and Wli~ analytic continuation, we complete the proof of (3.5. 2) for I'll < 1:1 -< 1/1,,1 and : ;. 1. 0

Theorem 3.S.2. Fur (3.5.5)

FI (:) _

1,1< 1:1 < 1/1,,1 and z!- i ,

"" (- , : +8 2: (_,'+)' -z .. _I

1) ,, _1 q" n

00 .. +4 L ~(:" +Z-")'

,"

.... 11-,..

P roof. We begin by $quilTing bot h 8ides of (3.5.2), and in doing 10,

we uae the elementary identitial

(3.5,6) (:

~ ;) (z-" -

z") _ (z"

+ , - ") + 2

(3.5.7)

(,_n _ z"){z-'" - '''') _ (,"+'"

.E(,·,

+ :_t ) + 2,

+ z-.. - "') _ (z"- '" + : "' -").

Copyrighted Material

SPIRlT OF RAMAN U JAN

69

It follows from (3. 5.2), (3. 5.6 ). and (3.5.1) that

1+ : ( 1- :

-7"'::-'''-1") ) ' - +' n_ LL-,q"-,(''-I+q"

(3. 1>.8)

DO

where the OOnBtants en, >1 ~ 0, are independent of ;. Now applying explicitly (3.5.6) and (3.5.1) in the expanBion of the left side of (3.5.8), we find that (3.5.9)

I +; + 2f: qn(: _n - Z") )' ( l -z ,,_ 1 I+q" = (I +,)2 + 4 f: ..L.. f (z" +z-n ) +2I:(z~ + z- l )+ 2} +4

f:

~_ L

"al '+qn l

1 -;

..L..~{(z ...·m+z-"-m) _(;n-m +zm-n)}.

",.,,_ 11 +qn I+q'" Equating coefficients of (zn (3.5.9), we find that

+ z-n ) On

the right sides of (3.5.8) and

For the purpose of ~implification, rewrite each of the three sums in (3.5.10) and arrive at

q"

q"

"" { qn+m

..,

C - 4-l+q"+ 8l _q" "L.. ft -

q,,+m } -qn(l+qn+"') - I +qn+m

q"" "-, { q' q"-' } ..E, 1 - 1+q"' - 1+q,,- m q"~{q.- + q""} - 8 -l _ qn E -l +q'" q"(l+q"+"') .

+ 41 _ q (3.5.11)

Cr$/n9trted Material

B . C. BERNDT

70

Obeen-e that t.he fint portion of th", 6.l"$t sum on the right side C&DCeIs wilh the IIeCOIld portion of the tbird sum on the right side above. After still further ean~Uation , ...." find that (3.5.11 ) reduca simply to

q"

q" q" q" 1) 1 - q" + 8 1 _ q,,\ +q"

e.. .. 4) + q" + 4(0 (3.5.12)

Ob&erve that the left side of (3.:;.8) VIUlillhe!l at .t .. - I . T b"",, setting Z " - \ in (3.5. 8), ...~ readily find that

(l .S. ll) Putting (U .12) and (3.5.13) in (3.5. 8), we complete th.! proof 0( (3.5.5). o

Theorem 3.5.3. We how

(l .U t ) Proof. Rewrite (3,5.5) in the form

(3.5.15)

F 2{:) .. (:

+ t)~ + 4 - :

f:. ~ (:" + :-" _

.... 1 1 - q..

Divide both .idet of (3. 5.15) by ( I L'II?101piuJ'. rule, _ find t hat

+ :)2

2(- 1)").

and let .: !.end to - I . U1ill&

(U .16) Letting'" _ - 1 in (3.5. 15), Wling ( 3.5. 16). usillf; the definition (3.5.1) on the left lide of (3.5. 15), IUld lastly employi ng (3.2.7), we oollClude that

(3.5. 17)

00( 1)"3" 'P'( - q)", l+16 L !'I q , .. _ 1

l - q"

Replacing q by - q in (3. tI. 17), " ·e ded uce (3.5. 14) to compleU! the proof. 0 COpyrighted Ma/anal

SPIRJT OF RAMANUJAN Theorem

3 . ~. 4.

For QlcA po$iri~

71 int~

n,

r. (n) _ 16( - I )" L (- l )~~.

(3.~. 1 8)

". Proof. From

(3.~. 14),

~

,l (q) - I + 16 L

~..

- I + 16

"'q'

1- (

Ef

q

)~

~qd(_q)_ =

1 + 16

4_ 1 ... -0

E(-I)" L(-I)~~q",

",01

~I"

... he", we pu~ n _ d(m + 1). The desired result now follows by equating ooefficient.s of qn. 0 ~ 1, on both aides abo\'~ . 0

3.6. Sums of Triangular Numbers In thill.bort lIeClion we provide eaII)' pn:w:>& of formuiN for 11(0) and 14(n). We begin with IOme elementary identities for theta functions. Exercise 3.6. 1. Prove that (3.6. J)

,.,(q) + ,.,( _q) .. 2,.,(q4),

(3.6.2)

",(q) _,.,( _ q} _ 4q~(q8 ),

,.,(q}oJ.o(q:l) _ 1/I1 (q).

(3.6.3)

Theore m 3.6.2. ReaUllhat dj.~(n) IS dtjintd in (3.2.3). Then for each polilltie mltytr n, (3.6.4)

12(n)

= d •.4(4n + I ) -

dl .• (4n + I ).

Proof. U.ins (3.6.1}-(3.6.3), ..-e easily find tlat (3.6.5 )

",:I(q) -

Sq"r..2(q4).

Employing (3.2.2) Ilnd using the generating function (1.2.5) for It (n), _tee IhM ~

4

L .... 1

~

(d ••• (m) - du(m»qm - 4

L

(d •.• (m) - du{m»{ - q)m

"'. \ ~

- 8 COpyrighted Ma/anal

..L,12(O)q'0>+1 .

8. C. BERNDT

72

Equating coefficients of q. ..... I, n

~

0, on both aides above, we con-

clude that

d 1•• (4n + 1) - d~ .• (4n+ \ ) '" 11 (1\), which is what we

WlUlted

o

to prove.

Theor e m 3.6.3. For t (lch po3itive

int~ger

n,

(3,6.6)

when! u (l\} '" L ll" d .

ExcrciSfl 3.6.4. In analogy with

(J.6. ~), pro~

/lw!

(3.6.7)

Proof. Multiply (3.6.:'1 ) and (3.6.7) together and use (3.6.3) to deduce that

(S.6.S) Now invoke Theorem 3.3.1 and employ the generating function (1.2.r.) for t. (n ) to arrive lit 8

f: L

dq" - 8

f:

L d (- q)"

= 16

f.

1. (I\)q2"+1.

Equating coefficient8 of qln +l on both sides &OOve, we conclude thlrt

L

.'

d

= t. (n ).

4112"+1)

t

Noting thllt the condition 4 d is superfluous for divisors of odd o integers, "'"ll complete the proof.

3.7. R e presentations of Integers by x 1 +2y2, x 2 +3y2 , and x~+ x y + y2 In our next theorems, we derive analogue!! of Theorem 3.2.\ for .ep" resentations of pOIlitive in~ers by the quadralic forms ",1 + 21/ 8.IId ",1+3V~ ' However . first "'e need a lemma which is a beautiful ident ity by itself.

Copyrighted Material

SPIRIT OF RAMANU J AN

73

Lemma 3 .7.1. Fo r lal,lbl
(H I )

,+2

f:

an + b" ,,_t'+{ab)"

= ( - a;ab)",,( -b;ob)",,(ob; ab)~ (a;ab)""{b;ab),,,,(

P roof. In Corollary 1.3.14,!!et (>

'"

{3 "" - I, q

ab;ab)~

= Jab, and : = ,jiIfb.

Then (3. 7.1) readily follows upon o bserving that

( I;ab)" (a b;ab)"

2 l+ (aW'

o

for each posit ive integer n. Theorem 3.7.2. We MW l }n(n+1 )/ ~q~" - 1

(3.7. 2)

1

q~

.. -

I

Proof. Set a = q Md b = q3 in Lemma 3.7. 1. Using (3.2.7) Or (1.3. 13), ( 1.3. 14), (1.3.32), and (1.3.30), we find that 1+2 00 q"+
=

(-q ; q~)"" ~( ' ) (q: q~}""

(q ) (- q)

= 'I'(q~)J ;~~) -/'P(

q)op(q)

- op(q) 'I' ( q~).

(3.7.3)

Expand ing 1/( 1 + q4n) in a geomet ric eeries on the left. side of (3,7.3), ..", arrive at the following array of powers:

,

" " "0' "

-,' _ q lO _q l~

-

," " ,"" ,"

_

q~l

- q"

i"

Copyrighted Material

,~

8 .C. BERNDT If we sum thie arrl,y by coo

L ... 1

(3.7.4)

lIU
q" + rf" 1+ q4n

"" (

=L

... 1

and rows, we find thac.

1)>O("H)/HI q2" -1 1 - q, .. - I .

If we use (3.7.4) ill (3.7.3), we complete the pcool of (3.7.2).

0

Theorem 3.7.3. Let r, .2(n) denote tM nllmber 0/ repre.ltnlatlOnl

0/ the p.:",twe Inleger Th en, for 'I ~ I ,

1"\
rl .,(n ) _ 2(d l •B(n)

the $Urn 0/ " 'qulIre "lid tWlU " lquare.

+ dJ,a(n ) -

d~,.( n)

- d , .B(n) ,

whet"\! dJ.I( n) , j _ 1, 3,05,7, ir defined bll (3.2.3).

Proof. In (3.7.2), apand 1/ (1 - q,,,-I) in a geometric series, aDd 0

the desired reI!lult follow3 .

Proof. [n Lemma 3.7.1. &et "

I + 2 ... (- q)"

~

I

=-

+ (- q' )" '"

+rn

(

q and b _ - q' to deduce that

(q:tt )",,(q': qJ )",,(tt : q3)~ q:q3 )",,( q':q3)",,(-q3:q3)l.o

(".I~(""' I~ ( q:q}""(-q3: q3 )",, = rp(- q)
(3.7.6) by (3. 2.7).

Nat, on tile left side of (3.7.6) , exl)&nd 1/( 1 + q~" ) in a g,eomet· ric &eries. invert the order of summation, and sum the two re!lu1tilli g,oometric 8('TiCl. Thia elementary tllSk then g,ives us

(3.7. 7) ~(

L

""'

q)"+(-q')" . 1+9"'''

"" _ I - H ( L

(

)

qJm+1

q3m+,)

1+q3",+1+ 1+9"""+2 coP~ted Ma/anal

'

SPIRIT O f.' RAM A NUJAN

75

lnaerting (3. 7.7) in (3.7.6), replac.ing q by - q, MId employing, for brevity, the Legend«! .ymbol (3), we find tha~ J

'f'{q)
(3.7.8)

~(",.. .

" , .. .

1+:2'::-0 1+ ( qjS"'+l - 1+ (_ q)3oo!+2

- 1

)

+' f: (~) ," ~_ I

3

1+ ( q )"

~('") -1+ 2L.. ~_I

3

~(:2n+l) -1 +qh --+' ,,--3q'" ..-0

q2 .. +l I _ q' ..+l

_ I +:2~ .f:I ('") 3 l - q'"

(-..t:...... - 2~) I - q· "

+ :2f;(2n+,) ~-O

3

q'''+l 1 _ q2~+l

f: (") "

(") ,"

~ 'I 1_ 0" - 1+' •• • 'I 1 _,.+ 4~ _ 1+:2~ (I

+

4 ~ (

.L

~;:+L

q4{~ .. +L)

I _ q"'{""'+ I )

- m;:+2) I

_

q4(3n+2)

I _ q"'t3n+' )

)

.

~

o

1'h«lrem 3.7.4 has the following arithmetical interpretation.

T heorem 3 .7.S. ut ru (n ) dt:nDte the number 0/ ~entatWn.l 0/ the pQ~it"le ",leg" n /)v %' + 3V', where % and V are intlger., and where repruenlalion.t with:l: and)' hailing different ligns are counttd 01 distinct. Reooll that dj..t«n) ..... defined in (3.2.3) . Then for ttWI po,.,ti"", .nteger n ,

Proof. If ,,~ex!)lUld each ofthequotienu I /{ I - q"' ) on the right ~ide of (3.7.5) Into geometric teries, collect the coefficienu of all common exponenu in each of tru. four ulllnK double sedell, and then eq\l.ll.te COpy ad Mmaoal

B. C . BERNDT

76

coefficients of like powers of n o n each side, we dedu.::e (3.7.9) to

oompww the proof.

0

We prove one further t heorem and its arithmetica.l consequence

in t hill chapter, but it ill closely re lated to Theorem 3.1.4. To effecl the proof, we firllt need a. beautiful analogue of Lemma 3.7.1.

Proof. We apply Corollary 1.3.14 with q and (3 = I /(ab) to deduoe thllt

= ab, Z = ~1 /(J2, = ab, Q

(3.7. 11)

«(ll ; a1b2)",,( bJ; (l1(,2 )""( ab; a 2/,J )""( (llb-"; 02112 )""

= I + of; (1/( ab); a1tl )"b2n + .... 1

(ab;a2 til) ..

~ I l /(ab) 2.. "" I + L... 1 - ("):ln ~ 1 b

n_'

I

= (1- ab) (

~

2 (ab; (l2b )n (12 .. (aJ ir' ;a1 b2) ..

+ ... L... 1 1

""

I _ ab

f

n_ 1

+~ I

ab 2.. (,, )2n +1 a

1

(1 2.. - 2

(abl"'-l -

,;,,-1 la

""

~

(ab)"'-I

I

Multiplying t he extremal sides of (l .?l l) by a/( 1 - (Ib),

.

we complele

the proof of (3.7.10).

0

Theorem 3.1.1. Ij t/J(q) U (l.7.l2)

)

d~fine4

by (1.2.3), then

, • = L- (q6nH

qt/J(q ).p(q )

1 _ q12n_l-l -

.. _ 0

1

qlln+5) q

12,, +10'

Exercise 3. 7 .8 . U~e Th""rem 3.7.7 to find a formula for the number of .-epre~entatiom of Q po&itiw: integer by the ~um 0/ a triallg1dar number and th~ time3 a Iria'lgular number. Copyrighted Material

SPIRlT OF RAMA NUJ AN P roof. In Le mma 3.7.6, put 0 ~

.=;

q2o> _ 1 _ I

.r( 2~-1 ) _

77

= q and b = q~ t.o find

that

q (q' ; qI2 )",,(q8; qI2).,.,(q'2 ; qI2)~ (q2;q I2)""(q lO; q I2),,,,(q6;qI2)~

q6{1n _l)

'. ' ) ( 12 . 12) _ q (q ,q "" q ,q "" (q2;q4)oo(q6;qI2)""

(3.713) .

= q'l'.J.(q').'.(q') . 'I'

It remains to !implify t he left. side of (3 .7.13). Expand 1/( I _ q8(2n - I)) in a geometric series and in''eTt the order of summation t.o deduce that

f

n

q2 _l -

... 1

1

ofI 2n - 1)

!I(1" - I)

f: f:

=

(q(6m+1lI2n - l ) _ q(6mH)(2"_'))

", .. 0 ... 1

(3.7.14) Substituting (3.7. 14) in (3. 7.13), we immedi(ltely deduce (3.7.12) t.o oompiete t he proof. 0 Theorem 3.7.!J. Recall that op(q) and .p(q) a~ defined bV (1.2.2) and (1.2.3), resp«tive1v. Then (3.7.l1i) _

(

+ 4qtfJ(q2),p(q6) = 1 + 6 L

3n +1

I q J,, +I

".0

q

P roof. Using (3.7.12) and the trivial identity x :I: x2 1 _ :l:2 - I _ x - I _ x2 '

q&n+~

I

'" L-

qI2"+11I

q&nH

+

I

(o'"H

"-.

I

q3n+1 -

I

0'"") qa,,+2

_ (q' (3" +I)

(3.7.16)

-L

1

)

q l2 .. +lO

q4(3 n + 1) -

q.{3.H 2) 1

t"ffpyrighted Material

q4(J .... 2)

) .

8 . C. BERNDT

"

Multi ply the eJ[tremai sidea of (3 .7.16) by 4 and IIdd the ruultin& equality to (3.7.!I) to complete the prOOf. 0

T heore m 3.7 . 10.

(3.7.l7)

;;..

~ q' J,b.-oo

'+J t .. t '

~ (~"l

- I +6 L

1

qS~+1 - I

i'' -t-2) q3n+2

.

.. _ 0

P roof. By Theorem 3.1.9, it suffices to show tbat

f

(3.7.18)

ql'+it+ t ' =
;.' _ _ 00

We begin by observing that

/ + jk + k' = 110 that if j _

Hi + k)' + 3(!i)' ,

2" + I,

Hi + .1:)' + 3 (b)' -

(n + k )(n + k + 1) + 3n(" + 1) + I.

Thus, J••

~""'q"+/H • • - C.~"" + ; .•~,Jq(lJH)·H(M· j.......

j odd

L

~

L

q{.. H)( .. + H 11.;.3..{.. +I)+ L

~

q"' {"HL)+, .. (.. +1)+l

"'." --00 - (.r )~(qS ). where we havt! ~ the biJ.ateraJ reprmenlation of J (q. '1') _ 2~q) ill (1.2.3). Now employ Theorem 3.7.9 on the rll. ri&bt aide aboYo!. This therefore completes the proof. 0 Theorem 3.7.10 hall the following immediate and heautifullLlith· metic interpreUltlon.

T heorem 3.7.1 t . ut r(n) deno! ~ th~ number 0/ nprnenlatiuns of th~ JIO$.hue tntegt .. " bov Ihe lII'admtic formj2+ jk+ k~, """ere difJ~ COpyrighted Material

SPIRIT

O~'

RA Jl.tANUJAN

aigou oJ j ond k )'«'/4 diltinct mUget' n .

79

rep7Ul!lltotlorll

0/ n . TIIr" Jor each

",",title

(3. 7.19)

P roof. Return to (3.7.1 7) IlI1d expand the aummands on the right aide into geo luetric series. Hence, ~

(3.7.20)

L

~

qi· • .i • • t '

-

1+6

...-or_'

J,t --"" Now equate ooefficientl of q" ,

~

L L (q(S"'+1lr _ q(3••H2Jr).

'I ~

tbe desired focmula (3. 7.19) follows.

I, on both

,id~

of (3.7.20), and

0

3.8. Notes The formulae for r2 {n), "4 {n), "a(n), and rs{n) gi~n in Theorems 3.2.1, 3.3. 1, 3.4.1, and 3.5.4, respectively, are due to J300bi [1311. All four theorem!! can be proved !U'ithmetlcaUy. For example, see Hardy and Wright '. book [112 , pp. 24 1- 242[ for an elementary proof of Theorem 3.2.1. In Chapter 6, w>e return to these problems a.nd give completely different proolli of Ja.cobi'. formull\ll for r4 (n ) and r, (n) arising out of Ramanujan '. theory of elliptic functions.

The first proof of Theorem 3.2.1 that we have given is due to M. D, Hincbbom [115[. In a late~ paper [119[. he p~ anothrr simple proof b.-I on J.oobi '!! identity, Theorem 1.3.9. One year later, Hirtchhom 1122[ showed that one can deri~~ further theorems about ,.,(n ) from T heort!m 3.2.1. The fim proof of Theorem 3.3. 1 that ..~ ha\'e gi~n is al80 due to HiI"8Cbbom [11 6], ... ho earlier (114) had given another proof. S. I3 hargava and C. Adiga [58[ have abo utilized the summation formula to give ~imple proolli of Theorems 3.2. 1 and 3.3.1. StlU'ting with Theorem 3.2. 1 for two equate., B. K. SpelU'man and K. S. Williama [2101 provided an elementary arithmetic proof of Theorem 3.3.1 for four lIquare!l. For a proof of Theorem 3.3. \ based on a recurrence and oomputer algebra, _ a pa~r by G. E. Andrews, S. B. Ekhad, and D. ~IifJaW,8IJJl~proof of Theorem 3.4. 1 i!I

I'"

B. C. BERNDT

80

taken from a paper by S. H. C han [72J. Chan used lhe SIlml! idea to allIo establish a formula for le(n) , namely,

(3.8.\)

18(n)

L

=~

(_ 1)(d+ I)/2 d l ,

d!4 Q+3

We are uncertain who first provro (3. 8. I), but an e<;JuiV\l.lent formula-

lion can be found in an unpubli!!hed manuscript of Ramanujan [194, p. 356), [19 , pp. 398-401J. T he first published proof known to U8 is by K. Ono, S. Robins, IInd P. T. WaM [1 77, Theorem 4J; th ls proofU8l!Ol the theory of modu lar forms. An arithmetic proof is given in [128, p. 262, T heorem 11] . S. Cooper a nd H. Y. Lam [80] al$(> employed the \1P\ summation formula \.0 prove Theorem 3.4.1. The proof of Theorem 3.5.4 that wc have given is due to J .- F. Lin [144]. who in IillOther paper 11451 gave 11. variant of his proof. Ail we have seen, Lin 's primary idea in [144J is based on squaring a particular Lambert series. This is the same idea used by Ramanujan in his proof of the fundamental identi ty (4.2.6) for Eisenstein series in Chapter 4. This idea was aLw employed and general ized by K . Venkatachaliengar [214 , p. 31) and elaborated upon in more detail by Cooper [76, Theorem 2.2]. The thoorem of Venkatachaliengar is beautiful, and we present Cooper 's formulation. Define, for Iql < I and (I 'I q2~ , .. here k i8 an integer ,

-

."

F«(I, t)·" .- n _'~ _ OO l _ (l q 'Po

T hen

(3.8. 2)

F«(I , t)F(b , t)

=

a

tJi F{(lb. t) + F (ab, t)(Pl «(I ) + Pl(b)).

Cooper [76 J offenl several formulas for the number of integers repr". 8ented by an even number of squares or an even llumber of t rilUlgUlar numbers. Cooper and Lam in [SO] use RamanUjan '8 nl' l summation formula and Vellkatacbaliengar'a fundamental idelltity (3.8.2) to provide proofs of forumlD.II for 8UJ1\8 of k lIQ uarelj and SUIIllI of k triangular Copyngnted MEifensl

SPIRJT OF RAMANU JAN

"

numbers, for k = 2, 4, 6, 8, For com pleteness, t he formula for Is{n) i1I giwn by (3.8_3)

L

L,{n) ..

,-

(n;I)3,

dl{n+l)

A, M, Legendre [139, p . 133] "'8.'i the first to prove (3.8.3). An ana_ lytic equivalent of (3 .8.3) can be found in Ramanujan 's lost notebook (194]. [19 , p. 401 ]. Ono, Robins, and Wahl (177, p. 821 have csta!;>lished (3.8.3) using the theory of modulll1 forms. An arithmetic proof of (3.8.3) can be found in [128]. We provide a proof of (3.8. 3) in Chapter 6; see Theorem 6. 2.12. The connection between formula.s for 8ums of squares and sums of triangular numbeTS w88 further solidified by P. Barrucand, Cooper, and Hirschhorn 132J, ",ho proved that

r. {8n+k ) = c.t.(n),

where

C_= 2.(I +~e) ),

l:S;k::;7.

ThUB. the study of I. (n), for 1 :s; k :s; 7, is reduced to the study of the subsequence r~ (8 n + k ) of r . {n). Annther elegant approaclJ wlIS giwn by L. Cllrli t~ (621. lie employffi 11 beautiful formula due to W. N. Bailey [3 11, namely,

zq" yq") xqn )l - (I yq,,)2 (xyq)",,(q/(xy) )oo( xq/V )",,(yq/x )""( q)~ (xq)~ (qlx)~(Y/ q)~( q/yllo to give proofs of formulllS for ro{n) IUld 1"6(n). An equivalent for_ mulation of (3.8 A ) can he found in N. J. Fine's book (94, p_ 22, eq. (18.85)1. Bailey's proof (3 1] of (3.8. 4) employs the WeierstrllSS p-function from the theory of elliptic functions. Shortly tbereafter. J. M. Dobbie [83] gave a shorter, more elementary proof of (3.8.4). Williallls gave an arithmetic proof of T heorem 3.::'.4 based on an extension of an identity of J . Liouville (225]. Ramanujan 's l06t notebook (194, pp. 3::'3-3::.::.1 contains" fragment providing manYdf~aiMd/e~rems on Lambert series.

82

O. C. BERNDT

This fragmen~ has been examined by Berndt 1391. with tbe arithmetical OOlIIIeQuenoes of Ramanujan 'lI Lambert ..ne. identities also d~ by him. See aIao Berndt'a hook with Andrewa [19, Chapter

In·

Andrew. [13) uxd the theory of hMie hypergeotnd. ric .mee: to give 11 uniform apPT(III(h to provinl Jacobi 's formulllll fot rn (" ), 1 !> ,I; 4.

s:

Formulas for r. (n) , when It is odd, have an entirely different Havor. GIIWIS found a formula for r3(n), which Wall put in a mOll! concrete form by G. Ei$elllltein [88J. [90, p . SO~I, who aJao gave $!I analogoUII formula for .)(n). Chapter 4 in E. Croeswald's book [101] is devoted to the study of r3(" )' while p8I!:e!I 128 and 129 in the same text provide information about rll (n) . For recent work on formuJas for r_ (n ), when k Is odd, !lee papelll by ~ ['7 1], [18J. Theornn 3.6.2 is due to Jaoobi . Tb.! lint proof of ~ 3.6.3 was found by Legend", [139, p. 133J. Further proofs ~ gh_ by Cauchy [65, p. 572[, [66, p. 64] and Plana [179, p. 147). For aD elenulntlUy proof ~ (ID an extension of !ton Identity of Liouville, SOle a paper by J . G. lIuard, Z. M. OU, 8 . K. Speannan, and WilIiams [128]. Jarobi [131] claimed tha.~ V. Y. Bol.lniakowsky first proved Theorem 3.6.3, bu~ he did not gi,-e a reference. LIIl [1 43] derh'ed a formula for the number of repre:lentation8 of a poIIitive integer by 8 triangular numbers. T he analogue of Lq;range·. theorem is the theorem of Causa [98, p. 497] stating that every positi\.., integer et.n be repr8ented III a sum of three triangular numbers. An elegant proof via q-aene. hM been gh~n by Andre .... [16]. Williams [2<16J d.!riYed Ilfi eqant formula for the number of representations of a poIiliw integer M a sum of two t riangular numbers pl\JII twice the sum of tWO tri&Jl&uIar numbers. Liouvi11e [146[, [147[ determined the positive integers 0, h, and c, 1 ~ a ~ b ~ c, such tla! tile polynomial a.o.l + 1>.0.2 + C.o.3, .... here Ill, .0. 2, and 113 denote triangular numbers, represenu all positive inwgers. They are

(a , h,c) _ ( 1, 1, 1), (I , 1, 2), (1, 1, 4), (1, 1, ~), (1, 2, 2), (1, 2, J), (1, 2, 4). Copyrif;lted Material

SPIRIT OF RAMANUJAN

83

For a.n exposition of Liouville 's methods giving fnrmulas for r1k (n), I k :s 5, see Chapter 14 in M. 8. Nathanson '! text (167J.

:s

Although not explicitly 5tated by him, ThO!Orem 3.7.5 is due to P. G. L. Dirichlet [82]. The first explicit statement of Theorem 3.7.5, hOWll\-.!r, is due to L. Lorenz in 1871 [152]. The proof of ThO!Orem 3.7.5 that "'-.! have given was independently gi,·en by Bemdt [35] and by Bharga,-a, Adiga, and D. D. Som!l5hekl'ra [59J. R. A. Askey [27] used Rama.nujan 's 1"'1 summation theorem to derive a formula for the number of ways a positive integer can be repre:\ented by a squ.are lUId twice a square. Although IIOt stat.ed explicitly by him, Theorem 3.7.11 is also due to Dirichlet [82J, ",ho proved a genero.l theorem for representations of integers by hinary quadratic forms. Thl.lll, every tlleofem in this chapter concerning representations of integers as 8U11lB of squares is, in fact , contained in DidchJet'a general theorem . Theorem 3.7.4 h!15 been enormously generalized by Williams [224J, ",ho derived a representation for certain sums of t he products op(of){f'(q'» in terms of Lambert seri6'l. The anslytic formulation of Theorem 3.7.11, that is, Theorem 3.7.10, can be found in a iet\.er from RamlUlujan to Hardy written on a Sat urday, probably in 1918, from the nursing home, F it zl"Qy House [194, pp. 93-96], [51 , pp. 196-198]. This letter is examined in detail in Berndt's paper [351. and indeed our proof of Theorem 3.7. 10 h!15 been taken from (35J. Hirschhorn [121] has used a general method involving partial fractiOns to give uniform proofs of Theorems 3.2.1, 3.3.1, and 3.7.5. Equality (3.2.8), equality (3. 3.12), Theorem 3.7.2, and Theorem 3.7.4 are given by Rarnanujan in Entry 8(i)- (iv ), respectively, in Chapter 17 of his &e<:Ond notebook [193J, [34, p. 1141. Theorems 3.5.3, 3.6.2, and 3.6.3, or equivalent formulations, are given by Ramanujan all examples in Section 17 of Chapter 17 in hill $eCOnd notebook [1931, [34, p . 139). We have concentrated Our remarks on rli (n), for k S 4. Most readers by now have probably asked, ~Are there formul!15 for k > 41" Formulas for k > 4 are more complicated than thOllC for k S 4. For t hose who have some familiarity wit h modular forms, we remark that t he generating function {f'2i(q) for r1. (n ) is a modular form of weight k. For k S 4, t he dimension of the spa« of modular forms in which Copyrigl!/ecj Material

SPIRIT OF RAM A N UJA N

89

Using the elLllily Vl!rified elementary iden~ities

q'Hk

(

l _ q.. H

") ," (qk q~+ k) 1+I _ qk = I-q" l _ qk - I_ qn+~

(I in (4.2.4), we find t bat

, _

e" -

q" 2(1 -

q")

q" 2(1

(402.5)

.,

,,_I (

" qnl ''--

2(1q" qn)

...

qn:;;" (qk qnH) '-- I _ q~ - l _ q"H

+ l - q~

+

qk q" _ ~) 1+ - - + - - 1- qk 1- q,,_k

(I ~"qn) 2

(11 - I )q" 2(1 - qn)

," (I n) l-q"-2" .

=I _ qn

Substituting (4.2.3) and (4.2.5) in (4.2.2),

WO!

find t bat

,

'"'

'"'

~ qkOO'l(kO) ~ 00t ~O ~ q.) sin(kO) ' .. (~ 00t ~O) 2+ '-1 k 4 2 + '-- (I_ qk)2 k. j q t_l

(

4

(4.2.6)

1

.,

kqk

+ 2" L I _ qk (I-COII(k8)) . QC

This is the first of tbe two primary trigonometric series identities that WO! need. Using (4.2.6), we e'ltablisb a recurrence formula for the functiol\!l Sr. Appearing in the recurrence relation are tbe fun<;liollll from (4.1 .3)

(4.2.7) TbeoreIJl4 .2. 2 . Fo .. n~ I,

8 .C. BERNDT

96

Hence, by (4.2.30) and (4.2.36), "-e deduce that

Equating coefficients of 9ln , n 2': I, on both sides ai>o>-e, we fi nd that

"

..

(o 1)(2,,+5) _ H 12(n+l)(2n + l)SlMJ- L

2n ) S

, (2k

which oomplete5 the proof of T heorem 4.2.7.

S 2Hl

In _2HI,

o

For example, putting n = 2 in (4.2.27) and using (4.1.6) and (4.1.8) yields (4.2.38)

0,

Putt ing n = 3 in (4 _2.27) yields (4.2.39 )

on

Using (4.2.27), the definitions (4.1.8) and (4.1.9). and induct ion we show t hat, for eacll IlOnnegati>"e integer r .

T,

(4.2.40)

wllere the number$ c... ,n are constants. It is dear that (4.2.40) is ,'&lid for r = 1.2 by the dclinition (4.].6) of 52. + 1. and for r = 3, .. by (4.2.38) and (4.2.39), respectively. Assume that (4.2.40) holds. We prOVi.! (4.2.40) with r replaced by r + 1. By Theorem 4.2.7 and the Copyrigl)/oo Materiar

8 . C. BERNDT

98

these formulaa form

Me

cryptic, The first il given by Ramanujan in the 1 _ :;1q _ T1ql + ... = P. I q q~+ .

in succeeding formulas, only ~he first t'olUQ terms of the lIumerator &r1! giwn, and in twu instances the denomina.tor is replaced by a. dash - . At the bottom of the page, he gives the first liw term5 of a general formula for T 2k .

In this section, we indicate how tQ prove these seven formul8ll and 000 corollary. Keys tQ our proofs are the pentagonal number theorem ( 1.3.18),

(4_3.2) (q;q)"" = I

+ f:(- l )~ {q~(3n_ l)/2 + q "(J"+1){2}

..,

= To(q},

where Iql < I, a.nd Ramanujan '. famoU8 differential equations (4.2.20)(4.2.22 ).

We

II(IW

state Ramanujan's six formulu fOT Tn followed by •

ooroJlary and his gener&! formula.

* .,

Theorem 4 .3.1. IfT2 defined bV (4.3.1 ) and P, Q, ond R art define4 by (4.1.7)-(4.1.9), then

(i) T2 {q) = P, (q;q).,., (iil

~. (q) =3p2 _ 2Q,

(q, q).,.,

(iii) ITo(ql) = 15pl - 30PQ + 16R , q;q 00 TS (iv) I (ql) .. IMP' _ 420p2Q

q;q

1'1 IT IO(lq)

q;q ""

(vi)

+ 44 8P R _

132Q2,

00

ITI1 {lq)

q;q "'"

'"'94Sp5 - 6300P'Q + IOOSOpl R _ S940PQ2 + 1216QR,

'"' 1039Sf"I - I03950P'Q + Z2176()P' R _ 196020pl Ql

+ 80256PQR -

2712Q3 - 9728Rl . Copyrigl!/ed Material

SPIRIT OF RAMANUJAN

'03

use Ramanujan's observation abm-e along with (4.2.23) and (4.2.2O)(4.2.22) to give simplified proofs of (2.1. 2) lI.IId {2.1.3}. The proof of (2. 1.4) is more tooiOWl, and we refer to Berndt's paper (40) for the proof of (2.1.4), which is precisely that of Rusbforth [201).

Theorem 4.4.1. For eodi IWflnegaliw inleg
p(~n

+ 4) =. 0 (rnod~).

Proof. From the definltiollll (4.1.8) and (4.1.9), respectiwly, (4.4.4)

Q =l +~J

R=P+~J,

sirw.:e n~ =. n(mod~) by Fermat's little tbeorem. I~ foliowl! from (4.4.4 ) and (4.2.20) that

Q3 _ R2 =- Q(I +5J)2 _ {p+5J)2 _ Q _ pl +5J (4.4.5) But, by (4.2.23) and the binomial ~heonlm, (4.4.6) Q' _ R2 = 1T28q(q' q):14 = 3q (q; q)~ + 5J = 3q (q~;
+ ;'J.

Combining (4.4.5) and (4.4.6) and using the generating function for II(n), we find that (4.4.7) We now equate those tcnllll on both sides of (4. 4.7) whose powers are of the form
ltamanujan's oongruence (4.4.3) follows immediately from (4.4.8).

Theorem 4.4.2. For wch no,megaliw integer n, (4.4.9)

0

w,

n.C.BERNDT

These formulas are analogoU!! to the asymptotic formula for p{n) of Hardy and Rama.nujan t hat was discU5!!ed in (1.1.6) and the fI()(eS for Chapter I. However, their forms and proofll are much different. For acwuntJI of thi!! work, see papers of IIl1Tdy and Rarnanujan [Il l ), [192, pp. 3]0-321 J. Berndt ""d P. Bialek (42J, and Berndt, Bialek, and A. J. Yee [43]. For an account of all of Ramanujan'$ thWTems OIl Ei$eIL'ltein series from biB 1000t notebook, otber unpublished papel'!, and letteTs, see the book (20) by G. E. Andrewll and Berndt. For an expository survey of many of Il.&manujan'. theorems on Eisenstein seriell from his Io6t notebook, see the paper (53) by Berndt and Yee.

In bis notebooka [193]. Rarnanujan made further claims about Ei!lenstein series. In particular, in Entry 14 of Chapter IS of bis 8eCand notebook, Ramanujan recorded another recurrence relation for Eiselllltein series; see Berndt's book [33, p. 332]. Also, Chaptel'$ 17 and 21 in Ramanujan 's second notebook contain many claims about Eisenstein series. See Berndt's book 134] for proof.'! of all t hese theorems.

An im port&llt fU llClion which \\'e ha\'e not di5cussed in lhUi book is the modular j-invariant defined by . Q-'(q) ](-T) := -. ( )'" qq ,q"" The values of j(T) and i\.l! Fourier or [XlWer series coefficients are of great importance in the theory of modular forma. At about the same time that Ramanujan discovered the differential equation (4.2.20 ) satisfied by P (q), in 1911, J. Chazy [74] con!idered the third order differential equation (4.5.2)

'I'" = 2'11/" - 3(11')',

where 'I = I/(t) and ~howed that one solution is ".jP(q), where q = If ...-e eliminate Q and R from the system (4.2.20)-( 4.2.22), 'NIl deduce the differential equation (4.5.2) for P In fact, the syStem of equatiolU! (4.2.20 )- (4.2.22 ) is equivalent to a system of three differential equatiolU! fifllt studied by G. Halphen [104] in 1881. The equation (4.5.2) is of fundamental importan!:e in the theory of integrable systems [1]. [2], [3]. [4]. Copyrigl!/ecj Material

e'·".

SPIRlT OF RAMANUJAN

107

The reults in Sections 4.1 and 4.2 "'" talcen from Ramanujan's paper [186], (192, pp. 136-162]. Exerci~e

dtduttd

4.5. 1. Show that Ramanujan'& key identity (4.2.6) can be Venka tacha/iengor'&ruull (3.8.2).

from

[n her thesis (183]' V. Ramamani extended Ramanujan's use of trigonometric ooril)Oj to study two analogues of ",(q), namely, ~

L

>ir, .• (q):=

(_ I) .. - Im'n'q""·

~

F, .• (q):=

L

(2m _I )'n' q(~m - l)nn,

...... __ OCI

where (q( < I. Ramamani employed the classical theory of elliptic functions in her work. In extending Ramamani '. study of (q), in her thl)Oji.9 [102], 11. Hahn chiefly employed ide ..... from Ramanujan's viewpoint in the theory of elliptic funct ions.

w, .•

H. H. Chan (68] deriwd an analogue of (4.2.2) .... ith cot replaced by C9C and used it to give new proofll of Ramanujan's famo\lll formulu

wd

.... here (i) denotes the Legendte symbol. These formulas lead, respectively, to proofs of Rarnanujan 's congruences p(Sn + 4) := 0 (mod 5) and p(7n + S) := 0 (mod 7). Z.--G. Liu [148] used the theory of ell iptic functions and associated complex analysis to deri>-e a trigonometric series identity invulviog thetB. functions that is analogoWl to Ramanujan's t rigonometric series identitiell. W hereall Ramanuj&n used his results to study sums of Squares, Liu used his identity to obtain repreJlentations for h.(n). The content of Section 4..3 i.9 taken from a paper by Derndt and Vee [53]. K. Venkat~3.8;~)ftJa~aflr;a~I -32] has gh-en

&.

similar

8 .C. BERNDT

112

Theorem 5. 1.6 (Landen'. Transronnalion). ForO

< :z:

<: I ,

K(:::) . . (1 +z)K(z),

Corollary :0. 1.7. ForO

(~. 1.8) ~Fl

1,

(, ' ('-')') 2''2; 1; 1 -

Proof. Observing that

I+:z:

.

(1+:z:)1",I -

(")

- {I +Z)l FI 2''2; I;r .

(,1+% -.), '

we _ lhal (5.1.8) follows immedi8tely from (5. 1.7). COpynghted Material

o

SPIRlT OF RAMAN UJAN

113

The version of Landen 's transformation that we have given is actually a special case of a mnre general Landen transformation, namely, (&.1. 9)

2F) (0, b; 26; (I

~X)2 ) = (I + x)2
(1 -

6+

~;6 + ~ ; X2) .

By Lemma &.1.3 and T heorem 5.1.6, the special case of (5.1.9) that .w. ba\"e proved is for (1 = b = ! . There is still olle further vcn;ion of Landen's transformation. To describe this transfonnatioll, "'"e need t.o defille incomplete elliptic im.egra.l!i. Definitio n 5. 1.8. 110 < Cl" :<::: 'If'/2, the incomplete elliptic integrol 0/ the fir!t kind u defined by

I." -';"l~d4>:;;k"'''i"'''''' We IlOW state Landell 's lraQSformatiOIl for illcomplete elliptic illtegrals of the first kind. Exercise 5. 1.9. Illxl < 1 and x sill 0 "" sin(2/3 - 0), prove that

(5.1 .10) When (r = 'If' and fJ = tr/2, (5.1.10) reduces to T heorem 5.1.6. We highly recommend that readers prm'ide a rigorous proof of (5.1. 10). Examining elosely the definition of 2F) (~,~; I; x) IIIl Z ~ I, we conjecture that a partial sum of 2FI(!,!; I; x) behaves like a partial sum of the harmonic series. We make this OOOervatioll more precise in the following lemma, which is used in the Ilext section.

Lenu"a 5. 1 to. A ! x - 0+, (&.1.11) ",here C i3 a can$tantcopyrighted Material

B. C. BERNDT

114

P roof. By Lemma. &.1.3, it suffices to prow that

2K(.fi} '" - lug(l - :.:) + C,

(:),].12) lIIIx ..... l-.

In the representat ion (a.1.3), let t

= sin~, $0 that COlI = ~.

Thus,

(5.\.13) Note that, for 0 < x < I , zdt

1- - = 1

(5. 1.1 4)

{} 1 - %1

- lQg(i - x ).

Now examine, for 0 < z < I ,

R{ , ) ,- [' {

.- Jo

,

.;r=tJ~

ObserVll that RC"' ) is continuous at

«(i. LIS)

RCI ) =

%

- - '

= 1-.

1- xt

}"

.

In fact ,

- -, - --- = ' } = 1'" 1'{' o

1-t

1-1

dt

0 1+ 1

log2.

Thus, as:t" ~ ' -, the two integral.'! in (5.1.13) and (5. 1.14) are as. ymptotic to each other, since their difference approaches a OOll.'!tiI.ot, namely, log:?, as x - ,- , Hence, (5.1.12) follOW!! from (5.1.14) and (&.1.15). 0

5 .2. The Main Theorem Briefly, Our primary goal is to prove that

whel'1l q is a certain {unction of :t. We shall e6tablish this relalion after a series of IlIlcillary lemmas.

Fi rst define . for 0

(&.2.\ )

< % < 1.

SPIRIT OF RAMANUJAN

i:

By oontinuity, tince lFI (!, -I; I; 0) - I and lFI (-I, I : z) - 00, as J: -> 1-, by Lemma 5.1.10, ~ (611 extend the definition of F (z ) t.o % .. O .... d;l: _ lbytetting

F(O) .. 0

F ( I ) _ I.

and

Lem ma 5.2. t. For 0 < % < I, (5.2.2)

F (%') - F '

Cl

:;l:Z}l) .

..

P roof. In (5.1.7) replace %by (I - %) / (1

, %

+ %).

Obeen:e that

- l -(I +%p'

u

,

( I+%), -> I - Z , 1 +;1:_ - '-.

1+.

Hence,

(5.2.3)

~

arrive .t

1 ~%2FI G , ~;I;I - (I :Z%)l) -

,F1

G'~ ; I ; I - zl).

Now divide (5.2.3) by (5.1.7) t.o deduce that

(5.2.4)

2

2F'(i , l ; I:I-~) (

,F1 1 , 1 ; 1 :(I !~j f)

-

,F1 ( , , 1; 1 _ %'

,F'I

, : 1;%2

Multiplyif18 both . ides of (5.'2.4 ) by - If , exponentiatif18 both tides, a.nd invoking the definition of F (%) horn (5.2.1), ~ complete the proof of (5.2.2). 0

Lemma 5.2.2. 1/%, 0 < Z < I, ", defined by (5.'2.5)

(5.'2 .6)

1- %

r:;:-; "

",2( _ q)

'P"(q) -: A - A(q),

116

8. C . DERNDT

Proof. By (1.3.32). (3.6.7), and the definition of A in (5.2.5). l' .,4(_,') _ { A (q ) - ..,.(q1) -

=~ =

11+» '

2,..{q)op( - q) } ' q)

~(,) + ~(

4~

(I+JT.)

,_ I _ z 3.

o

This concludes tbe proof. Lemm a 5 .2.3. 1/"

(5.2. 7)

F

= 2"',

when! m

U alll!llallllly(ltive illllyu,

(_'..,.(q) (-0») _F" (_'""{q") 1-,"»).

P roof. ~laeing z by (I - z)/( 1 + z) in Lemm.. 5.2. \, find that

1Io"l!

readily

(5.2.8) Applying Lemma 5.2.2 to (5. 2.8), _ find thllt (S. 2.9) Iterate (5.2.9) to deduce that

F(,\'(q)) _ F Z (AZ(qZl),. f'2' (,,2(l'») _ .. _ F2~ (,,3 (q'~»), that i$ to say,

o Lemma 5 .2.4.

1/ n

_ 2"' .

..men! m U anI! nOf\llt9(1111~ IIlllye,., IAen

(5.2. 10)

Proof. From the definition of F (<<) in (S.l.l) and Lemma 5.2.3, Ml that if ,.,4(_q")

!lee

coP;;';'J,,\!;J,;.,

SPIRIT OF RAMA NUJAN

,,.,,

3Fd , ! ; l ; l - ZL) _ 2 f'L(! ,!, I, I - Z.. ) n 3FL H, , 1; ZL) l FL (!, ~; I; z .. ) We rewrite the IlL!!t equality in the form

.

3FdL, ; 1; zL) , i ; I; I - z L

2F,(

-

21",(1, , I,z.. ) 2F,( , , 1, I - z .. )

But thill implies tbat F"( I - ZL) _ 1"( 1 - z .. ), whieb i8 the _ (5.2.10).

all

0

Theorem 5,2,5. lYe have (5.2.11 )

F

'-P" (q) ) = q. ( "(-01 1-

proor. Wrile Lemma 5. 1.10 in the form (5.2.12) wbere 0(1 ) denotes a funetion which tend.!! to 0 as z _ 0"' , not ~ sarily the same with each appearance below. Usinl (5.2.12) and the de6nition (5.2.1), we see that (5.2.13 )

F(z ) ,.. e loi r _C.

'"

-!L)

= A zeo{L) =

A:r:(1 + 0( 1»,

",' (- q" )

(5.2.14)

Z,, :_ 'P'(q" )'

Note that:.:" lends to I 118 n __ 00. We now take the nth root of both

sides of (5.2.10) and Hlt n tend to 00. We neM to take care on the ri&ht aide when letting" _ 00. Thus, in the notation (5.2.14) and with the use or (5.2.13), _

F(I-

find that

.-

.. Urn ylA(1

Zft ) z .. )(l+ o( l ))

.. tim .:11 - z •

• -- ~c-c-~~ . • ",' (q. ) - ",' t - q" )

.. hm Copyrighllia''f,f· ferial ""' (q" )

D. C. BERNDT

liS

.--

=Iim" (5.2. 15)

='1,

""bere in the penultimate line we ll8ed (3.6.8). ThiII oompletei the proof of Theorem 5.2.5. 0

Lemllll' 5.2.6. We /tape (5.2.16)

1 I

ZFI (

2' '2; 1; 1 -

'1'4( _ '1 ») 10'4('1)

») .

3

",Z(q) (1 1,. 1"4(_ '1 .. op2(q') ,F1 2'2' I, \ - 'P" (tf)

proor. From (6.2.5) and (5.2.6),

(l + r )2 .. I +Z( 1 _ ~) = # ('1 ) '1'4(-'1') 1_ z .p2( _'I) r,l'4{q') ~ (q) ~(q)""( - 'I)

= cp3(_q)

(5.2.17)

'P" (if )

'1'4 ('1 )

-
by (1.3.32). Now substitute from (5.2.5), (5.2.6), and (5.2.17) lnw 0 Corollary 5.1 .7, and the result (0Ilow5.

Lemma 5.2.7.

IV~ Ita~~

(5.2.\8) Proof. Iterate (5.2.1 6) m time. IQ obtain the equalitiN I 1

2Fl (

2' 2; I ; 1 -

(5.2.19)

_q») -

'P4( 'P'(q)

...

~(q)

(I I

'P4(_ q4 »)

2' 2; I : 1 - 'P"(tt )

(I I

= ,pz (q2~) , F\ 2' '2 ; I; 1 -

I". (-~-») 'P" (q2- ) .

Now let m _ 00 in (5. 2. 19). Ob8erving that
(i.!; \;

a!!

We are now ready j.O view the Main

&~nl .

COpyr;ghted Matenal

SPIRIT OF RAMA NUJAN

119 defin~ ~ (~.2. 1 ).

Theorem 5.2.8. ReaIlllhat F {,z) is I,

FqrO

< 2: <

(~,2.2O) CI'f", In

other

word,. If : I:I-,z» ) , , ; I;,z

(5.2.21)

(5.2.22) P roof. We begin by summarizing first Lemma 5.2.7 &Dd III!CODd Too. orem 5.2.5. If

,", (5.2.23)

.od F (2:)

(5.2.24)

= F (I -

uj.

The lMt eq\l&lity (5.2.24) implies that (5.2.25)

,F

1 (

,

:

I: I - :1:)

,F1 ( ,,: I; ,z)

~,F~,~(-\,-,,,-,,,,,-, =, F1 ( ,!: I ; I - ur

Suppoee tha.t we tan show that the denominatOf'll in (5.2.25) ue eq\l&l. i.e., that (5.2.26) Then it will follow from (5. 2.23) that ~2(F(,z»

= zF1 U, 1; I:,z) ,

i,e., (5.2.20) holds. We sbow that (5.2.26) eMily follow~ from the mOllOtonlci1y of ,Ft U,!: I:,z) on (0, I ). Suppoeoe there illlIOme poiot "'0 $uch thllt (5.2.26) does oot hold . Let Uo :- u(xo). Without 105S of generality, _ume 1hat I - Uo < 2:0, 110 that, by the monolooicity

8. C. BERN DT

120 l~

follows from (5. 2.25) that 2F, U,~; l;Uo) < ~F,

U'!: I: 1 -

:to),

which implies, by rnonotonicity, that Uo < 1 - :to, which is incompat_ ible with the previous condition 1 - Uo < Xo. Hence, our assumption that there exist.!! 11 point Xo such that (5.2 .26) does not bold has producN a contradiction. 0

We place Theorem 5.2.8 in the context of the classical theory of elliptic fUllctiolls and summarize some of Our principal result!! in this chapter. Set x = k 2, where, all 8~, k i! the modulus. Also set (5.2.27)

so that the complementary modulus k' is given by (5. 2.28 )

/(.'

= ~ =
Then, by Theorem :;.2.8,

(5. 2.29) where

(5. 2.30) q

= e-~ := exp (_w2f:,~~(t:: :;~2~2) ) = exp ( -w ~)

,

where K = K (k ) and K' "" K ("').

5 .3. Principles of Duplication and Dimidiatio n In this section we show how a formula inVl)lvi ng x (or ,(2 ), !I {de6ued by (5.2.30)), and : (defined by (~. 2_29)) ean be CQm'erted to IlD equa.~ion involvill& 2y or y/ 2.

Define, for 0 < x' < I,

(~.3.1)

x'

from which it foU(JW3 that (~.3. 2)

=

C: ~r

SP IRlT OF RAMAN UJAN Fbrthermo~ ,

define

(~.3.3)

e - " :a F (:r')

121

1_ , FL !!,!: I::r').

and

By (~.2. 1 ), (5.3.2). Lemma ~. 2 . 1 . and (5.3.3),

(U4 )

e-~ _

F (:r)

=F (

4~

) _ ';F(:s:')

(1 + :s:' )'

~ e-~'/'

H~,

(5.3.5)

h'·

r-

Furthermore, by (5. 2.29 ). (5.3. 2), (:>. 1.7). and (:>.3.3),

: = , F L G' ~; I;Z)

- , F O, ~; I: ( I :~),) L

_( I

+ J%ihFL G , ~;I;l':')

_ (1 + H)z'.

(~.3.6)

Th ..... solving (5.3.6) for : ' and Wling (5.3. 1). we find th&t (5.3. 7)

,: ::=~=

1 + v:s:'

:: ~

1+ L-

L+

I( ~I __ ,l+V I _ :r:.

_ I

-I

Them 'em 5.3 . 1 ( Princi ple of Duplication ). Supp!»e that tlllO .ret8 of parumdl!l"l, :r, r . and:: ami T , V and:'. are retatM /lr the cqualtOfl.f (5 .2.27), (5. 2.29), and (5.2.30) WltlL:r, 1/, and: rep/aud b1/%'. V, and T. re.opectlllO!l,. SuppoJe thelllatuh an equallon of /he form n {:r',

v,

t /) _

O.

aml:r L.!I retatM to:r' It, (:>.3.2). Then. by (5.3,1), (5.3.5), and (5.3.7), we obtain an equation of tilt foroL (5.3.8)

n (( 1-11=» ' I

+ .,JI

:r

I I +v1-.o:): ~ ) _ 0. , 21/,-,(

T heorem 5.3.2 (Principle of Dimidi&tion ). A I In the premotll theorem, , uppose that two ,et8 of poromettrl, :r. r, and: and:r' ) ami T , are ntalM 1>11 the ~1l!r}M&klbJl29), and (5. 2.30) ...~th:r,

v,

122

D. C . BERNDT

V' and :r replaced bv z.', an etj'U4to.cm o/liIe lorm

11.

and %', rup«t'w/y. SUPpole they " tuf)

U(%', y' , i): O. and

Ill\!

re!lerU lIIe rotu 01 z , 11, and

%

""Ill /hole 01 z' ,

11,

and

%', re-'J'«hw/y. Then, by (5.3.2), (5.3.5), and (5.3.6), we abtom an equotlon olllle larm (S.3. 9)

' I" 1 ) o ( (I + J%)1' 21/, (1 + v'i"):r

-

o.

5.4 . A Catalogue of Formulas for The ta FunctiollS a nd Eisens tein Series l/sin& (5.2.:19), (5.2.27), the princi ples of duplica~ion and dimidia~ion from tbe previoua KCtion, and element,.,.y theta function identities, such as thale liven in T heorem 1.3.10, we can derive a plethora of evaluation. of the functiolUl~, ,p, J, and X lit d ifferent ~n of the argument q in lerTlU! of %. z, and q. l/!ing the tool! mentioned above, rea.den should be able 10 easily derive each of t he formulll!l below. Proof, of all t he reIIults in Theorems 5. 4.1- 5.4.4 can be found in [33, IIp· 122- 125, Entriell 10-12]. After we state the formulll!l, ...·e offer a few proofs in iIlWltration.

Theorem 5 .4 . 1. Ilz. q, and (5.2.30), tlIen (il (u)

(Hi) (iv)

(v) (vi)

(vi i)

%

ore nUltttl

bv (5. 2.27), (5.2.29),

tp(q) - .;;.,

tp( - q ) _ ';;'(1 _

<;)(-q')

=

.ly/~,

';;'(1 - Z)I/I,

..fiV1 (I + JT'="i), (p(q.) = i..fi(1 +(I _ X)1 14). (p( v'V = ..fi (I + Ji) 1/' , (p(q') -

<;)(-v'V=..fi(I - Ji)I /'. COpyrighted Ma/anal

and

SPIRIT OF RAJl.1AN UJAN

123

proor. We offer proofs of (ii H iv ). To pro','e (ii)-(Iv), employ (i) and respectiYf!ly, (6.2.27), (1.3.32), and (3.6.7). 0

1Ifie,

6. 4.2 . 117:, q, and (5.2.30), then

T~re:m

(I) (il)

~

are ,dated boy (5.2. 27), (5.2. 29), and

~(q ) _ ~(x/q) I/', ",(- q) -

vr; ! vr; {(

(x{1 - x)/qJI/I ,

(iii)

\I!(q') - h'':(zfq)I/4.

(iv)

,,(q~ ) -

(v) (vi) (vii)

1-

...'f="i) I q} 1/2 ,

= !v'i{ 1 - ( 1 - X)I /4}/q, = .ji (Ht + Ji) } I /~ (zfq)llil, y,{ - ..;q) = .ji {Ht - Ji)} I /~ (7:/q)l/ll. f,o(ql)

Y1{..;q)

Proor. We prove only (i}-(iii). To p~ (Iii), use (3.6.8) along with Th«lrem 5.4.I(i). (ii). To p~ (i), use (iii) and the principle of dimidiation. Part (ii) follow. horn (1.3.30) and Theo~1ll.! 6.4.I(i), (ii) and 5.4.2(;). 0 Theore m 5 .4.3. II x, q, and ~ an rflated by (5.2.27), (5.2.29), and {5.2.30}, then

(I) (il)

I (q),., v'i2- 1/ 1 (x(1 _ x )/q)1 /34 ,

(ili)

I{ - q) _ .ji2- 1/ 1 (1 _ %)1/1(%/ '1)1/24, I (-tl) "" .ji2- 1/ 3 {%(I - %)/ql l/l~ ,

(iv)

1(_q4 ) = ."Ii4- 1/'(1 - %)I/:M(x/ q)I/'.

proor. We prov'e (i}-(iii). Both (ii) and (m) follow from (1.3.34) and previously pl'(lVe(! result.a . Next employ (1.3.30) to p~ (i). 0 Theorem 5.4.4. 11[1.. q, and z arf related ~V (5. 2.27), (5.2.29), and (5. 2.30). then

(i)

(H) (iii)

B.C. BERNDT

124 Exercise S.4.5 .

Pro~ $om~ furlh~T

rt$ult" in Theorems 5.4.1-5.4.4.

We oonclude lhis chapter by deriving reprel!ent atiQlllI for P(q), p (q2), Q (q), Q (q2) , R(q), and R(q2) in terms of x and z . HoweVl'f , first we need t he following t WQ impOrtant theorems, the first of which ~aders can elLllily prove.

Theore m 5.4.6. The /unction z /cnmtial equation

tP:

(5.4.1)

=

2F,(!.!; I;z ) $Q ti.!jiu the d;!· d:

1

x(l - x),n1+(1 - 2x)d.z - 4" z=O.

Exercise 5.4.1. The differential equotion (5.4.1 ) ha.! a regular.ringUlllT pomt at % = O. Solve (5.4.1 ) by the method Q/ fh:>be,,;"-! to find that one of /he two linmrly independent 6o/utiQn.I 1.'1 2 FI(~'~; 1; x). Those readers unfamiliar with solving ordinary differ(!ntial equa· t ions with regular singular points may use the definition of : = 2F, (i , i;l;r) t(l e3IIiJy check that: = 2FI(i,i;1; x) is indeed 8 solution of {S A. I ). Theo rem 5.4.8. If 11 i, defined by (5.2.30) and ~ if defined b~ (5.2.29), then (5. 4.2) Proof. Uaing (5. 2.27), (5. 2.29), (1.2.4), and Theorem 5A.3(H), (iv), we can resdily esta.blish the identity (5.4.3) Take the losarithm of both aides of (5. 4.3 ), differentiate with resp«t to q, and multiply both side! by q to deduce that q

Ih

00

4nqh

00

nqn

- <' - 8" - - +8 "_ x(l - x )dq ~ I_q"n ~ l_qn n_ 1

((;. 4.4)

= I+8L 00

.... I

""

n_ l

n q"

~ q

= ,.,4(q) = z',

Copyrighted Material

SPIRI T O F RAMA NUJAN where we have employ«! the fifth line in the display (3.3.7), and (5.Z.Z9 ). Using the tri vial fact dq d, d:J: = -q dx

o

in (6.4 4). we com plete the proof. T heore m 6 .4.9. With : d~finw. by (5.Z .Z9 ),

(6.4.6) (6.4.6)

,

d,

P (q)=( 1 -6z)z +IZx ( I -:t")z dx'

, d, P (q1 )=( 1 _ 2:t") z +6x( I - z )z d:J:

P roof. We pJ"O\'e only (5.4.6); the proof of (6.4.6) is similac. Applyill& t he definition of P(q), Thoorem 6.4.3(iii), the defini tion of I ( - q)

from (1.2.4), s nd lastly t he chain rule along wit h Thoorem 5.4.8, we find t bst

L

12-d -

=1-

log{l - e-2n ~ )

dY .. _I

=l -

IZfv

Jo.g(e-1~ ;~-2~)oo

"" _~ log{ ~ - ~ l 2( _~ - 2~l} d, d

= __ log{ e-~z6 Z- ' z( 1

d,

-

x )~~ }

= _ _d log{ zax( l - :t"»

d,

= 1'(1 -

d

1'):1 dx log{ z5 Z(1 - x l}

d, ,.. ( I _ 1' ):2 _ n 2 + 61' (1 - z ):-

d.

= (I -

21'): 2

d,

+ 61' (1 - :1:):
Copyrighted Material

o

\2,

B. C. BERNDT

Exercise 5.4. 10. Altenwtiwlv, apply the (5. 4.6 ) to derive (5.4.5).

p~u

of dirnidiation to

Th eorem 5.4. 11 . lYe halH!

(5.4.7) (5.4.8)

+ 14% + 2:2),

Q(q) '"' , · (1

Q (q2) '" : . (I - :r; + %2),

proor. We first prove (5. 4.8). Se<Xlndly, we establish (5 .4.7) by ar>plying the p rocellll of dimidiation to (:'I.4.S).

,ho,

From the differential equation (4.2.20) and the chain rule, we find

(&.4.9) From the definition of q, i.e., (5.2.30), Theorem 5.4.8, and the chain rule, we lIeJ
(5. 4.10) It follows from (5.4.9) and (5.4.10) that

(5.4. 11 ) We next use (5.4.11 ) lLod Theorem 5.4.9 to establish (SA.S). Differentiating (5. 4.6) with respect to rule several times , IlIId simplifying,

""i!

:1: ,

applying the product

dedu<::e that

(&.4.12) dP(q2}

'" ,

* + 6x (1 - .:El (*)' + x)z dx' ~, d') + 6r(1 - %) (d')' ~, = - 2. (z - 4(1 - dx + 6% (1 - %)zdx dx' ,p, (d')' ,p. =-2z . 4x(1-:r:)dx,+6z(I-:r) dx +6x( I-:r )zdx2

= -2z + 8z(l -

2%) dx

dx

2%) -

= - 2z:r:(I -:r) dx,+6:r(I:r:) ,p,

dx ~'I.' a er/al

Copyrighted

6:1:(1 -

SPJRlT O F RAMAN UJAN

,p, - 3 =- Zz(l - 'I' ) ( zdJ:~

127

( "dJ: ) ' )

'

... here in the antepenulti mate step we employed the differential equation (5A .\ ). Using (5. 4.12) and (5.4.6) in (504 .11), we find that

all ) = (I _2.c):2 +6z{I -

z )z

!:) 2

- 6,(1 - , ),' ( - " (I - . ) ( ,

= ( I _ 2'1'f z·

~: -

3 ( : )') )

+ 12'1'(1 _ '1')( 1 _ 2'1' )z3:!: +

12x~(1 _ 'I')~z3 ~;

= ( 1 - 2'1' )2: 4 + IZ'I' (I _ '1')( 1 _ 2'1' ):3 : + 12'1'( 1 - '1'):3

(- (1 - 2Z)~ + ~ z)

_ ( I _ 'I' + : 2)z' , "'here in the penul timate line we utili:red the differential equation (5.4.1). T his oompletes the proof of (5.4.8). We now use the Principle of Dimidiation to establish (5A.7). A ccordingly, from (5. 4.8),

I"') (I + v='l'/z" )= (1 - (1 +4",.;i)2+ (1+,fi)4

Q(q

=( I + lb +'I'2)z' ,

o

after simplification. T heorem 5.4.12.

IV~ h411
(5.4.\3)

R(q)

(5.4. 14)

R(q2)

= z6( 1 + '1' )( 1 _ 34'1' + : 2) , = zG(1 + '1')( 1 _ ! 'I' )(1 - 2'1'j.

Proof. We begin by provi,,& (5.4.14) and proceed a8 ...-e did in the derivat ion of (5.4.11 ) of the p~i,,& proof. Using (4.2.21 ), (5.2. 30), (5.4.10), Theorem 5A.8, and the chain rule, we deduce that (5.4.15)

3'( I _C"'\l2~~2) = 2P((, ') Q(,') 0PyfIg_ d Ma/eri"sl

2R(, ') .

B. C. BER NDT

128

Solving (5.4.15) for R(q~) and emp\Qying (5.4.6) and (5. 4.8) (twice), we find that R(q3) =

P(q~)Q(ql) _ ~Z{I

_ X)Z2dQd{q2)

2

•

... {{L _2% }Z2 +6Z(I - Z)Z:!:} z4(1 _ %+ %') =

_

~;z( l - xl:' {4~J: (1- x + %') + :4(_1 + 2%)}

z6

{(I - 2l:)(1 - :z: + ];') + ~x(i -

1'}(1 - 22:}}

= z6(l -2x)(1 + x) (1- ~x) , which completes the proof of (5.4.14). Applying the Principle of Dimidiatioll to (S .4.14), we find that R(q)

= :8(1 + y'i)ti

(1 + (1 :~p) (I - (I !~p)

X(l - (1!~p) '" :8 (I

+ 6,,1% + z) (I

- 6.ji +:1:)

(\ + xl

= zG(\ +x){l _ 34% + x'), which completes the proof of (5.4.13).

o

E xe rcise 5.4.13. Prow; that p(q4 )

= z2 (I _ ~x) + 3x(1 _ xl: : '

Q(q' ) '" z' (I - x

+ t6",2) ,

R(q' ) = :6 (1 - ~:l") (I - z - -hx2).

5.5. Notes We have follo,",w Ramanujan's presentation in Chapler 17 of hi~!IO!C ond notebook [1931. as presented by the author in [34 , pp. 91-10'2, Sectioll5 2- 6], in our dewlopment of T hoo"",m 5.2.8, Ho...-ever. the clever proof of Theorem 5.4.8 that we have gh~n was comm unicated to us by 11. 11. Chall and is m"ch si mpJer than the proof ill {34, p. lWj: Copyrighted Matenal

SP IRJ T OF RAMANUJAN

\29

his paper with Y. L. Ong ]11]. The Princi plet of Duplication and Dimidiation aTe due 1.0 J acobi 113 1J. Ramanujan l!.aleS the Principle of Duplication in Entry l3{vii ) of Chapter 17 in his eeoond DOtebook ]193], (34, p. 127]. It is clear that Ramanujlln frequent ly II!Jed the Principle. of Duplication and Dimidiation in his "'"Ork OIl elliptic runetiom. Our pre:!!entation of these principles is similar to that given in the author', book ]3-4, pp. 12&-1261. lee

For Ramanujan , it WIIII nawral to IIII k whether there were other theoriea of elliptic functiom in which the bypergeometric function 2FI (!,!: I:z ) il ~plaoed by other hypergeometric functions. In his famous paper, ModtJ4r tqUGtWM Gild Gppro:nmotiGn..I to ,or [184], [192, pp. 23- 391, Ramanujan wrote, "There are corresponding tboories in which q is replaced by one or other of the funct;OfIB ql _ e - ·K:~fKI ,

lh _ e- l.K'/( K, .I5I,

-ho<, K\ _ I F\
-

l

F I Ci,i:l;z),

Ka - 2Fl (!,1; I;z)." Pages 257- 262 in RamlUlujan's se-cond notebook are devoted to de""loping these theories. T he many clainu on theee aix page!! were firn ~ by BerOOt, S. Bhargava, and F. G. Garvan [411. See also Chapter 33 In BerOOt'. book ]38] . H~r , Ramanuja.n '. tbooTies are by no means complete, and since the appearanoe of ]411. m.any other papers have been written on Ramanujan '. alternath" theories, .. ith Aill much remainint; to be acoomplisbed. The formula (tI. 1.6) is due to t he Scottish mathematician C. Madaurin 11531 in 1742. Our prOOf of Landen '. tra.nsformation , Theorem 5.1.6, is allllOllt identical to the proof t hat the English astronomer J . Ivory ]1291 ga,'e in 1796 fOl" (5.1.6) . When Ivory lubmi t~ ted h.i.s pllpef to editor John Playfair, he naturAlly Included a cover letter, but, rllther Burprlsi ngly (at least to 118), the cover letter WIIII published aJOIlI with the paper. In thiBletter, he relalel how he di&CO\"red hil formula. ~ H avi"" lIB you know , beII~ .. good deal of time and attention ~~~,+'W~lnep.rt of physiCAl astronomy

8 .C. BERNDT

""

which relates to the mut ual disturbances of the pJ6flelS, I ha'~, naturally, been led to oon.,ider the various methods of resolving the formula (a1 + b 2 _ 2ab 0084»n into infinite series of the form A + B 0084> + C COII2q; + k c. In the oourse of these investigations, ... series for the rectification of the ellipsis occurred to me, remll1kable for its simplicity, as well lIS its rapid convergency. As [ believe it to be new, I send it to )":Ill . Landen's transformation was introduced by J. Land(m in a paper written in 1771 [137] but developed more completely in hi:! pa_ per [1381 publ ished in 1775. This trlUlBformation was crucial in our proof of the fundamental Theorem 5.2.8. The importance of Lan· den's traruformation is conveyed by G. Mittag-LefHer, who, in his survey sUicle [l60J on elliptic functions written in 1923. emphasizes, uEuler's addition theorem and the transformation theorem of Landen and Lagrange ""ere the two fundamental ideM of which the thoory of elliptic functions was in po5IIe!i5ion when this thoory WM brought up for renewed consideration by Legendre in 1786. ~ Born in 1719, LandeR was appoiRted as the llUld-D.geRt to the Earl Fitzwilliam, 11 pOI>t he held until his retirement t"u years hefore his death in 1790. According to an edition of Encyclopedia Britannica published in 1882, ~He [Landen] lived a very retired life, and saw little or nothing of society; when he did mingle in it, his dogmatism and pugnacity caused him to he genera.lly shunnM." Landen made se,~r&l contribut ions to the Ladies Diary , which was published in England from 1704 to 1816 and "designed principally for the amusement and instruction of the fair sex." As was common with other contributoTli, Landen freqlllmtly used pseudonyms, such as Sir Stately Stiff, Peter Walton, WaltonieTL'lis, C. Bumpkin, and Peter Puzdem, for problems he proposed and solved. The largest portion of each issue was de,uted to the presentation of mathematical problellll5 and their 10Jutiollll. Despite i19 name , of the 913 contributors of mathematical problems and IIOlutions O"I~r the years of ill< publication. only 32 were women. For additional information about Landen and the Ladic6 Diary, see a paper by G. Almkvist and the author [11]. Reader!! are also recommended to read G. N. Watson's article. The marqui6 and the Iand-lIgent; a tale ollhe eightunth century [21 7]. (You know the

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SPIRlT OF RAMANUJAN

131

identity of the 1and-agent; to augment your curiosity, '"' relTain from ~n, you the identity of the marqui$.) Ext!ciaoe 5.1.9 is just one of many beautiful transformation for_ !!Iul.. for elliptic intep-a1s, many o f which are due to Jaoobi and/ or Ramanujan [34, pp. 104-113J. We offer a few of thete transformation formulas as exen::i8etI. Exercise 5.5.1. I/O

.,'

< o,p
~ ~= (£)B1

Exerel.se 5.5 .2.110

11

< o ,fj

11

~how

~

in the theory

.. +J.' Vi ..

v l - uin'41

v'T"="i lan p,

t -0/'"'-_:'::.'.'-''''''

T he next exen::ise is a form of the addition of tlliptk funCl.iona.

..,[

..

r sin'41

_ ~'F,(". b) 2 2 ' 2 '"

The follOlVing exercise gives the elasllical duplkalion /ormula for elliptic integrala, Exercise 5.5.3.I/O < 0, {J< t". ondc:ototan{{J/2) ..

Vl

r!in'a,

proll!! that

J.

2 11

0

dO

Vi :uin'l~ =

EJeerclse 5.5,4 , I/ O < a ,fj < (I + 2O).i no , ,.-ow Mat

J.' Vi .. OIl

!.-

2Otinitjl '

and ( I

+

2Osin 2 o )3inP

The t ransformation abow; la called GaU$ll's trlllUlfo rmation alld is similar in foml to Landen', t ransformAtion given in Exercise :>. 1.9. Historically, the t heory of elliptic functions 1lf0lM! from t he prolJ.. Itm of in''erting i nro~~~, such as the incomplete

B. C. BE RN DT elliptic integral of th'" first kind given in Definition ~.1.8. These inversion problems "''ere motivated by the well_known inver!lion of the trigonometric integral

. J.'

# O< x <1. ~' o Vl -~ Landen's transformation for incomplete elliptic integrals is only one example of many beautiful relations thM exist between elliptic integrals, many of which are due to J acobi /13 1] and Ramanujan /193, Chapter 17, Section 8J, {34 , pp. 104- 113]. The text [180. Chapters 24] by V. Pra.solov and Y. Solovyev is a ~uperb introduction to elliptic integrals describing the fundamental theorems of Legendre, Jacobi . and N. H. Abet arCSmx=

Lemma ~.1.I0 is a primitive, special case of a much more general asymptotic formula for Gaussian hypergeometric ~ries; for example, see (33 , pp. 77-78]. T he catalogue of formulas for theta functiOnll given in Section 5.4 is taken from Sectionll 10- 12 of Chapter 17 in Ramanujan's se<:ond notebook (193], (33, pp. 122- 124]. The fonnul,.... for Q (q) and R (q) in Theorems ~ .4 . 11 and ~.4.12 can be found in Section 13 of Chapter 17 in Ramanujan's second notebook [193], (34 , pp. 126-1271. Although the formulas for P(q) in T heorem ~. 4,9 are not found in Section 13, from the appearances of other for mulas for P(q) in that section, it is clear that Ramanujan knew Theorem ~.4,9. Ramanujan u.sed his fonnulas for P(q) , Q(q), and R(q) to derive a multitude of elegant evaluations of infinit'" ~ries in Sections 13-17 of Ch"pter 17 [34, pp. 126-139]. For example,

'od

~

(- qt (2k+l )

1 2

~ cosh{ I(2k + l )y} = 2'%

Jx( 1 - :1:).

See al80 Corollary 6.2.10.

Copyrighted Material

Chapter 6

Applications of the Primary Theorem of Chapter 5

6. 1. lntroduct ion Our goal in this chapter is to provide some applications of Theorem 5,2.8 and the several representations of theta functions and Eisenstein series in term.9 of z and z that !U"OOIe from Thoorem 5.2.8. We first demolllltrate in Section 6.2 how the formulas we deriYed for Eiaell!ltein series in Chapter 5 can be applied to giW! simple proofs of the formulas for r.(n) and ra(n) that we pl"OVf!d in Chapter 3. We also deri~ an elegant formula for 18(n ). In Section 6.3 we define one of the most important ooneepts in the theory of elliptic functions. namely, a modular equation . Lan. den's tra.nsformalion, Theorem 5. 1.6, caD be thought of as a modu· Ia.r equation of degree 2. However , the theory of modular equations more properly began in 1825 with !.egendre's discovery of a modular equation of degree 3. The construction of modular equatiol1ll is not easy. There is no single method that onc can use to produce modular equations. In Section 6.3, we shall derive some modular equations of degree 3.

Copyrighted Material

133

D. C. BERNDT

134

6.2. Sums or Squares and Trian g ular N umbers We lint offer aomeooroUarie!I ofTbeor~m S.4,9, Reeall thAt 0 < :r: < I and q and .I; .n defined by (5.2.30) IlIId (5. 2.29), respeet.iwly. W~

Corollary 6 .2 .1.

haw

(6.2. 1)

Proof. Applying

Thoo~m

5.4.9, ...~ lmmediately

ded~

that

2P(q') _ P(q) .. ,2(1 + :r:).

(6.2.2) On the other hand, (6.2.3)

2P(.r)- P(q) _ 1+ 24

nq~) ~ ",. L~ (2"q'" -~ + - - _ 1+24 L -1 -' l - q I - q" .. _1 +q"

.. _ 1

D

Combininc (6.2.2) and (6.2.3). we deduce (6.2.1).

C orollary 6 .2.2 . We MW

- ,.

2P(q4) _ p(q2) _ 1+24L: ~ _ z2

(1)

.. _II+Q"

I --z

2

.

Proof. Ufting the pTOCeS!l of duplicat ion alons wi t h Corollary 6.2.1, we find th.M

1+ 24 ".f 1 I::" q _41" (1+~' (1 + (11+';1-% - ,tr=r)')

2P(qt ) - p (q2 ).,

_ Z2(1_~Z). D C orollary 6 .2.3. We /law

1+8 f (~,) .. nq~ _:'( I _ z ). +," COpyrighted Material

SPIRIT OF RAMANUJ AN

135

P roof. Using Cor(>llaries 6.2.2 and 6.2. 1, we find that

o

from which Corollary 6.2.3 is immediate. Theorem 6. 2.4. FOT tach po&itive (6.2.4.)

inl~er

r4(n ) = 8

n,

L d.

.,",.

First Proof o f Theore m 6 .2 .4 . From T heon'm 5. 4. I(ii) &rid Corollary 6.2.3,

thst is, with q replaced by - q, ~

","

<;,4(q) = 1 + 8L 1 + ( )", .. _ I q which s rises fr(>m (3.3.12) and (3.3. 13) in our second proof of T heorem 3.3. 1, i.e.. Theorem 6.2.4.. TbWl, the remainder of the proof of Theorem 6.2.4 is identical to that of the aforementioned proof. 0 We now gi''il a second proof of Theorem 6.2.4, which is & variation of our first proof aixn-e. Second Proof of T heorem 6 .2.4. Recall th8t o(n) = Ldln d. We extend t he definition of dn) by setting o (n) ., 0, if n is IIOt an integer. With this in ~Ya~m 5.4.1, Theorem 5. 4.9,

8.C. BERNDT

136 and Exercise :;.4.13, we find tha.t

L- r.(n)q"

• = 'P'(q) = :2 = 'JP(q4) -

3"1 P(q)

"~

=j (I-24f:O{n/4)qn) - 5(1 -24f:O(njqn)

-

=1+ L

".,

.... I

.. _ I

(80(n) - 320 (n/4» q",

Equating coefficients of qft, n 2: 1, on the extremal sides above, we find that

rt(n ) = 8o(n) - 320{,,/4)

..

=8 L

d,

'1"

o

which completes our second proof of T heorem 6.2.4.

To ~eri\'e Jaoobi's formula for T8(n), we need the follO"o';ug consequence of Theorem 5. 4.11.

Proof. By a direct calculation. 16O(q2) - Q(q) = Hi + 16·240

= 15

""32"

'"'3"

L

L

n q". - 240 ~ .. _ 1 1 - q .._ 1 1 - q"

"q . (1+16 L-( - ,1)".''') q ft _ I

On the other hand, U1Iing Theorem 5.4. 11, "'~ find that 16O(q2) - Q(q) = 16:'(1 - x

The desired resul t

flOW

z' (l

+ 14x + xl) = IS: ' {l - :lY

followa from the last

Theorem 6.2.6. For MCh (6.2.6)

+ ,.,2) -

po~itiw inJ~r

tW(l

",

r8 (n) = 16(- qn ~) _ I )dttl.

""

Copyrighted Material

identitia.

0

SPIRIT OF R AMANUJ AN Proof. From Theorem ::,.4.I (ii) and Corollary 6.2.S, 'P'(_q)_:4( 1 _'I')2 _ 1 + 16

~ _ (=!, I )c·;n'~": :L 1

~ -I

q"

Thus, replacing q by - q, "''e deduce thAt

.. hieh il precisely (3.5.1 4) in our previoUll proof of Theorem 3.5.4, or Theorem 6.2.6. The remainder of the proof is then the Mme !Ill

0

~~.

Exercise 6 .2 .7. GIVe IInother proof I)f Theorem 6.2.6 In the frXWw'119 rnonner. F.,-,t, """9 furmu14. for Q(q), Q(q2 ), IIlld Q(q' ) from 1"Mon!m S.4.1I OM &err:ue 5.4. 13, r"f:,pa;ti~", ,,-ewe tJw.t ~

L ... (n )q" !Mtr"f:

C4UU

~

:'

= 1 + L: (16o'3(n) -

32cJ,(n / 2) + 256u, (n / 4» q~,

."'

uJ{m) - 0, .f m ~ not on integer. S«01Id, ~ ro.".dering the n wtn and " odd uparutely, 1Iho", that

16u3(n) - 32o'3{n/2) + 256u,(n/4) - 16( - I)~ We IIOW tum our

IIttenlion to

L:) - 1)~fil. 'I·

turns of triangular numben! . First,

we gi\'e A proof ofThoorem 3.6.3 along the lines of the previous proofs in thiIJ !ledion. ~ following simple exerciae will be \lied in our proof "Iow. Exercise 6 .2 .8 . I/m

~

on even po.otl.w: ;nleger, the!

arm) - 3u(m/ 2) + 2<7(m/4) ., O. T heorem 6 .2.9 . For eo"h nonnegatiw: mltger n ,

t.(u)

=

u{2n+ I).

8.C. 8ERNDT

138

Then, using (6.2.1) in oon,junction with Corollarie!i 6.2.1 and 6.2.2 and Exercise 6.2.8, _ deduce that ""

1

1

I

L I4(n)q2nH ... - U-P(q) + SP(q2) - T2P(q') .~

..L-,

=L

_.,(q(m) - 3u(m/2) + 2n(m / 4))q'"

=

u(2n + \ )q7n+l .

Equating the ooefficient.$ of oomplete t he proof.

q2n+ 1

on the extrem&.l sides aoo.-e , we 0

In the notes to Chapter 3, wt! mentioned a fonnula for Is(n), wbicl\ we now prove. We first need another corolla.ry. C orollary 6.2.10. We ham: 00

kJ

1

L sinh( ky) == 8%':1:· ,., P roof. Using the definition (5.2. 30) and Theorem 5.4.11, we find that (6.2.8)

00!;:3 si nh{ky)

L

k_ 1

00

,,3

= 2 kL_ L ek~ e-'~

k)e- ~~

""

== 2

L

' _ 1

1

e-u~

o Corollary 6.2. 11. \Ye haue ""

k 3 t/'

q~S(q)= L ~' k_ 1

-

q

Copyrighted Material

SPIRI T OF RAMANUJAN

139

Proof. Ftom Theol'em .M .2{i), 1 1 4 %. .wo'{q) _ _

16

Employin& Corol.lary 6.2.10 and (6.2.8), we complde the proof.

0

Theore m 6 .2.12. F'q r eGch pc)lIflt>e mtqrer n ,

- - -

Proof. By Corollary 6.2.11,

~'(q) -

2>V L qn... = L: L ~.I

-

... 1

tPq".

"I"

" I d odd

Equatill8 coefficients of qn+l on both sides above, ...,., complete the proof. 0

Recall that in Theorem 4.2.4 we establ ished the following fundamenu.l theorem.

T heorem 6 .2. 13. lYe have (6.2.9)

\0

Our representations from Theor=15 5.4.11 and 5. 4. 12 enable us give a very simple proof of Theorem 6.2. 13.

proor. It will be simpler from

~1nI

to UIe the argument ql insttad of q. Thus, 5.4.11 and 5.4.12. ,,"l! find that

Ql(q1) _ n2 (q2) .. ZI2(1_ %+ %1 )J _ Z12(1 + %)2( 1 _ !%)2(1 _ 2%)' _ -13 ' : 12%2( 1 _ %)2, (6.2.10) after a doNge of elementary algebra. On the other hand. by Theorem 5.4.3(iii), lns.? (q2 ;q2)!! .. 1728q2{"(_q') '" Z& . 3J q 2 z I22- 'z2(1 _ z )llt? (6.2.11 ) '" *3' : 11:.: 2 (1 _ %)'. Combining (6.2.10) ltDd (6.2.11 ) and replacing q2 by q. tbe proof.

Copyriglted Matena!

""l!

complete

0

B. C. BERNDT

140

6.3. Modular Equations Definitio n 6.3.1. Ld K , K' , L, and L' denQte oomplde elliptic integrat.. (If the jiNt kmd auodaled with the moduli k , k' := v'f="P, I, and ~, respectively, where 0 < k,t < L Suppou that K' l/ (6.3.1 ) n- = -

r:=

K

L

for lome poritive mteger n. A ",,'alion l>etw«n k and I induced by (6.3. 1) if called a modular equanon 0/ degru n. Following Ramllllujan , set 01 _ /,;'

~d

We o!'ten !lay that {J ha.s degree n o_-er a. for

UBing Lemma [>.1.3, we may replace the defining relation (6.3.1) modular eqUlltion by the equ ivalent relation

11

(6.3.2)

~F,(!,!; I ; 1

n

aj , F,(!,!;l;Q) ,..

,F,(!,!; 1; 1 -

P) ,Fl(!.!;I;,Bl .

Using (52.29) and the formulas from Section 5.4, we see !.hat 11 modular equation can be considered as IUl identity amongst theta functions with arguments q and theta functions with arguments qn. In fact, most often One e/jt ablishes a modular equation of degree n by 6r8t proving the requisite theta function identity. Then we use the formulas from Section ;'.4 to express theta funct.ioll.!i with argument q in terms of a , % = ~1, and ()XlSI:Iibly) q, and the theta functions wit h argument q~ in term. of (J, z.. , and (possibly) q", where (6.3.3) The multiplier m of degree n is defined by

(6.3.4)

m

=

'f','(('))

VI q"

= .:!.. z..

We note t hst the method of establishing modular equations briefly dt)8(:ribe
SPIRIT OF RAMANUJAN

141

d~'f:r

or OOll.'!truct modular equ(ltioll.'!. One needs t o be relIOurceful and use a variety of tools_ Generally, all the degree of tbe modul ar equation increases, the difficulty of e!tablishing modular
tJ

P roof. If """ mske the indicated robfltitutions in the definition (6.3.4) of the mult iplier, we find that [6,3.5)

n

2FI( . ; 1; 1 -(3)

;;; =~ FI (! ' ; 1;1 - 0-)"

Rearranging (6.3.5) and using (6.3 .4 ), we ded uce that

lFI( , 1; 1; 1 _ 8) ~FI( !'

:1 ;11) ,

i.e. , """ obtain the defining relation (6,3.2) for a modular equat ion.

0

We now discuss modular equations of degree 3. We need several identities for Lambert series.

Theorem 6.3.3. If {j} denote" Me ~re ! ymbol, th( n q""H I

)

ql2n+IO '

(ii ) (iii )

(iv) Part (i) ill identical to Theorem 3.7.7, and part (iil Can be found in [3.7.8).

Copyrighted Material

B.C. BERNDT proor o r ( Ill). RePJac;1II q2 by q in (1.3.60) and employilll the J. cob; triple product identity, equation (1.3. 11). we find that (6.3.6) ~

".,'(-q)/(-q) _

= d:d

L

(6n + I )q(h'+oo)n

{z/(q',8,q/ ,8J}1'. 1 d

- / (q, q2 ) d, (IOS {tl(q1:8 .q/:8)})1'.1 - / (q,q1)

(I + : : (loc( _ q/ z8; q' loo( _q2t';~)00{q3; t¥')00)1._1 ) '

Ullilll the Maclauri n teries for log(l + ,). inverting the order of SUDr m.tion, and then summing the resultin& geomeuic &eries. we find that, for 101 < t , ~

(6.3.7)

IOS(-o; q3)"" _

L log(1 + aq3.. ) .~

00 "" (- I)"'(atf")'" ~ - LL m .. _ 0 ... _ 1

00

(_ Cl )'"

-- _ L.,m (1

"-r

U8;1II (6.3.7l in (6.3.6), we find that

'P'{-q)/(-q) _ I (q,q' )

(1 - f: ~

(- q:-I)"

d: .. _ I n(l

q3" )

(6.3.8) where we eJi:pllnded the summand.!! in goometric ..rie- and then iD''ert«!. the order of tummation. Now , by Theorem 1.3.9, (1.3. 15), and COpyr;ghted Malenal

SPIRIT OF RAMANUJ AN

143

(3.2.7), ~(q)/(q)

I(

-r(q)' -q; -q)"" (q; q3)",,( q' ; q-l )",,(-q3; q3)-

q,q~)

_~'(q )(_q,_ q)"" (q, q)"" I (q; q)"" (q; q3)",,{ q'; q3)"" ( q3; q3)""

' I ) {q~; q3)"" -~ q ( q3; q')"" _

(6.3.9) ~lacing

q by - q in (6.3.8) and then using (6.3.9),

~ complete the

_~~ ).

0

proor or (Iv ). The proof is similar to the previous proof. U~iog (1.3.61 ), the J aeobi triple product identity in equatioo (1.3.11 ), and (6.3. 7), ..-e find that

(6.3.10) w(ql)/'( _ q) _

f

(3n + l)q3"'+2"

d - dz {:/(q6:1,q/ z'

.. I {q, rt)~

lll"'1

(10& {z/(qSt ' , qlzl)}) 1,,,1

.. I (q, rt) : z (log {z( _q~z'; q6)oo( _qlz';q')",,(qf.; ql )",, })I ' ~I - / (q, q$) (

-{

Oh"

... . , } )

1 - 3~ l +q$"+l- I +q8"H

.

Using (1.3.11), (1.3.14). and (1.3.1&), _ find that

1/,(q')/' (q) (q< ;q< )""( - q; q' );,(q'; q' );, I ( q, ~) "" (q2;q~ )""(q;q8)""(q~;q8)",,{q8;q8),,,, _ (q.;q4)""lq'; q4);,(q' ; q' );,{r ;"e)""

copY"
".

B. C. BERNOT

ReplacinS q by - q in (6.3. 10) and then usin& the c:aJculatioo alxM:, we finish the proof. 0

In the theory of modular equatiollll of degree 3, it is ad vamageow. to introd uce the parameter p defined by (6.3. 11 )

m = I +2p.

T heorem 6.3.4 (ModuJIU" Equations of ~ree 3). u t {J mUle 3 /l1JO!r 0 , tInd k l m dmote tAt; mulhpl~r 01 dl!g'l'te 3. Thrn

d~

(i)

( ~ y/' - (1\ ~1J) 1/' .. I .. el]~~l) 1/. _(~ )'/' ,

(ii)

(0,8)1/4 + ({I - 0 )( 1 _

(Hi)

(iv )

__')'" (P')'" , -3= 1+2 (1_1-_0) /1 ,{(O')'" ) (0')'" 7i If -p,

m _ I+2 -

o

m

P)P,4.. 1,

m

- 0

""

.,

SPIRJT OF RAMANUJAN

Proof of ( I). Uain« (Hi) of Theorem 6.3.3 t ... ice, ooce with q replaced by - q1, ,,-e find that "'(q)

"'(-0')

- (.....

'Ptr) +op(_tf)- 2 + 6 L

"-,

I

.....)

.r+ I + l +qeo.+2

(6.3.12) by T heorem 6.3.3(iv). Employing Tbeo~ms 5.4. 1{i), (iii) and 5.4.2(i), we tranllCl'ibe (6.3. 12) into the form

:;3/2 :L. z~/'

+

:'/'{1 _ a )' /S

I

: !'2(1

/ q)3/' - 2' 2-1/2z312(a --::.c~,,,,,,,L, 2-1/2z!I2(IJ/qjl,t'

{J)I/e -

which upon llimpli fication yields the first equality of (I). The eeoond equality of (I) is the reeiprocal of the first. Proof of (ill. Prom Theorem 6.3 .3(ii) , ,

J

+( ..~, (q' (")

op(q)op(q ) - op( - q)"'( - ., ) "" 2 ~ "3

..,

1

- ( qGn+1 "" 4 2: I q12n +'- ,

(6.3.13)

(-q)')

q)" - I + qn

0

8 . C. BERNDT

146

by Theorem 6.3.3(1). Com'Wti~ (6.l. 13) into .. modular equation by applicalions of Theorems 5.4.I (i), (ii ) and ::'.4.2(iii), we dedoce that

Jiiii - Jiiii{(l -

0 )(1 - p)}'/4 _

Jiiii q(o / q)I/4(ftl tl )I/\

from whicll (ii) follows.

0

Proof of (Ill). Using Theorem 6.3.3( iii), ~(q) .r(-q')
~

find that

(_ l )"q3nH)

(_I}"q'" .. l

00 (

= 3+6~

wt!

1+( q)300+1 + 1 + (_9)3 n +2

(qG"'U

qSo.H)

+12 L -'+q8""'+'+q8nH ""'

" (I+,f:G) I +:~q)") "-, ·3
(6.3.14)

by Theorem 6.3.3(ii). Using 'Theorem 5.4.1(1), (Hi),

"'e

may rewrite

(6.3.14 ) in the form ,~n

:.m

,1n(l _ 0)3/S

+:2 ~/1

3,,1ZiZi.

-

:3 '3 (I-{J)'/' Simplifying, we obtain the second eq uality of (iii). The first equality of (iiI) i! the r«:iprocal of the SO!(:Ond. 0 Before embarking on the proofs of the remaining modular equaprovide IIOItle useful parameleTiZlltiolUl that greatly facilitions, tate the derlvtltion of modula:r equations of degyft 3. From Theorem 6.3.4(iii),

"''ll

(6.3. 15)

Thus, from T heorem 6.3.4(i) aod (6.3.1:)), (6,3.16)

(

I-IW )1/8=1+(",)1/1 . ",+1 . 1- 0

0

Next, from Theorem 6,3.4(iii), (6.3 .11)

(1- ():)3) 1/' • ~

, ~.l:L? .. . opY1lfjm d Mater",,1

2

SPIRIT OF RAMANUJAN H~Da!,

147

from Th
(6.3.18)

",)1 / _ I + (I - 0)3) 1/' _ 3+ m . I -IJ - 2m ( IJ

Taking the product of the cube of (6.3.18) and (6.3. 1~),

(6.3.1 9)

.-

(m- l )(3+m)3

"=

(m- l )'(3+ m)

1ft!

find that

16m 3 ' while taking the product of the cube of (6.3.1:;) and (6.3. 18), "''e find

,ha,

(6.3.20)

16m

p

'

Next multiply the cube of (6.3.17) by (6.3. 16) to deduce! that (6.J.21)

1 - 0= (m+l ) (3-m)' 16m 3

'

aDd lutly multiply the cube of (6.3.16) by (6.3. 17) to fiDd that (6.3.22)

1_P

=

(m

+

1.'::: - m ).

pl'OQr of ( Iv), By (6.3. 19), (6.3.20), and (6.3. 18),

o

&.:!m which (Iv) is immediate.

P l'OQf of (v ). Substituti~ (6.3.11) into (6.3. 19), (6.3.20), {6.3.21}, and (6.3.22), _ readily deduce t he four formulas In (v). 0 Pl'OQf or (vi). Uli", fint (6.3.19 ) and (6.3.20) and 8O!(X)ndly (6.3.21) and (6.3.22), 1ft! find, re!lpecth'ely, that

(6.3.23)

"'" (6.3.24 )

~)1 /3

(o

_m{m - I)

I_P) I12 _ ( 1 -0

3+m m(m+ 1). 3 m

If we lubstitute (6.3.23 ) and (6.3.24 ) in the right lide of the first equality of (vi), 1ft! easily Vl'Tify itS troth. The $
B. C. BERNDT

148 P roof of (vII). From (6.3.19) and (6.3.20),

(o pS)I/A

=

~(m ~~I~)'~(3~+~m~)

'm

&nd from (6.3.21) and (6.3.22),

{(I - 0 )( 1 -

p)~P/Il = (m. + 1~~3 - m ).

HMce. (a p$) I'1 + {(I- 00 )( 1 _

p)~ } l /B _ 3 + m' 'm _ 1 _ (3 - m)(m - l )

'm

__ ( IPt!

(6.3.25)

1

by (6.3. 19)-(6.3.22). This

pro\'l'!S

0 (1

0'JJ))3)'"

t he first equality in (vii).

By the principle of reciprocation, the fI(!OOtl(! equality folloQ from the tint. Lastly. from (6.3. 19)-(6.3.22),

(6.J.W)

1 + {a p» )/ 2 + {( I - a ){ l - {J) IL /'

(m

_ 1+

1)'(3 + .... )'

16m'

+

(m + I l , 3 _ m)' (m' + 3)' ' 16m' 8m' .

If we multiply both sides of (6.3.26) by ~, take the ik[\UI.fe root of both aidm, and appeal 10 tho! first equality in (6.3.25), _ compiew the proof of the third equality of ( vii). 0 The next t heorem giVe! t.;o further modular equ.eot iona of ~ 3 that ~ prol.Otyl)('!l of moduw equatiollll of further or~n. Theo~n'

6.3.5. 1/

P :_ {160P(l _ 0)(1 _ /J)}I /S and Q ._

(6.3.27)

P(i - Pl) '" ( --) ( -01 - 0

SPIRIT OF RAM A N U JA N

149

If

P :. (0 ,8)1/1

Q - b = 2(P - ~) .

(6.3.28)

P roof. From (6.3.19)-(6.3.2'2) , we easily find that

p2 _ (m 2

1)(9 _ ml )

-

'm'

Thus,

m2

PQ .

9 _ m2

P

_1

and Q - 2m3Jf

2../2

E1iminalinl m from thilJ Last pair cl equations

2../2 -

P

yi~lds

PQ

Q = 2PQ ../2+ " Reart&lllinl thilllul equality, we r~&dily dedOOl! (0.3.21). The proof of (6.3.28) ill similar, From (6.3. 19) &lid (6.3.2Q),

~ _ ( m - I ){3+m)

4m [t

and

Q2 ,. m(m - I) . 3+m

follow. that

,

PQ _ m - I

P

3+m

Q - 2,;"" '

Eliminating m from thilllast pair of equations, ~ find that

P

Q

2+PQ 2PQ+I '

which upon r~arrang~ment yi.elds (6.3.28).

o

With the use of (6.3.11) in oonjunction with (6.3.16) and (6.3.11), it follow. that p > 0 and p < 1, respectively, or equivalently t hat 1 < m < 3. Prom t he formulas fo r 0 IUld (j in Theorem 6.3.4{v), we readily find that, for 0 '$ p '$ 1,

do 2(1 _ pf(2 + p)2 -., >0

8.C.BERNDT

150

dP dp =

Spl( t + p)2 0 ( I + 2p)2 2: .

There ill consequently a one-t~one oorrespondence between a and p and also between f3 and p when a ::; p::; I.

T heore m 6.3.6. Let P be

d~fintJ.

bV (6.3.11 ). Then

(6.3.29)

IF,

O,~; i :p c:~r) = (I + 2phF, G ,~; l:pJ U::p))'

Proof. Recall the definition (6.3.4 ) of the multiplier m and use The. orem 5.2.8 and (6.3.3) to write this defini tion in the form

2F,(!.

i; 1; a ) '" m 2F,(!, t: 1: /3).

Now use the repNl5enta.tions for rn. 0 , Md (3 given in (6.3.11) and Theorem 6.3. 4(v), and (6.3.29) follows immediately. 0

6.4. Notes We are grateful to K. S. Williams for providing the second proof of Theorem 6.2.4, the proof of Theorem 6.2.9, and Exerdse 6.:!.1. The t heory of modular equations begilUl with Legendre's 1139, voL I, p. 229] modular equation of degree 3 in 1825. namely, (6.4.1)

(a O)1/4

+ {(l- o){l- P)}'/4 = I.

In t ile century that followed, several mathematicians. including A. Berry, A, Cayley, A. Enneper, E . Fiedler, R. Flicke, C. Guetzlaff, M. Hanna, C. G. J . Jaoobi, F. Klein, R, Russell, L, Schliifli, H. Schroter. L. A. Sohn<;ke, G. N, Watson, Md H. Weber, contributed to the growing list of modular equations. 1I0we,·er. the mathematician who diS<;QV\'red far more modular equations than any of these mathematicians was Ramallujan. who constructed over 200 moduIa:r equatiOll!!. As indicated immediately a bove. the form of the modulllJ' equat ion given in Theorem 6.3.4(ii) is d ue to Legendre (139. vol. I, p. 229]. and can also be foundd3pWtJh~'i;'t~'~iBl text on elliptic functiom

SPI RJT OF RAMA NUJAN

151

(67 , p. 196] and in Jacobi's epic work (13 1, p. 68). This type of

modul .... equation hM been established for lIe\-eral other de&r-. 10 particul.... , formulas due to SchrOter in his dis8ertation ]200] are useful in eIItablishing lucb formulas; _ also ]34, pp. 66-7'2]. Th_ modular equations are abo called ~of Russell-type,~ after the English mat hematician R. RWl8CIl. who deriwd !IeV1lral modullU' equations of Ihi9 80ft (202). (203). For example, the modullU' equations of d~ [; and 7 of th it type are gi,-en by, respecti vely (34, pp. 280, 314J,

(O/,W I2 + (( I - 0)(1 -

(otW/S

tiW l2 + 2{loo,B(1 -

+ {(I

- 0 )( 1 -

0-)(1 - P)}I/S

.8W" -

=

1

I.

The parameterizations for 0 and.8 given in (6.3.19) and (6.3. 20), m1pectively, ~re first dilJco\-ered by Legendre [139, vol. I , p. 223) and mJilJco\-ered by J acobi (131, p. ~J. As ~ hao,-e eeen. Ihe parameterizatkms of <) and .8 in tenns of the multiplier m are ex tremely useful in deriving mooular eq uations of d egree 3. Similar paramelerizations exist for <) and ,B in the theory of modular equationll of degree [; [34, pp. 280-288), bo t for higbel' degrees "-e do not know of any further parameteriZlltiol1ll, which refloct.s t he fact that wi th illCreasing n, the SMpe8 of modular equatiolUl become increlLllingiy more complex, and finding them becomes increasingly more difficult. The paper [IOlt! by M. Hanna providea a .ummary of much of the work accomplished on modullU' equations up unti l 1928. RamantUatI .ummarized much of his work on modullU' equatiorul in fTagmenu that "-ere published with his Jo.t notebook (194) on pagea M and 350-352. For IIoC(:OUnt.s of these (r"menU. _ Chapter 17 in the book by And rews and Berndt [19). In particular, the formulall for III in Theorem 6.3.4 (vi) are exam lllell of a large cllI8II of similar formu las for m. Although ",-e haV{! provided a 8ill1111e proof based Oil paratlleteriUl.tiolls for Q and p, in ge ner"', "''e do not kllOw fuun anujlln'. methods for deriving them. In our book [34), we used the theory of modular forms to W!rify .'lever'" of them ; RamatltUatI malt likely did not ~ Ma/anal

B. C . BERNDT

'"

The modular equation in (6.3.21) is of "Schliill.i-type" [204). and R.Aru&Ilujan recorded mlUlY modular equations of this sort in his notebooks and IOBt notebook [193), [194). See [19, Chap . 17). [34, Chap!!. 19-20), and 137, Chap. 25]. In this sbon monograph providing an introduct ion to Ramanu. jan's theory of theta functiollll, we do not have tile apace to give applicatioIl8 of modular equations. The proof:! of Ramanujan 'a congruence!l for p(n) modulo arbitrary pov.~ of 5,7, IlIld 11 depend upon modullll" equations [194[. [50), [218J. [28]. /IIany of the reBults in the following C hapter 7 crucially utilize modular equations of degree 5.

Copyrighted Material

Chapter 7

The Rogers- Ramanujan Continued Fraction

7.1. De finition and Histo rical Background A

continued fraction ill an expre8lion of t he Il0l1

(7.1.1 )

"'+ ;~:==== + _'='~===== _ "'0'_,,+ -'-'III

0.1

b,+

b~

+ ...

wllicb ill commonly written in the more eompact. form

(7.1.2)

The continued frao;tion (7.1.1) or (7.1. 2) may terminate, i.e., th'l froclion.!!

i:" do not continue indclinitely. For ex8.tl1ple,

ill a ~inatin, continued fraction . More interesting are lhc.e contin.ued fractionl that do not terminate, i.e., infinite continued fractiollll.

Copyrighted Material

153

B.C. BERNDT

154

Suppoose

~bat

we define the sequences p.. and Qn. n

p.. - b.. P.. _1 + a.. P.. _2 , Qn

= b.. Q.. _ l + a .. Q.. _~, Q_, = O,

P_, = l ,

~

- 1, by

n ;:: 0, n;:: 0,

Po - boo Qo = 1.

Exercise 7. 1. 1. Prove that (7.1.3) T hen if tirn p.. ,,- co Qn

exists, we say that the continued fraction (7.1.2) converges; otherwise it di,"'rges . Cleacly, t he first task in developing a theory of continued fractioIl.'l is to derive criteria for collvergence and diw:rgence. In par_ ticular, when the numerators a .. and denominators b.. are functioDII of a complex variable, an extensi~ t heory has been dc...",loped, and it continues to evolve, It is not the purpose of this monograph \.0 develop such a theory, and so we refer readers to the excellent te.>:t by L. Lorentzen and H. Waadeland [151. Chapter I] for many criteria for convergence and divergence. In particular, see [151, p. 35, Thoorem

31· M""t mathematics students first encounter continued fractions in a course in elementary number thOOTY. T he first infinite continued fractinns that students may be a.sked to e\'lIluate are those in the following exercise. EX(lr<:ise 7. 1.2. Prove that

(7.1 .4)

'"' (7.1.5)

I

1+

1+ 1+ 1+ I

J"S+ 1

.

1- 1+1 - 1+·

~ -2-

..rs - 1

~-2- ·

Theoe are, in fact, special cases of poerhape the mOllt interesting continued fraction in mathematics. the Rogers-Ramanujan continued fraction. which first ap~ared in a paper by L. J. Rngers 1198] in 1894. Copynghted Material

'55

SPIRlT OF RAMANU JAN

Definition 7. 1.3 . The Roger, - Rarr"-Hlujan continued jroclion R (q) iI defi ned by L /~

(7.1.6)

R (q ):=

L

~

~-lI

q

...

1 +1+ 1+1 + '

provided that it converge.. FUrthermore. , et

(7.1.7)

'''' (7.1.8)

T (q)

Re&delll will immediately &'!k, UWhy does ql/~ appear in the defi. nition of R(q)?~ The Te880n is that R(q) belongll to the ,,; orld of theta funetiolllL, and R (q)'s modular properties are more symmetric and el. egant with q l/~ appearing in its delinition than if q l/5 were absent. However, there are occ&'!iolllL when the factor ql /5 is not helpful, and so we then use one of the representatiolllL (7.1.7) or (7.1.8).

7,2. T he Convergence, Dive rgence, a nd Values of R(q) In the definition abovt! , we have not indicated where R(q) converges or where it diverges. By standard theorems [151, pp. 35, 941. R(q) converges in the complex q-plane for Iql < I , and it d iverges for Iql > I. What happens if Iql = I? Taking the reciprocals of (7.I A ) and (7.1.5), "'-e find, respecti \-ely, t hat (7.2.1)

R( I ) '" ../52-

1

and

- R( - I )

= ,fS2+

I,

where we ha\'e taken the root (- I )l/~ = - I in the definition of R(q). Furthermore, L Scltur [207, pp. 319-321], [208, pp. 117- 136] and Ramanujan in his notebooks (193 ) established t ile following theorem [38, p. 35]. Theore m 7.2. 1. RUGIi th" t T (q) u defin ed bV (7. 1.8). Let q ~ a primitive mth root of unity. If In U a multiple of fo, T (q ) diverye$. Oth erwi.fe, T (q ) converge. and ( ) 7.2.2

T {n} =QT (n \n( I -Qr)/~

r:;f'JPYrIfFlr~Maferral

'

IS' where

B.C. BERNDT 0

re.ndue

deflote, llIe ~re ,ymbol t7Wdulc 5.

(1'-)

ond p

lot

tM ko&l positiw

0/ m

ThUfJ, for. root of uni~y, we know wbeD R(q) con\'t'r~ and .. hen it divergea. What .bou~ other points on [ql _ 11 We do not mow the MlSWer in ~neral , but D. Bowman and J . MeLaughlin [6 1J have found an uncount&ble eet of measure 0 on the unit cirde (not indud· ing 5nth rootl of unity) where R (q ) diverge. . It Is conjectured that R(q) d ivergetl on the unit cirele except for thOlM! point.3 of convergence

described in Theorem 7.2.t. Aa mentioned above, the Roger.-Ramanujan continued fn.ction WNI first defined by Rogers, who pr(lVf!C\ .. rew of its properties. Howel/ff, IllOIt of the resut", t hat "'<e kTlO'to' about this continued fraction are due to Ramamijan. Se.'HllI. theorems about R{q) appear in his notebook. [19 31. but his lost notebook 11941 OOOtailU considerably mort! material on R (q); /lie!! [19, Chapters I--5J. In his finn. 11\'0 letters to G . U. Hardy [192. pp. xxvii, xxviiiJ. [S I , pp. 29, 57], Ramanujan OOIIImunicated 5eW!ral thoonlms on R(q). In particular, in his fim letter [19 2 , p. XXViii, [5 1, p. 29]. RamllIlujan sa"" the first OOIll'te. memary e\'IlluatiolU of R(q) , namely,

(7.2.3)

,

,

R(e - 2-) '"' /5 +.;5 _ .;5+ I

(7.'2.4)

(7.'2.5)

S (q) = - R(-q).

In his seoond letter to Hardy 119 2 , p. xxviii ). 15 t . p. 571, Ramanujan further &S8erted that

In both leue~, Ramanujlln clllimed thllt [192, p. xxvii), [5 1. pp. 29. 57). "It iI lIl ....ays po8Sjble 1l> find elt&ctly t he value of R(e--';;;)." . GOpyrighJed Matenal

SPIRIT OF RAMANU J AN

157

The mea.ning of this IlI.'jt statement W!L'! not clarified until 1996 when Berodt, H. H. Chlln, and L. - C. Zhllng [46] demonstrllte
5(e-./~) ""

(&.;'5 - 7 + /3.5(& _ 2.;'5))

L/~

where 5(q) is defined by (7. 2.&). For proo!8 of these c!aitruL. see (44 ),

[461, and [19, Chapter 2]. Those readers with some background in Blgebraic nwnber theory and who havtl been exceptionlllly observant will ha\~ obseml<1 that each orthe value!! for R(q) that we ha\-equoted is 11 unit in some lIIgebraic number field. This is true for all values of R(C·""), when n is a poiIIitive integer; see [46] for a proof. We record one further result on R(q) communicated from India by Ramanujan to IIardy. Let (> and P be poiIIitive numbers sllch that o{3 "" ".1. In his second letter to Hardy, Ramanujan [192, p. xxviii) , [St , p. &7] claimed that (7.2.6)

( 1 +2,j5: +

R(f!-~"))

(I

+2,j5: + R(f!- lIJ) ) ... 5 +2,j5:,

which was first proved in print by Wat.son (216). This result is also reo;.rded as Entry 39(i) in Cha.pWr 16 of Ramanujan's second notebook; see ]34, pp. 84- 85) for another proof and further reference!:!, IInd also see [19 , p. 9'\!;-~$1IPt.; iWgP3,fflOWS the ,-a.] ue of R(C''')

B. C. BERNDT

'"

for a certain Q , then (7.2.6) enables one to immediately calculate anotbeT value R(e -''') . ThWl , besides being a beaut iful formula, (7.2.6 ) enables one to obtain two values from one.

1.3. The Rogers- Ramanujan FUnctions The Rogers-Ramanujan oontinued fraction ill intimately connected with the famous Rogers- Ramallujan fUDCliollB G(q) and H (q), which are defined by (7.3.1)

Glq) ,~

f: I q"')

~ _o

q, q

,"d

~

H lq) ~

<»

qn(nH )

L -Iq,-.qn-) .

n_O

Bot h Rogers [198] and Ramanujan [193, Vol. lI, Chapter 16, &<;t. 1:>1, [34 , p. 301 proved that (7.3.2) which we n ow prO\-"e . In fact, we fi rst prove a fin ite form of (7.3.2), which can be found 88 Entry 16 in Chapter 16 of Ramanujan 'a seo:lnd notebook [193], [34, p . 31). and from which (7.3.2) follow$ as an immediate corollary.

Theorem 7.3.1. Far each fWflllegative integer n , let (7.3.3)

~ := p. .. (o , q):=

1(.. + 1)/2)

L

i_O

(7.3.4)

v:= v" Ia , q) :;

~ I.'

IO q (q)t (q).. _2Hl

(q) .. _ H

,

I~I (q).. ~ ~a~q- (-+l)

f;o

(q)k(q)..

lk

'

whtrt r::t:] denolu the greatest integer less /f"", or equal to::t:. Then, / orn ;:O: l , (7.3.5)

!:: = 1+ ~ 1

v

+

a.q2 1

aq"

+ . .. + - ,- .

Proof. For each nonnegalive integer r , define l(n ~ .+I )/21

( ) k k(.+k) q n ~ . t+ l a q . b O (q)k {q) n - ' ~ 2HI Copyrighted Material

F. := F.{a, q):=

L

SPlRlT OF RAMANUJAN

Obeerve

'"

tha~

(7.3.6)

H - I'.

Fo- Jl.

F~

(7.3.7)

_ I

F" _ I_ I +aq" .

We now develop !\ recurrence relation for Fr. When we combine the tWO sums in the first step below, we use the fact th!t.t 1/(q)_1 _ 0 by (1.3.35). To that end, U ~ -'+I)I11 ( )

L

Fr - Fr +l .,.

~ t (rH )

q .. _r_ t +I O q (q).(q) .. - r - 2HI

t -o

1( .. - r)/l] (q),,_._kClq-"(HIH)

-E

(qJA(q),,_r _U

.

(q)t (q) ..

L,

.

(q) .(Q) ...

~

{q)1_1(q) ,,_r

1(.. - r+I )/2I (q ).. _r _kakq-"(rH ) ( I _ q.. _r _l+ 1

- L,

_._u

I

(1 - qk)

1( .. - <+1 )/11 (q) .. _r _lllkqk(r +k)

=

-.-2.

)

q.. _._n+I - t/

(1 _ q,, - r -2HI)

U ~ -r+l)/l] (q) " r_lr.a.ql (r+.)

-

~HI

(( .. _ r _1)121 (q).. -r-i _ I 0,+1 qU+I)(r+ .i+l) -

:;

_ oqHI

(7.3.8)

(q ),(q ).. - r-2, - 1

L

. 1I' q-' «+2+ , ) . -(. +2)-,+1

J-G

(q),(q).. -(H2)- 2.i+ 1

1( .. _r_' )/2\ (q)

_ lIq,+lF.+2 .

Using (7.3.6), (7.3.8) repeatedly, and lutiy (7.3.7), we oonclude that

~ -= Fo_ FI+QqFl _ I+ ~ " FI FI/ F~ F,

- 1+

aq

aq

- 1+ -

1.E&i1YFWl'ilI~renaJ

1

aq'

-+ F2 / Fs

B.C. DERNDT

160

04 .. - 1

+ --+

aq" I

o Corollary 7 .3 .2 . F'Qr any

compl~

number Cl IInd

Iql <

I,

(7.3.9)

p roor.

Le~

n-

00

in (7.l.:».

o

The~tlnued fraction in (i .l.9) is called the Gtntrcl/utd R....Ral7la"u)O" COfItmuu fiuction ,

The renowned Rogen-Ramanujan identities, which we do prove in ~hia book, art: !liven by

DOt

(7.3. 10) I

Usill8 (7.3. 10), we immediately deduce tbe elegant repr(':fl('ntatKm for R{q) in the next theorem.

T heore m 7.3.3. lYe Mill! (7.3. 1I) P roof. Set (I .. I in Corollllry 7.3.2, take the reciprocal of both sides, use the dc6nitioIU (7.3.1), and lastly Ill!e (7.3,10). 0 The reti ul u. in this eection can be greatly generalized in that many more general OXIlltinuoo fTl,,: tiona can he repm.ellt.ed as quotients of two q.8erieII. ~. in particula:r, 119 , Chapter 6J and 134, pp. 30-31]. Copyrighted Matenal

SPllUT OF RAMA NU J AN

161

1.4. Ide ntities fo r R(q) Ramanujan d~red two beautiful identities connecting R (q) and I/ R(q), which "'f: prove in this IM.'C.tion. T he first i9 given below. Theorem 7 .4 . 1. Recall that T(q ) ". defined in (7.1.8) .

T (l }- q -

(7.4.1 )

L

T (r)

_

(q, q )""

(q~,~)oo

Comparing (7.4.1) with (2.3.13 ) in tbo! proof 01 Theo<'f:m 2.3.4, that tbe unidentilioo series Jt (q) and J, (q) IlI'f: equal to T (q) and - II T (q) , respecti'.'f:ly.

we

flee

We IlOW prove a more general t heorem from whicb Theorem 7.4.1 folloowl by spccializ.ation. It will be convenient to introduce tbe notation

{aL ,a" ... ,a... ;q)"" - tal ;q)""{a, ;q)"",,, (a.. ;q)"". Theorem 7 .4.2. For anll compltz numwa, (7.·U )

(a, o? , ql a. qlo? q;q)"" ($ 5) (a~q ; q5)""(a-lq~ ;r/)",, (a5q';q5)""(a- Sq3; q& )",, - q ;q.., (q;.f)""(q4 ;q5)",, - a (q':.f)..,(ql ;q5)..,

'7';11")",,(0. -Iq'; 11").., (q' ;q5 )".,(of' ;q' )""

2 (0. 5

-a

+

J

a

(0.5 q4;.r )",,(a-&q;.f)oo ) (q;q5)",,{t(':q'l.., .

Before proving Theorem 7.4.2 , we .bow that Tbeorem 7.4.1 fol·

krws Immediately from Theorem 7 .4.2.

by.r

Proof of Theore m 7 .4 . 1. If we replao:;e q in (7.4.2). then set 11 _ q, realiu that (I; q'5)"" _ O. and recall (7.3.11 ), we deduce (7.4.1) forth .... ith. 0 The dclcompOl'litioM in TheortlIlll 7.4.1 and 7.4.2 8I'f: called ~ d.."ed,olU. because in the former theorem, the aerit:s terlIlll 01 {q;q ).... 81'1!! aepa.rated out in powers of q according to their residue c ' modulo 5, and in tbe lau.e r t heorem, the terms are lleparated out in J)O\\'er! of 11 accordi~lt"""'-~ mooulo 5.

B . C. BERNDT

162

Proof of Theorem 7.4 .2. UBi ng the Ja.cobi triple product identity (1. 3. 11) twice, we find t hat 2

( 1 I /1 .) _ (a,q!a, q;q)""(a1 ,q!a ,q;Q')",, a,a ,'1 a, q a ,'1,'1 "" ('1 ;'1)""

~

f; (_lra

_1_

('1,'1)"" ' __ 00

r q (r' - r J/2

f

(-i)'a 2 ' q['>-')/ :

• _ _ 00

L ~

~ _1_ (_ It "'"'a r .H ' Q(r' - r +.,- .)/2 ('1,'1)"" ' ,'__ 00 ~

(7.4.3)

L

""

where, for

- 00

<

a ~c,,(q ), fI

<

00,

c,,(q)

,

We now determine <:"('1 ) according to tbe residue class of n modulo

First, consider the residue class O{moo S). Replace n by 5n &.Dd make the change of variables r = n - 2t and $ = 2n + t. Note that r+2 5 _ SOl. Then, simplifying and applying the Jaoobi triple product identity (1.3.11 ), we find that

~ .. (q) = _

L

~

1_

( _ lj"H q (n -l"'-I.. -21)+(2,,-t ,j' -

12n.H )n

('1 ;'1).., , __ ""

= (-

1 )"q(~n' _Jn)/2

('1;'1)""

=

(_ \ )"q{~"' -3")f2

('1;'1)00

L""

( _ I)'q(~"H)J2

, .. _<><>

f( - rjl,

_ '1 2 )

( -I )" q(~'" - 3,,)/2

(7.4.4)

(q; q3)",,(
Second, oonsider the n'!iidue class I (mooS). Set ~ = n - 2' + I and ~ = 2n + t, !)O th~rlptr/!;flmcfl4fdrJnaThen . upon sim plification

SPIRIT O F RAM A NUJA N and

~he U!IOI!

or the Jacobi

~ ..+I(q)

~riple

163

product idelltity (1.3. 1I). we _

that

L

= -I-

( _ 1)"+1 ... 1 (q;q)"" ' . _00 x q{("_21+ I)' - (,,-21+ 1)+(2n+l)'-p.. +o))I2

= (- 1)"+1 q(.ift'-,, )/1 (q;q)oo

=

(_ I )"+l q(~"'-")/2

(q;q)""

L

..,

' __ GO

(_ I)'q(&<'-3I:)/2 4

I(-q,-q)

( _ 1)''+l q (1)oo'-'')12

= (Q2;~)00{tf'; q~) ... · It sbould

now be clar how

to calcul&.te the three remainiq CNeS.

Exercise 7.4.3. Prol!e that (_ I ) "+ lq(~' +")/2

(7.4.6)

~+2(q) - (q2;~)oo(q3;~)oo'

(7. 4.7)

C31o+3(q)

(7.4.8)

ot...H(q) = O.

(- I )"q(&OO' +310 )/2

= (q;q5) ... (q4 ; ~ )",,'

Substitute (7.4.4)-(7.4.8) ill (7.4.3) Md U!IOI! ~he Jaoobi tri ple prod uct idellti ~y (1.3.1 1) four timefl to conclude that (0, 0 2,q/ o, fI / a2, q;q)"" _

1

f

(q: q~)",, {q4; If)"" ...

_ 00

(_ I )" 05"q(31o' - 300)l2

B. C. BERNDT

164

From Theorem 7.4.2, we ing R(q) or T(q).

deri~

the second major identity involv_

Theorem 7.4.4. We have ~

(7. 4.9)

~

q!O

T"'(q ) -llq - r->(q6)

(tf;q~)~

= (q15;q15)~'

P roof. T he proof ill almO$t identical to a portion of the proof of Theorem 2.3. 1, We therefore forego most of the detai\5. Write (7.4.1) in t he form (7.4.1 0)

. ~I ( q,q l....l. , q-, q ,9 ~ ,9 ""

{q2~:q26)""

_ T(

-

'1 __ _ q_'

q

q

T(q6j'

Let '" be any fifth root of unity, and replace q by (Jq in (7. 4. 10) to find that (7. 4.11 )

Now multiply all five equalities (7.4.11) together. On the left side of Our product , use precisely the same argmnellt that W8Il used in (2.3.18)- (2.3.21)!.O arrive at the right- hand side of (7.4 .9). for the

right-hllnd side of our product of the expressiol18 in (7. 4,] \), em ploy the same argument that was used in deducing (2.3.23) from (2.3.22). We t hen obtain the left-hand side of (7.4.9) to C(Implete t he proof. 0 Although the proofs are essentially the same, the use of Theorem 7.4.4 leads to a somewhat cleaner and more satisfying proof of Theorem 2.3.1 than t he proof employing the pentagonal number theorem, Jacobi 's identity, and the division of power series in Chapter 2. We C(Implete our work on t he Rogers--Ramanl\ian continued fraction in thili section by giving thili alternative proof of Theorem 2.3. 1. Theorem 1 .4 .5 . For C6ch (7.4.12)

nO'H1egahv~ H1teg~r

vl5n + 4} =: 0 'modS). Copyrigh ed Malerial

11.

SPIIUT OF RAMA NUJAN

'"

We 001" exlrllCt those t.erIlllj from both sidea of (7.4.1 4) that involve only the powers tf"H," 2: 0, to d eduoe that (~

""

2~)l

"' Pt:5n+ 4)qr...H_Sq,; q

(7.4.15)

~

"".

(q~ ; q~)!.

Dividi"g l>oth side!! of (7.4.15) by q' and replaciut! q& by q, we OOIDplete the proof. 0 T heorems 7.4.1 And 7.4 .4 were found by C . N. WatllOn [21&[, [21 6J in Ratuanujan'& notebooks and pnn"l!d by him [215] in order to e!ltabli3h (7.2.3) and (1.2.4 ). Oblerv!! tbat if we C3Il evaluate the ritlht side of either (1.4. 1) or (1.4.9) for A oertllin value of q, then we un 6nd t he Ytllue of R{q) by ,imply IIOlvi"t! a qu&dratic equation.

Exercise 7.4.6. The lhi/tttJ FilIon
(704. 16) E414b1i8h nllmbers:

/ .. th~

- 1.. _1+1.. _2,

n 2: 2.

Iollowing generating function for ~

L: /,,·~

". 0 lAJP

th~

I.

"Mil'eoal

1, 1< 1

.hifted Fibonaa:i

D. C. BERNDT

166

Next prove /hilI

C()flc/ude that h"H ;;; O(mod5). In JI(Ir'!icwar, f~ = 5, /e = 55, = 610, = 6765, Imd h. = 7W25. Lastly, proll
"e

h.

The previous exercise relating p(Sn

+ 4)

to Fibonacci numbers

was suggested by M. O. Hil'llChhom (125J. In [124J, he used the fact 51/~"+t and a variation of the argument we gave aoo..~ in our proof of Theorem 7. 4.5, t.o give a similar proof of Theorem 7.4.[>. See also (2.3.25). It is remarkable that /f>rIH and f'V> ..+ 'U obey the same CQngruencell as p(Sn + 4) and p(2:'m + 24), respectively. Ramanujan's

original conjocture (2.1 .5) in the case that.s = 53 is p(125n + 99) = o (mod 12[,). Unfortunately, in general, it is not true that fn~ ..+flIl is divisible by 125, and so the analogy f&iLs for 53,

7.5. Modular Equation s for R(q) Recall that a modular equation of degree n can be thought of Ill! a relation among theta functiol\$ with argument q and theta functions with argument q". We dose this chapter by offering some of Ramanujan's modular equatiol\$ for R(q). First, on page 326 in his seoond notebook [1931, [38, p. 12J, Ramanujan recorded a relation between R(q) and R (q~ ) in different notation , which Will! also proved hy Rage", [199, eq. (~.4 )]. Thoo rem 7 .5 .1.

ut .. := R(q)

andv: ""

R(q~ ).

Th en

Also on page 326, Ramanujan adroitly defines the parameter

(7.5.1) and states the rollowing two elegant and symmetric relations [38, p . 13] . Copyrighted Material

SP IRlT OF RAMANUJAN

167

Theorem 7 .5.2. With k d~fined. by (7.5. 1),

Jl&(Q) _ k(I - ,)2 I +k

anG

W(Q2 )_ k7(1+k) I- k

.

In h~ 10et notebook 1194), Ramanujan recorded te'-eTal exqu~ite identities for theta fuDC;tions in the arsument k 01 (7.S.I ); _ 119,

Sects. \.8, 1.9}. The next beautiful modular equation of degree 3 is found on page 321 in R&manujan 's socond IlOtebook 1193}, [38, p. 17), and 'NIIS a1so efltablillhed by Rogera [38. p. 392). T heorem 7 .5.3.

ut u :_ R(q) and 11 :_ R(q3 ). ell _ u 3)(1

+ 10113) _

TMn

3u~1I7 .

Lastly, we conclude with 11 modulu equation of degree 5 for R(q) thllt R&manujan oommunical.ed in his finlt letter to Hudy (107. p. xxvii), [5 1, p. 29] , and that is found on page 289 of Ram&.nujlUl 's teCOlld notebook [19 3], [3 8, pp. 19-20). Again . this modulu eqllll-

lion WM first esta.blisbed by Rose'" [199, p. 392J. FOI' reference! to fw1her proofs, !lee [38. p. 20] or [51 . p. 43]. T heor e m 7 .5.4 . ut u: " R(q) "'nd v:. R ("). Th~n " 1 - 211+4v3 _ 3v3+v·

u - Il l +311 + 4v2+2v3 + ... .

7.6 . No tes The Roge.-.-R&m!l!lujan continued fraction _ one 01 R&manujan'. favorite functions. We have relatO!
B . C. BERN DT

168

urging him to write ... paper about it. (11 ... letter probably written On 24 De«mber 1913, Hardy exhorted [51 , p. 87)

If you will send me your proof written out carefully (so that it is eMY to follow ), [ will (lIS8uming that I ~ ,,';th it--of which I have very little doubt) try to get it published for you in England . Write it

in the form of a paper "On the continued fraction

giving a full proof of the principal and mO'lt remarkable tbeorem, vi~. that the fraction can be expre>sed in finite t.emlll when 1: = e-· ..... , when.!! is rational.

However, Ramanujan never

folk",~

HllTdy 's a.hice.

The iLisUJry of the famoWl Rogers- Ramanujan identities (7.3. 10)

i!I now weLl known. They were originally di.scov<:'red by Rogers (198) and rediscovered by Ramanujan, who at fil'!it did not ha\~ proofs of them. Que day while at Cambridge, Ramanujan wlIS perusing back I.. sues nfthe Procuding' of the Land<1n Malh ematical Socielll and found Rogers's paper (198) giving proofs of (7.3.10). Ramanuj&n soon found his own proof! and published them in (189), (19 2, pp. 21 4- 213). For further historical BCCOunUl, see Hardy's book (107, pp. 90-99), Andrews 's text (14. Ch apter 7), or Berndt '8 book (34. pp. 77- 79). M8JlY proof! nf t he identities now exist; a description and classification of all known proofs up to 1989 can be found in Andrews's paper (17J. The identitie-s (7. 3.10) have beautiful combinatorial interpretatiollli. [n the definition of G(q) given in (7.3.1), write n 1 = I + 3 + ... + (2n - 1). The first identity in (7.3. 10) is equivalent to the 8SIlel"tion that the number of partitions of a pOSitive integer N into distinct parts with differences at [ea5t 2 equals the number of partitions of N into parUl congruent to either I or 4 modulo 3. For example. there are five partitions of 9 into distinct parUl with differences between parUl at least 2. namely, 9, 8+ I, 7+2, 6+3, and &+ 3 + I. The five partitions of 9 into parts congruent to either I or 4 mooulo & are 9, 6+ I + 1+ I, 4+4+ I, 4 + I + I + 1+ bt~y~Rt~~ For the second identity,

Marei"i"t!.

SPIRlT OF RAMAN UJAN

169

in the definition of lI (q), write n(n + I ) = 2 + 4 + .. ·+ 2n. The second Rogers-RanllllHljan ide ntity in (7.3.10) is an analytic statement of the fa.ct t hat the number of Pl'rti tioIlS of N into distinct ptU"t.s with differences at least 2 a nd with no 1'8 is equal to the number of partitioIlB of N into Pl'rts congruent to either 2 or 3 modulo 5. For example, the three Pl'rtitions of 8 into distinct parIS with parts differing by at least 2 and with no l's are 8, 6 + 2, and 5 + 3, while the three Pl'rtitioru; of 8 into p&rt8 congruent to either 2 or 3 modulo 5 are 8, 3 + 3 + 2, and2+ 2 +2+2. Roger~- Ramat\uJ
Exercise 7.6. 1. Prove lhat the

Theorem 7.4.2 ill due to Hirschborn [123], and the proofs of The7.4.1 and 7.4.2 t hat we have given are also due to Hirschhorn (123). The only other proof of Theorem 7.4.1 known to US is by Watson ]2 15], who employed t he quintuple product identity. Hirschborn 's proof is somewhat simpler. The proof of Theorem 7.4.1 given by Berndt [34. p . 267] is similar to that of Wal.8On. Our deduction of Theorem 7.4.4 from Theorem 7.4.1 is the same IL!I that given in t he aforementioned WQrlul of " 'al.8On, Berndt, and Hirschhorn. 0Te1tl5

Exercise 7.4.6 arose from the combined effortB of Hirschborn ,

P. Crutci>er, O.- Yeat Chan , and t he author. An approach to the Rogel"lt-Ramanujan rontinued fr!l.(;tion via mod ular forltl5 has been written by W. Duke [8 5].

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Copyrighted Material

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