Number Theory in the Spirit of Ramanujan

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Number Theory in the Spirit ofRamanujan

1+

q

1+

ql 1 + q' 1 + ...

@AMS _ _ ... "".... .....

Conte nts

Preface

lX

Chapter L

Introouction

§U.

Notalion and Arithmetical Functions

1

§1.2.

What are q.-Scries and Theta F\mctions?

6

§1.3.

Fundamental Theorems aoou t q-Series and Theta Functions

7

§1.4.

Notes

Chapter 2.

22

Congruences for ",(n) and r(n)

27

§2.1.

Historical Background

27

§2.2.

Elementary Congruences for r(n)

28

§2.3.

Rarnanujan'sOlngruencep(5n + 4}=O(mod5)

31

+ 5) = 0 (mod 7)

39

§2.4 _ Ramanujan 's Congruence p(7n

§2.5.

The Parity of p{n)

43

§2.6.

Notes

49

Chapter 3. Sums of Squares and Sums of Triangular Nwnbers §3.1.

5f>

Lambert Series

a5

§3.2. Sums of Two Squares

56

§3.3.

Sums of Four Squares Copyrighted Material

" -

B . C. BERN DT §3.4.

SUtl'Ul of Six Squares

63

§3.5.

SUtl'Ul or Eight Sq\Ull'e!!l

67

§3.6. Sum! of lliangular Numbers

71

§3. 7. §3.8.

Representations of in\.ef:ers by :z:2 +

2v2, :z:2 + 3~,

and

:z:2+~+y2

n

Notes

79

Chapter 4.

Eisenstein Series

"

§4.1.

Beruoulli Numbers and Ei.senste:in Series

§4.2.

1Tigooometric Series

"87

§4.3.

A CIILSS of Suim from RamanujlUl '. Lost Notebook Expressible in Terms or P, Q, and R

97

§4.4.

Proo& of the Congruences Pl5n p(7n + 5) '" 0 (mod 7)

+ 4) _

0 (mod 5) and

§U. Not.eII Chapter 5. §5. 1.

102

'"

The Corlneo:;tion Bet...-een Hypergeometric FUnctions IItId Theta Functions 109

§S.2.

Defi nitions of Hypergeometric Series and Elliptic Integrals 'nle Main Theorem

109 114

§5.3.

Principles or Duplication and Dimidia Lion

120

§5.4.

A Catalogue of Formulas for Theta Functions and Ei3enstein Series

122

Notes

128

§5.S.

ChapleT 6.

Applications of the Primuy Tbeor(!m of ChapteT 5 133

§6.1.

Introduction

133

§6.2.

Sums of Squares and lliangular Numbers

~6.3.

Modular Equations

1" 1<0

§6.4.

Notes

1511

Chapter 7. The Rogers-Ramanujan Continued Fraction §7. 1.

Dclinition IUld Hist,orical

B~!lnd

Copyrighted MBtenal

153 153

SPIRl T OF RAMANUJ A N

§7.2.

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

Hi5

S73

The Rogers-R.amanujan F\"."tions

>58

§7.4.

Identities for R(q)

161

§7.S. Modular EquatioIl.'l for R(q) §7.6. Notes

166

167

Bibliography

171

Index

185

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Preface

Generally acknowledged IL'i India's greatest mathematician, Srinh'BSa Ramanujan is most ofull thought of as II Dumber theorist, althoU&h he made substantial contributions to a.nalysis and several other areas of mathematics. For most number theorists, when Ramanujan's name is ment ioned, the partition and tau functions immediately come to mind. His ;nT,erest in these arithmetic funetio"" was inextricably intertwined with his primary interests of theta functions and other qseries. In fact, most of Ramanujan 's research in number theory arose out of q-se:ri ... and theta functioIL'l. Theta functioll.'l are the fundamental building blocks in the theory of elliptic functions, and Hamanujan independently de\'eloped his own theory of elliptic [unctions, wh ich is quite unlike the classical theory. We do not fonnally define an eHipti\: function, but, roughly. elliptic functi011S are merom orphic functions with two linearly independen~ periods over the real numbers. The ooncept of double periodid~y is not used in this book, and, 1.0 the belt of our knowledge, Ramanujan ne"er utili~ this idea. The purpose of this book is 1.0 provide an introduction 1.0 this large eJ<pallSe of Ramanujan's work in number t heory. Needless to say, we shall be able to cover only a wry small fraction of Ramanujan's work on theta functions and q-selies and their collJlections with numoor theory. 1I""..,ver, aft", developing only a few facta ..bout

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B.C. BERNDT q-series and theta functions, we will be equipped to prove many interesting theorelns. T he arithmetic functions on which we focUil are the partition function p(n}. Ramanujan's tau function 1'(n ), the OutDber of representations of a posith"e integer" as a sum of 2k squares denoted by r2~ (n), and other ari thmetic functions closely allied to T2k (n). Most of the material upon wbi ch we draw can be found in Ramanujan'. published papenl on p(n) and 1'(n). the later chaprers in his second notebook, his lost norebook, and his handWTitren manuscript on p(n) and 1'(n) published with his lost norebook. We emphasize that Ramanujan left behJnd few of his proofs, especially for his claims in his notebooks and lost notebook. Thus. for many of the theorems that we discuss, we do not kno"" Ramanujan's proofs. This is particularly true for the theorems OIl sums of squares and similar arithmetic functions that 'I!."e prm... in Chapter 3. The requirements for reading and understanding the material in this book are relatively modest. An ur.dergraduare course in elementary number theory is advisable. For some of the analytic arguments, a solid undergraduate course in comp!ex analysis is essential. Howe\"Cr, the occasions when deep analytical rigor is needed are few , and so readers who do not have a stroog background in analysis can sim· ply verify formally the needed manipulations. Our intent here is not to gi'''' a rigorous course in analysis but to emphasize the most impor· tant ideas about q-series and theta functions and how they interpla}' with number theory. This book should be suitable for junior and senior undergraduates and beginning , raduate students. Since many readers may not be frnliliar with Ramanujan 's life, 'I!."e begin with a short account of his life where readers learn about the r>Otebooks and lost notebook in which he recorded his theorems owr several years. We provide brief lustories, first of tbe M ordinary" notebooks, and second of the lost notebook. After these biographical and historical uarrations, we provide short summaries of the book's $C"en chapters. Rarnanujan was born on December 22. 1887 In the horne of his maternal grandmother in Erode, located in the southern Indian state of Tamil Nadu. After a few months, his mother, Komalatlllllmal. returned with her son to her home in Kumbakonam approximately 160 Copyrighted Material

S PIRI T OF RAM ANUJA N miles south·south"'....st; of Madras. whfre her husba.nd was a clerk in the office of a cloth merchant. At the age of twelve , Ramanujan bor. rov.'ed a copy of the second part of Loney's Pwne 1ligonomelry (1491 from an older student and worked all the prohlems in it. This longtime popular textbook in India has much mOfe in it than its title suggests. For example, infinite series and elementary functioDl! of a CQmplex ..,..riahle are two of its topics. At the age of about fifteen , he oorro"''ed from the Kumbakonam College library a ropy of C. S. Carr'$ A Syn · OP$i$ of Elementary R esul/d in Pure Malherrwlics (64J, which scn'ed fIj; his primary source for learning matllematics. Carr was a tutor io London and compiled this compendium of 4417 results (with very few proofs) to facilitate his tutorini1:. At the ai1:e of sixteen, Ramanujan enter«! the Government College in Kumbakooam. By that time, Ra· manujan was completely devoted to mathematics and consequently failed his examinations at (.he end of his first year, because he would not study any other subje.::t. He therefore lost his scholarshi p and , because his family Wflj; poor, was force;! to terminate his formal edu· cation. He later t,,~ce tried to obtain an education at Pachaiyappa's College in Madras. but each time he failed his examinatiOIlll. After leaving the Government CoHege in Kumbakonam, Ramanu· ja.n de\"Oted all of his time to mathematics, recording his reBUlts with· ,.,,,!, "........r.. in nOf....hooho It w"" proba bly around th .. ag.. of sixt.een that Ramanujan began to record his mathematical discoveries in notebooks, although the elllries on magic squares in Chapter 1 in both his first and second notebooks likely emanate from his school days. Liv. ing in lXl"erty with no means of financial suppor t, suffering at times from serious illnesses, and "'"(Irki ng in isolation. Ramanujan de,"Oted all of his efforts to mathematics and continued to record his discov· er ies without proofs in notebooks for tbe next fi\'\) years_ In 1909, he married Janaki. who was only nine or ten years old . With mounting pressure to find a job, Ramanujan visited V. Ramaswami Ayyar. the founder of the Indian Mathematical Society. Ayyar contacted R R.am1>Cha"d.a R.o.o . who agr~ to gi"<1 R.t..nanujan, who now hM ffio--W to Madras , a monthly stipend so that Ramanujan could con~inue his mathematical res<>arch unabated. Copyrighted Material

B . C. BERN DT After being supported for about fifteen months , for reasons that a-re unclea.r, Ramanujan rcfuwd furthfi financia-l as:;istance and be-came a clerk in the Madras !'ort nu,t Otfiw. This turned out to be a watershed in Ramanujan '6 career. Several pwple, including S. Narayana lyer, the Chief Acwuuta.llt, and Sir Fi-ancis Spring, the Cha-irma-n, offered suppart, and Ramallujan was persuaded to write English mathematicians a-bout his mathematical discoveries. Two of them, H. 1". Baker and E. W. li obson, evidently did not reply. M. J. M. Hill replied but was not enwuraging. But on J anuary 16, 1913, Ramanujan wrote G. H. Hardy, who responded immediately and encouragingly, inviting Ramanujan to come to Cambridge to develop his mathematical gifts_ Ramanujan and his family were Jyengars. a conservath'e branch in the Brahmin tradition. Travelling to a distant land would make a person unclean, and so Ramanujan's mother was particularly adamant about her son's not accepting Hardy's in· vitation. After a pilgrimage to Namool with S. N. lye.- and after Goddess Namagiri appeared in a dream to Komalatammal, Ramanu· jan received permission to t ra\'eL So on March 17, 1914, Ramanujan boarded a passenger shi p for England. At about this time, Ramanujan evidently stopped recording his theorems in notebooks, although II fe>,>- entries in his third notebook were undoubtedly recorded in England. That Ramanujan no longer conwntrated On logging entries in his notebooks is evident from two letters that he wrote to friends in Madras during his first year in England. In a lett.er of November 13, 1914 to his friend R. Krishna Ran [51 , pp_ 112- 113], Ramanujan confided, "I have changed my plan of publishing my results. I am not going to publish any of the old results in my not.ebooks till the war is over." And in a letter of January 7, 1915 to S. M. Subramanian [5 1. pp. 123~1 251 , Ramanujan admi~ted, ~I am doing my work v.:.ry slowly_ '-Iy notebook is sleeping in a corner for these four or five monLhs. I am publishing only my present researches as ( ha\"e not yet pron.rl the results in my notebooks rigorously." Ramatlujan soon bocame famous iOT the papers he published in England , &:Jme of them coauthored wilh Ha.rdy. Olle of his most im· partant papers is [186), 1192, pp. 136-1621. in which he introduced Copyrighted Material

S PIRI T OF RAMANUJA N

xiii

his famous tau function r(n), discussed in Chapter 2 of this book, and the Eisenstein series, P , Q, and R, which are introduced here in Chapter 4 and which were key players in so much of his research. In their paper 11091, 1192, pp. 262- 275], Hardy and Ramanujan la unched the ne'" field of probahilistic number theol)', which became an important hranch in number theory. In another paper 11101. 1192 , pp. 276309J. in the course of ohtaining an allymptotic series for the partition function p{n). Hardy and Ramanujan introduced the circle method, which still today is the primary tool for analytically attacking prob.lems in additive number theol)'. The genesis of t he circle method can be fouod in Ramaoujan's notebooks 1193]. but unforlunat.ely it is most freQuently called the Hardy-Littlewood circle method today. [n the latter part of his Slay in England, Ramanujan wrote his famous papers on congruences for p(n) [188),1192, pp. 210 - 213] and 1190J, 1192, p. 230), about which much is written in this book. Altbough Ramanujan ne,..,r doubted his decision to accept Hardy's invitation to Camhridge. not all Wall well with Ramanujan. World Wac I began shortly after his arrival, and being a ~trict veg~ etarian, he could not always ohtain familiar food and spices from India. On March 24, 1915, near the (lIld of his first wint.e. in Cambridge, Ramanujan wrote his friend E. Vinayaka Row in Madrall [51, PI' 116- 117). " 1 "''''" not well till the hrm ~ng tn the weatber and consequently I couldn't publish any thing for about 5 mOnlhs.~ By the end of his third year in England, Ramanujan Wall critically ill, and, for the next two years, he was confiued to sanitariums and nun;iug homes. R.amanujan's health turned slightly up"'ard when in 19 18 he became the second Indian to be elecU.>d as a Fellow of the R.oyaI Society and the first Indian to be chosen as a Fellow of 'Tlinity College. Af~r World War I ended, in ]919, Ram&nujan TCturned home, hut his health continued to det.erior8te, and on April 26, 1920 RamlUlujan died at the age of 32 _ Doctors in both Eng]and and India had difficulty diagnosing Ramanujan's illness. He Wall treated for tuberculosis, but a se~..,re vi_ tamin deficiency. liver cancer, lead poisoning purportedly from not properly cleaning his cooking vessels, and a rare tropical disease were Copyrighted Material

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other diagnOlSeS. However, D. A. B. Young 152, pp. 65- 75) made a careful examinatinn of al! extant records aud recorded symptoms of I{amanujau's iUuess and convincingly concluded that KamanuJan suffered from hepatic amoebiasis (a l/aIasitic infection of the lh'er). Not only do a ll of Ramanujan 's symptoms suggest t his disease, but Ramanujan's medical history in India &Iso favors this diagnosis. Amoebiasis is a protowal infection of the large intestine that gi\'
SPI RIT OF RAMANUJA N Hardy sellt the first notebook t.o the University of 1>ladras where Ramallujan'sother notebooks and papers were being preserved. Plan8 were undertaken to pUblish Ramanujan's collated papers and , 1KJSSibly, Illi! notebooks and other manuscripts. Handwritten copies of the notebooks were sent to Hatdy along with other manuscripts and papers in 1923, but the papers were never returned to the Unh-ersity of Madras. It traJl.'ipired that Rarnanujan·s ColI«led Papers [192) "-ere published in 1927, but his notebooks and other manuscripts "-ere not published. Sometime in the late 1920s, G. N. Watson and B. M. Wilson began the task of editing Ramanujan 's notebooks. The second notebook. being a revised. elliarged edition of the first, was their r>rirnllry focus. Wilson was assigned Chapters 2- 14, and Watson was t.o examine Chapters 15- 21. Wilson devoted his efforts t.o this task until 1935, when he died from an infection at the eatly age of 38. Watson wrote o'·er 30 papers inspired by the notebooks before his interest evidentl)· waned in the late 1930s. T hus, the prQjoct W8.'l ne~-er completed.

It was not until 1957 that the notebooks were made available to the public when the Tats Institu~ of Fundamental Researeh in Bombay published a photocopy edition [193), but no editing was undertaken. The first notebook W&II pubEshed in volume I, and volume 2 compri""" the """"nd ru>d third nO\.(lbooka. Tbe p.-.:tlCnt author un dertook the task of ed iting Ramanujan's notebooks in 1977. With the help of SC\-eral mathematicians, the author completed his work with the publication of his fifth volume [38) on the notebooks ill 1998.

In the spring of 1976, George Andrews of Penn6yl\"ania State Unj'-ersity visited Trinity College, Cambridge, to examine the papers left by Waison. Among Watson's papers, he found a manuscript containing 138 pages in the handwriting of Ramanujan. [n view of the fame of Ramanujan's notebooks [193), it w&ll natural for Andrews to call this newly found manuscript "Ramanujan's lost not.ebook. ~ How did this manuscript reach 'n-inity College? Watson died in 1965 at the age of 79. Shortly thereafter , on separate occasions, J. M. Whittaker and R. A. Rankin visited Mrs. Watson. Whittaker was a son of E. T. Whittaker , who coauthored with Copyrighted Material

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WatsQn probably the m(>lt popular and frequently used text on analysis in the 20th century {221]. Rankin had succeeded Watson lIS Mason ProfffiSOr of Mathematics at the Unh~rsity of Birn,ingham, ",here Watson sen-w fOT most of his CLreer, but was now Professor of Mathematics at the Uni,~rsity of Glasgow. Both Whittaker and Rankin went to Watson 's att,<; office to ex&mine the papers len by him, and Whittaker found the aforementioned manuscript by Ramanujan. Rankin suggested to Mrs. Watson that he might sort her late husband 's papers and send those worth preserving to Trinity Col. lege Library, Cambridge. During the next three yean;, Rankin sorted through Watson 's papers sending them in batches to Trini ty College Libr3l')', with Ramanujan 's manuscript being sent on DeoJmber 26, 1968. Not realizing the importance of Ramanujan's papers. neither Rankin nor Whittaker mentioned thelll in their obituaries of Watson 1195), 1222). The next question is, How did Watson come into ~on 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 papers to Hardy_ Most likely, this shipment contained the "lost notebook." Of the O\'t'r 30 papers that Watson wrote On Ramanujan's work, t"'X1 of his last papers were de'X1ted to Ramanujan's mock theta functions. which Ramanujan discovered in t he last year of his life, ,,·hieh he described in a letter to lIardy only about three months before he died 151 , pp. 220--223), and which are also found in the lost notebook. In these two papers, WatSOn made some conjectures about the existence of certain mock theta functions. [f he had the lost notebook at that time, he would ha'"e seen that his conjectures were COTn!Ct. Thus, probably sometime after Watson's intere;;t in Ramanujan's ,,-ork declined in the late 1930s, Hardy passed Ramauujan's papers to Walson. In early 1988, just after the centenary of Ramanujan's birth, Narosa Publishing House in New Delhi I'ublished a photocopy edition of the lost notebook 1194). Included ill this publicatinn are partial m,,,,.,,,,,,ripl<', I"""", papcl1l, and frngmer.t<J by Rnmanujan, .... well .... letters from Ramanujan to Hardy writ«n from nursing homes during the last two yean< of Ramanujan 's sojourn in England. Copyrighted Material

SPIR I T O F RAMANUJAN The first chapter of this book is de\'Ot.OO to basic facts about q-series and theta functions, includi~ the q-binomial thoon:m , the J acobi triple product identi ty, the per.tagonal number theon:m, Itamanujan's IIp! summation theon:m, and the quintuple product identity. Many of the theorems pro....ed in Chapter I can be found in Chapter 16 of Ramanujan'a second nmebook (193], (34). Chapter 2 focuses on congru,.,nces for the partition fun<;tion p(n) and Ramanujan's tau function T(n). Uuch of this material is taken from Ramanujan's handwritten manll- contains a large number of such theorems. Eisenstein series permeate Ramanujan 's notebooks (193] and loot notebook (194]. Much of our exposition 011 Eisenstein series in Chapt(!r 4, ho...-ever, is taken from Ramanujan's epic paper [186], [192, 136- 162). One of Ramanujan's approaches t.o congruences for p(n) is based on Eisenat,.,in series, which we demollStrate at the doee of Chapter 4. In Chapter 5, we introduce readers to hypergoometric functions and elliptic integrals. OUT goal in this chapt(!r is to prove one of the most fundamental theorems of elliptic functions relating hypergeometric functiollS and elliptic integrals t.o theta functions. This theorem enables us \.0 express theta functions and Eisenstein series Copyrighted Material

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at various argumentiJ in ltlnns of certain elliptic parameltlrs. Our exposition is derived from Chapter 17 of Ramanujan's second notebook (19:1], [:14, Chapter 11), wheu, through" ","ci"" of preliminary lemmas, Ramanujan leads us to the aforementioned key theorem. Applications of the aforementioned representations for Eisenstein series and theta functions form the w ntent of Chapter 6. First, we retuTll to the topic of sums of Squares and demonstrate how the formulall for Eisenstein series lead to short proofs of some of the results from Chapler 3. However, most of OIapter 6 ;., de\"Oted to modulILT equatioJtll, a topic to which Ramanujan made more contributions than any other mathematician. Chapters 19-21 in hili second notebook are devoted to modular equations, and Our short introduction to this wpic is drawn from our previous account of Ramanujan's work iu these chapters [34]. One of Ramanujan's favorite topics was the Rogers-Ramanujan conti nued fraction, the foc;us of Chapter 1. Because we wish to share $0 mucl! about this wntinued fTaction with readers and because the length of the chapter would be prohibiti'", if we proved all theorems offered in this chapter, we forego $Orne of the proofs. H(U,:e\'el", we do prove t",-o key thooTeIilll relating the continued fraction with its reciprocal. These theorems are then used to give an alternative, cleaner proof of an identity of Ramalluj&ll in Chapter 2, yielding immediately the congrueuce p(5n + 4) = O{mod5). T he famous RogersRamanujan functions are also discUS$Cd, and, in particular, "'.., pTO\"e that the Rogers-Ramanujan continued fraction can be represented u a quotient of the two Rogers-Ramanujan functions. Our f",,-ent wish is that our sampling of the m&lly beautiful properties satilified hy this continued fraction will motivate readers to turn to original sources to learn more about it. Ubiquitous in this book arc

produ~ts

(1 - a ){1 _ aq)( 1 _ aq2 ) . . . ( l &$

well

&$

of the form aq ~ _ l ) = :

(" ;q)..

th .. ir infinite versions

(I - a) ( 1 - aq)(1 -

ail··· (I -

aq"l· . =: {a;q)"" ,

Iq[ < I ,

whicll are called q-products. Although we lISSume that readers of this hook are familiar with infillite series, it may "..,11 be that some Copyrighted Material

SPIRIT O F RAMANUJAN a!"1l not familiar with infinite product.. A reader desiring to learn a few basic fact. abont the convergence of infinite products may consuit" good text On complex analysis, such as that of N. Levinson and R. Redheffer [142 . pp. 382- 38,)], for basic properties of infinite products. In particular, all the infini te products in the present \.ext converge absolut.ely and uniformly on compact subsets of Iql < I. In particular, taking logarithms of infinite products and diife!"1lntiating the resulting series t.efmwise is permitted. At first, you may flOd that working with the products (a ;q)n ane:: (a ;q)"" is somewhat tedious. In order to verify q-product identities or to manipulate q-products, it may be helpful to write out the first three or four terms of each q-product. This should provide the needed insight in order to justify a gh-en step. After working with q-products for awhile, you will begin to handle them more quickly and adroitly, and no longer need to write out any of their terms longhand. When you reach this stage, you should feel quite comfortable in I:l3nipulating q_series. It is assumed throughout the entire book tba, Iql < 1. O,...,r 50 exercises are interspersed within the exposition. The author is grateful for the comments of graduate student. at the Universities of nIinois and Leece "here this material was taught. Y.-S. Choi, S. Cooper, D. Eichhorn, A~[S copy editor Ill. Letourneau, J . Sohn. and K. S. WiWams provided the author with many helpful sugge:stions and corrections for which he is especially thankfui. T he author also thanks N. D Baruah, S. Bharga\"a, Z. Cao, II H. Chan, W. Chu, AMS Acquisitiol.lS Editor E. Dunne, M. D. Hirschhorn, /It . Sornos, A. J. Vee, and the referees for their suggestions and corrections.

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Chapter 1

Introduction

1.1. Notation a nd Arithmetical Functions in this first section, we introduce not3110n that wi!! be used throughout the entire monograph. s.ewndly, we define the arithmetical functions on which "'~ will focus for most of this book, and which can be st udied by employing t he theory of theW!. functions and q-series. A brief introduction to theta functions and q-series will be given in the next section, to be followoo in Section 1_3 by a few of th'" moot useful theorems about these functions

Definition 1.1.1. Define

(l.l.I)

(alll:= (a;q )o := I,

(1. 1.2)

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

(a)""

:=

(a; q)""

:=

"-...n,

(1 - (til) ,

..n,

(I -

llill,

n

> I,

Iql < 1.

lVe call q the /xue, and if the identifica.!iQIl of the ba&e i.s clwr, we often omit q from the lIotation.

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-,

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2

Replacing II by 0, and using L'HiiI;pital's rule while letting q tend to I , "-e lind t hat

( L I.3)

. (
. l - q" 1 - '1<>+1 1 _ '1 0 +" - 1 hm ... ' C';'::-;:_ . -1 I q 1 q - I q 01(a+ l ) ·· ·(a + n - l ).

The expression On the right side aoo..'e is called II rising or shifted factorial, and so...-e see that (a; 'I ).. may be considered as an analogue of t he rising factorial. We next define the q·binomial coefficient, or Gaussian coefficient, I::.], which is an analogue of the ordinary binI). mial coefficient (::').

De fi nition 1.1 .2. LeI n and m denol€ ini£ger,. Then Ihe Gou.ssian rot:ffic;ent ~ defined by

(0). (1.1 .4 )

[ J ;

:=

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

ifO<m~n.

Exercise 1.1.3. lj,ing ( L L3) thru timu. Ulch time with II = [, show that

r [,,] ~~ no =

(") ,.,. .

Thus, the q-binomial coefficient te.,ds to tbe ordinary binomial coefficient when q _ \. From the definition (1.1. 4) it ill not obvious that the q-binomial coefficients ru-e polynomials in q.

Exercise 1.1.4 . Using !he definitwn (1.1.4), readers shotdd first pro~ the first q-onologue 01 Pascal 'slormula given below. Le m ma l. 1.5 . For n ;:-: l.

(LUi) Exe rcise 1.1.6. Second, employing umma 1.1.5 mId induction on n, prove that 1::.1 u a W1yno'!lial jn.1l of 4egree m(n - m ). Copynghled MafenaJ

S PIRIT OF RAMANUJ AN Exercise 1. 1.7. Es !obluh whiJ;h i.! .qiVf!n for n > 1 bU

0

urooo

3 q·onologu~

of PlJSoo/.'s formula,

E xercise l. 1.8. Prow that, for rod. pair of nonnegatiw integer6

m,n,

t [m+jl,; ~ [m+n+ll · jag

m+1

J

We DOW define four primary arithmet ica.l funClion!.l on which we focus t hrollRhout the monOl:raph . Definition 1.1.9. Ifn is a positive it"!leger, let p(n) denol~ the number of um'{!Stricted repre&~ntatiorl$ of r. ..... 0 sum 0/ po6ifillf! inttgfr8, lL"'ere repn!sentatioru with different on/ers of the same summands are not regarded as distinct. We 00/.1 p(n) the parldion function. For example, P(4) = 5, because there are 5 ways to represent 4 as sum of positive integers, namely, 4 = 3+ I = = 2+ 1+1 = 1+1+ 1 + 1. Readers should check t hat p(5) = 7 and p(6) = 11, but readers should not check (at least by hand) tha~ p(200) = 3, 972,999,029,388. A f.. w mom .. nt~ of r .. Auction convinC
a

2+2

(l.l.6)

p(n)_

4n~exp(nJ23n),

i.e., the ratio of the left and right sides of (1.\.6) tends to I as n tends '" 00.

T he generating funclion for p(n), due 10 Euler, is gi'"Cn by

1

(\.\.7) I

"

where we defi ne P(O) = 1. To see this, obsen "C that the factor

\ _,,L 'l''k ~

1 ~ Copyrighted Mflterial

8 . C. BERNDT

4

generates the numbe!' of t 's that appear In 1\ particulM partition of n. Each partition or n appears once ud only once on tbe right side 01 (1.1.1) when interpreted in this manl~.

The generating function for the number of partitions of n into difitinct pIU'tI, deoot<':d by pd(n), is gi\''eIl by ~

( 1.1 .8)

L>d (n)q~ = (- q: q)"" _ ( \

..,

+ q)(1 + ql) ... (1 + q~) ...

,

becaUlle IIny particular integer Can occur at most onee in any given representation of ft. We now show that, by elementary product manipulatiolUl, we ca.n deduce the followi~ famous theorem of EuleT. Thoor e rn 1.1.10 (Euler). The " .. mller (>f po>rloh" .... oJ .. 1'..,,,1,,,,, ml~r n into dilliru:t pam ~ the 'Iurnbu 0/ parliliornl (1/ n IIIW odd pam. denotcd "II p,,(n).

proor. From (1.1 .8), (1.1.9)

~

().

(

)

(",q')~ "( . 1')

t... Pdnq = - q;q",,= ( . )

n~

q, q ""

q, II

~ ) "" t... po{nq" 00

n-li

Equating coefficients of q~, n ?: I, on both aidell of (1.I.g). we romplete the proof of Euler's identity. 0

For example, Pd (6) = 4, becaUSO! 6 has 4 representations iDio distinct partll, namely, 6 == 5 + I = 4 + 2 _ 3 + 2 + I. On the other hand, Pe(6 ) "" 4, because 6 has 4 ~pretCllUl.lioll$ imo odd paI't$, namely, 5+ 1 - 3+3 = 3+ 1 + I + I _ I + 1 + I + 1 + 1+1. E"erc ise 1.1.11. Evkr's i.dmldy Cdn be .I.gllily refinal. Prove Uut/ the number 0/ parldlOn8 0/ n into on el", (odd) number 0/ odd J>(1rlJ ftllUIU the ntlmber 0/ parlltlOn8 0/ n 111/0 dl.ttHlcf pan.., ILfttre lilt ntlmkr 0/ odd pan.. is ew:n (odd). E"crcise 1. 1. 12 , Pro~ that the n~mber 0/ partlllO"" o//he positiue integer 11 into pariS Iho/ are not dWi..!lblt bll :.I loS equal 10 the number of partitlolU of n ;" which no part oppt:lr$ more than IwiG#:. E"crcise 1. 1. 13. Eult:r'$ identlill Cdn k refinal.n lIet onolht:r tooll. Ltl k and n be pru,ti"", ;"Ugen with k > 2 .. ~ Iho/ tht number of COp'fri{1lted Wienal

SPIRlT OF RAMA NUJA N

5

partitions of 71 in which no part is ditliHble 6y k eq"IUIU the number of parti/io'lS of n such thot there lire strictly less than k copit:3 of each part. Note thai when k = 2, we obtain Euler's identity. The case k = 3 is the previous aertise.

Using elementary product manipulations as in the proof of Theorem 1.1.10, readers are encouraged to prove the theorems in the following exer<;ises. Exercise 1.1.14. Let A(n) denote the ntlmber of partitions of the pasitiue in.teger n into parts congruent to 2, 5, or ] 1 modulo ]2. Let 8(n) denote the number of partitions of n into distinct parts congro.... 1 to ~ithtr 2 , 4 , M 5 modulo 6 P~ lh
A(n ) = B (n). Exercise 1.1 . 15. Prot!e thaI the ntlJJllr
(1.1. 10)

q(q;q)~ =

L r (n )q",

"-.

Iql <

I.

Then r(n) is called Romanujan's tau functioo .

For example, 1"(2) .. - 24 and r(15) = ] ,2 17, 160. A glance at a table of 1"(n) shows that Ir(rt)1 grows quickly. It h8!l long been conjecwn:d, but never p l"OVOd, th"t "(11) " 0 for all n > O. h may seem at first sight t hat the definition cf r(n) is artificial. Ilo,,'Cver, it is one of the moot important arithmetical functioJ15 in nu mbolr theory and arises in many contexlS, in particular, in the theory of modulllT forms. We study this eb~Wffi.}jf$b~WIBl in Chapter 2.

B. C. BERNDT

6

Definition 1.1.19, Fur po$iti~e illt~ers II and k, let Tk(n) denote the number of representations of f\ os 4 slim oj k sqwlres, where representations with different orden and diJJ~t signs are CQ.mted Q8 distinct, By convention, Tk(O) = I.

For example, r~ (2) = 4, because 2 = 12 + ]2 = 12 + (_1)2 = (_ 1)2 + 12 = (_1 )2 + (_ 1)1; T, (9) '" 4, because 9 _ 32 + ()2 = 0 2 + 32 = (- 3)2 + 02 = 02 + (_3)2; and T2(7) = 0, because there aTe no~ any ways we can write 7 as a swn of 2 squares. Definit io n 1.1.20. For n > 0, the tri6ngul4r numbeJ"$ are the ",im· ber, n(n + 1)/2. If nand k are poSItive integers, let Ik (n ) denote the number of rePf"senlation.! oj n a.8" sum of k tria".q!daT number" where representations with different ,rden an: CQ.mtea as distinc/. Set ' k(O) = I.

For example, ~(1) = 2, Ix!eause 7 '" 1+6 = 6+ 1, while /,(16) = 4, because 16 .. \ + 15 = 15+ I = 6+10 = 10+6.

1.2. What ar e q-Se ries and T heta Functions? Generally, a q-series has expressioJl.'l of the type (a; q)~ in the summands. In Section 1.3, we consider /I few of the most important exawpleo. lIow."..,r, m....y q-"",r;"", do not have auch pwduct.o in their su mmands, but they may arise as limiting cases of series containing products (1.1 .1), or their theory may be intimately intertwined with such series. Theta functions are such examples.

Defini tio n 1.2.1. RallUlnujan'$ genemllheta funct ion I(a,b) u de· fined by (1.2.1)

I(a,b)

labl < I.

:=

Thru speciaJ ca.se$ are definetl. by, in Ramanuja n', '1otalion. ~

(1. 2.2 )

",(q) :=/{q,q) =

L

qn',

~

(1.2. 3)

I/J(q} :=/(q, q3 ) = Lqn{n + L)12,

Copyrighfed'Material

SPIRlT OF RAMANUJAN

7

L ~

I {-q ) := f( _ q, _ q2 ) =

(1.2.4)

. _

(_ I )n qn(3~ ~ 1)/2.

~DO

Exercise 1.2.2. JustilY !he s«.o:>nd equality 01 (1.2.3). (1.2.7) 01 Umma 1.2.4 below.

Su IWO

The numbers '1 (3'1 - 1)/2. n ~ 0, are the poffi tagonalllumber$, and the numbers '1 (3'1 ± 1)/2, 'I ;:: 0, are the generalized poffi/agonal number$.

Now that ~

(1.2.5)

,.,k(q) =

L Tk(n )qn

md

-".

T hese generating functioll8 wi!] be used in the sequel to find formulas for r k(n ) and Ik (n ), for certain positive integers k. ReAders may wonder why the fundioos defi ned above are caned theta functions, sinw no theta appeaf$ in tbe notations . In the elassical definitions, e.g., see Whittaker a~d Wat'lO~ '$ text [221 , p. 4641the notation,'} is employed. We emphasize that Ramanujan's def· inition is different from the usual definitions, bul the generali ty is ide~tical to that in (221 ), i.e .. all of RJunanujan 's theta fu~ctions fall under the purview in (221], and conversely. i':xercise l.:l.a. Proue the properiier 01 theta functiolU.

Lemma 1.2.4. We

/ollowm~

!:oa.ric and enonnou.sly useful

ha~e

(1.2.6)

I {a, b) = I (b,a),

(1.2.7)

1( 1, a) = 2/{a, ( 3 ),

(1.2.8)

1 (- I.a ) = O,

and, il n U allY integer, I (a, b) = an( n+1 )/2 bn ( n ~ l)f2 I (a( ab)ft . b( ab)~ ~). ( 1.2.9)

1.3. t'undamc ntal The ore m s about q-Serles and Theta Functions Perhaps the most importaot theorem in the subject of q-series is the q_binomial theorem. Copyrighted Material

8

B.C. BERNDT

Theorem 1.3.1 (q-analogue of the binomial theorem). For I,

f: (a)~:~

(1.3.1 )

"~o (q)"

=

Iql, It I <

(at)"".

(zl""

ProoL Note that the product on the right side of (1.3. 1) eon\'erges uniformly on wmpact subsets of 1:1 < I and so represents an analytic function on Izi < 1. Thus, we may write

(a: )"" ~ " III < I. F (l):= (,) .. ~A .l , "" " _0 From the product representation in (1.3.2), we can readily verify that

(1.3.2)

(1 - l) F {z) = (I - az )F{q: ).

(1.3.3)

Equating coefficients of : ", n > I, on both sides of (1.3.3), we find that A n - A,,_I = q" An _ aqn _ 1A,,_I, 0'

1- aqn_1 A" = I q" An_I.

( 1.3.4)

fI

> 1.

iterating (1.3.4) and using the value A u = J , which is readily apparent from (1.3.2), we deduce that

A" _(a)" (q) .. '

( I.. 35)

n

> O.

Using (1.3.,,) in (1.3.2), we complete the proof of (1.3. 1).

0

To understand why Theorem 1.3. 1 is called the q - (lnalogu~ of th~ ~inQf1lial theoTWI, replace (I by if' in Theorem 1.3.1. Arguing as we did in (1.1.3), we find that

)" I·lin (if' (q)..

~_1

=

~(~o-,+C" I )C'C"f(o,+"-""CO,,,,) n!

.

Assuming nOW that II is a positi,"\! illWger, and remembering that a has h
"" L

.".

ala + 1)··· (a

+n

- I)

:"=

~rr -'

nl 1 Copyrighted Malenal

."'

1 :

=(1-:) - ·

'

S PIlUT OF RA MA N UJA N

9

which in cakuJus is often called the generalized binomial theorem. Corollnry 1.3.2 (Eul~r ) . For Iql..:.: \,

Izi <

(1.3.6)

I,

"od (1.3.7)

~

{_z)"qn(,, - I)/2 _

L.,

- (':}oo ,

( )

q

n . (I

Izi <00.

n

Proof. Equality (1.3.6) follows immediately from (1.3.1) by setti ng

0 = 0. Replace" by o/b and z by liz in th~ q- binomiaJ theorem. Thus, for Ibz l < I,

(1.3.8) We Jet b ~ O. Now,

Iim (o/b)"b" = lim

&_0

(1.3.9)

(1 - ~) (1 - "'I) ... (1 - "'"-') b" b II

6_ob

'" ( _ a )"qn{n- I I/2.

Equality (1.3.7) now follows upon setting a = I.

o

Alert readers will have immediately noticed that we have taken the limit on b under the summation sign without justifying it. In the theory of q_series, taking limits under th~ summation sign, I\Ij we have done, is always assumed to be justified, but hardly anyone ever does it. Nonetheless, a rigorous argument almoot always ruIll! along the $aIIle lines. In the case at hand, cltoose a number AI such that Iql < Ai < I. Lett> 0 be given such that 0 < 2( < I - Ai. With Ibl < t temporarily fixed, let No be that unique positive integer such that O$k
Ibl + I, IM " '

<"

COpynghled MateoaJ

8. C. BERNDT

10 T hen, for Ihl :s;

£

and

fI ;::

No ,

(alb)~b~l..: ('l)n

n;:.; ((bl + !aIM~)

-

(I

M )"

< "('c+'-'iI·CIl= N-i '(~2,,,)_"-_N_' -

(I

M )n

~('+ I.I)N'( 2, )" 2(

I

/If

SinceZ«l - M,

T hus.

f:

(a/b} n (h)"

n~O ('l)n converges uniformly for Ibl :::; ~ by tl:e Weierstrass M -test, and 50 letting b ..... 0 under the summat ion sign is justified. NJ the q-anaJogue of the binomial theorem is perhap6 the most fundamental result in the theory of q-snies, 50 the Jaoobi triple product identity in the next theorem is likely the most important and useful result in the theory of theta fUllCtions.

Theorem 1.3.3 (Jacobi Triple Prod 'J d Identity). For z # 0 and

Iql < I , (1.3.10)

"- -"" [n Ramanujan's notation (1.2.1 ), the Jacobi triple product identity takes the shape (!.3.11)

I {a, b) = (-(I; abj",,( -b; ab)",,(ob; ab)"".

Proof. In (1. 3.7), replace q by q2 and z by -;q to deduce that ..,

n

n'

: q ( ~q; q 2 j"" - """" L., ( ,. ')

~~(lq.qn

(1.3.12)

SPIRlT OF RAMANUJAN

11

since ('lhH; q' ).,., = 0 when n is a negath~ integer. Now apply (1.3.7) again with q replaced by q2 but this time with z .. q2"+2. Thus, [rom (1.3.12),

by (1.3.6) with" replaced by -q/z and 'l replaced hy q1 , and therefore with the restriction Iq/ zl < 1. Rearranging abo\.~, we complete the proof of (1.3. 10) for Iq/zl < I. Ho"."ever, by analytic continuation, (1.3.10) holds for all complex z t- 0, and so the proof is complete. 0

Corollary 1.3.4. Reatlilhat Ramanujan'&/hela /nne/ions

Proof. T he product (1.3.13) follows from (1.2.2) and (1.3.11 ) by settinga = b=q. ~tting a = q and b = if in (1.3.1 J) and using (1.2.3) and (1.1.9), we find that ( LJ .16 )

,.

. ') "" ( - 'l ,'l ')"" ('l '.,'l') "" ¢o('l ) = (-'l;'l ')""( - q~.,'l ') DO ('l ' , 'l ') 00 = ( - q,q =

' ') (q2 ;q2)"" (- q;'l ) 00 ('l ;'l "" - .{ 'a2 1 . Copyrig'llred Maferial

B. C . BERNDT Lastly, set a = - q.1> = - q~ in (1.3.1 1). T hen. from (1.2.4), f{-q)- {q:Q3)"" (q2:Q3)"",(qJ:Q~ }",, = (q;q)"".

(1.3.17)

o Combining ( 1.2.4) and (1.3.15), ,,-e obt.a.in Euler 's pentagOnal number theorem, wb ich, because of its importance, ....-e sta.te in a separa te corollary. Co rollary 1.3 .5 ( Euler's Pentagonal Number Theorem). ~

( \.3 .18)

L

(_ 1)" q"{3 n- l l/2 =

~

L

.--

( _I)n q" pn +l)/2 = (q; q)"" .

Note that the second representation in (1.3. 18) arises from the former by replacing n by -n in t he first sum. In tbe sequel, we shall use tbese representations intercllangeably. From Corollary 1.3.5, we can ellllily derive a re<:urrence formula. for calculating ~-alues of p(n). C oro llary 1.3.6. For I>reuity, set Then (1.3. 19)

p{n)=

L

w,

=

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

(_ I )i+ 1p (n _wj).

0",-"" :;,,

Proof. By Corollary 1.3.5, (1.3.20)

m.' Now equate coefficients of q" , n <:: I. (m both ~ides of (1.3.20). The recurrence relation (1.3. 19) then immed iately follows. 0 E xer cise 1.3. 7. There i.J arlO/her simple recurn-nce rdalion for p{n) thai i.J due to E"k;r. Consider the genemtin9 function (1.3.21 ) Logarithmically differentiate both sides lif (1.3.21 ). Then expand tk re,u/ang denominators lit! the righ t·hand side into 9eometric !erie-. Copyrighted Material

S PIRlT OF RAMANUJAN

13

'f O'(n) := 2:dl.. d, deduce that, for Mm integer n > 1,

.-.

np(n ) = 2:>U)Cf(n - j ). ,~

Euler 's pentagonal number theorem has a beautiful oombinatorial interpretation given in the next corollary. Corollary 1.3.8 (Comhinatorial Version of Euler 's Pentagonal Number T heorem ). Let D~ ( n) denol" th" "umber of partitioll$ ofn i" to an even nllmber 0/ distinct parts, and let D~(n) d"note the number of partitioll$ 0/ n into an odd nllmber 0/ distinct parts. Then (1. 3.22)

D~(n)

P roof. Let

I.Il<

-

D~{n)

=

(-i Y

t

0,

'

if n =

j (3j ± 1)/ 2,

otherwise.

begin by closely examining the product

(1.3.23) which "'-e!!lee ill similar to the generating function (1.1.8) for Pd (n), except that in (1.3.23) there are minus signs in the product. When we multiply the terms of (1.3.23) logether, we obtain a plus sign if we multiply an even number of pOwers of q. and we obtain a minus sign if we multiply an odd number of powers of q. T hus, partitions of an integer n into an even number of distinct parts are weighted by a plu.s sign, while partitions of n into an odd nwnber of distinct parts !lTe weighted by a minus sign. Thus, from the pentagonal number theorem (1.3.18), ~

1+

L {Do(n) -

.-.

The result

nOW

~

D o(n)} qn - (q; qj"" =

follows.

L

j _ _ oo

(_ 1)'q'{3.i- 11/2.

o

FOI .,,,,,,,,pie, D «6) Do (S) _ I , ~i nce S _ 2(3·2 - 1)/ 2, i.e. , J = 2. Indeed, the partitiollli of S into distinct parts are ::., 4+1, and 3+2. Second , D« 6) - D~(6) = 0, since 6 is not a generalized pentagonal number. Indeed, the parti t,oIL'l of 6 into distinct parts are 6, 3+ 2+1 , 4+2, and Stopyrfghled Material

B. C. BERNDT The next thMrem is fundamental, and "'''' shall times in the sequel.

-.., L-

USE:

it several

Theorem 1.3.9 (J acobi's Identity). lVe h411e

(1.3.24)

2 ) - 1)" (20 + l )q~ (" · I )/~ = (q; q)!,.

Proof. In (1.3. 10), replace (1.3.25)

.t by Z2 q to

deduce tbs~

.I?"q~'+" - (_z2q2;.2)00(_11:2;~)00(q2;~)00.

"--00

Divide both sides of (1.3.25) by 1 + l Iz' to find that (1.3.26) Eoo :2"+l q"'+. " _ _ 00

_

z + I/z

(_ z'q2 ; q2)00(_q'l z2 ; ~)00 (.l ;

q' )<XI'

Obser'", thst, by (1.2.8), ( \.3.27) Thus, letting z ~ i in (1.3.26), employing (1.3.27), and applyin& L'Haspital 's rule, we find thst (1.3.28)

L

2"1 -

(- l n2n

+ I )q" ("·... ) = (q2 ;q' ):".

" _ _ <XI

We now divide the sum a\)m", into two paru, -00 < 0 S -1,0 :s; o < 00. In the former sum, replace 0 by - 0 - 1 and simplify. Lastly, replace q2 by q in (1.3.28). We thus arr;'", s t (1.3.24) to com plete the proof.

0

Besides the functions 'PCq), lI1(q), and f( - q), Ramanujan defines one further function (1.3.29) which is l>OI. '" the\.8 function but which plays " "mminent rol. in the theory of theta functions. T hese four functions satisfy a myriad. of relatiolUl. We offer he"" mOflt of thcee recorded by R.amanujan in Entry 24 of Chnpter 16 of his second notebook 1193); see 134, pp. 39401 for proofs . Copyrighted Material

SPIRIT OF RAMANUJAN

15

Theorem 1.3.10. (1.3.30) (1.3.31)

I{q) "'('1) d,,) r;<:;) I( q) = ¢( '1 ) = X( q ) = V ~'

I (q) ,.,{q) 1(- '12 ) X'1 = I ( '12) = 1(1/ ) = I/J( q)' ()

(1.3.32)

",2 (-l

x( q)x (- '1) = .\ (_'1 2 ),

(1.3.33) (1.3.34)

) = ""(-'1 )",,{q),

IJ( _ '1 1 ) =

li l( - l

)v(l l _ 1"( - q)IP 2 ('1 ).

Exercise 1.3.11. Use the /unctio'fl9' product repreuntatians ta prove ~"mc

oJ lite idcntilie:.

"I Th""",m

1.::1.10.

The q·binomial l hoorem is perha~ the mos~ widely used summation formula in the theory of q-series. Ramanujan 's ,IP1 su mmation thoorem is a bilateral (i.e., the summalion index TUII.'i from - 00 to 00 instead of from a finite integer to 00) analogue and is undoubtedly the most useful bilatera l summation iJ q-series. In both (1.3.1) and (1.3.36) below, the series on the left sides hR,·e exactly One q-produet in the numerator and one q-product in the denominator, and both theorems give product representatioll.'i for the series. The prefix 1 On '" indicat ... that there;" one q_product in the nwnerator, and the sur_ fix I on '" indicates tha~ there is one q-product in the denominator. Before we stat(! and prove Ramanujan's summation theorem, it will be convenient to define an analogue of (0: q) .. for negati.-e integers n. So, for - 00 < n < 00, define

I""

(1.3.35)

(o; q)oo ( ) (1;'1 .. = «(lq":q)oo ·

Ob6erve that ( 1.3.35) agrees with De6nition l.1.l when n is a nonnegat ive integer. T heorem 1.3.12 (Ramanujan·s iand l'1 I
,1/1,

Summation). Far Ib/ol < Izl <

B. C. BERNDT

16

Proo f. Define fez) ;-

( 1_3 _37)

In the annulus,

Ibfal < 1:1< 1, f ez) is aoalytic, and so therefore has

II Laurent expansion ~

J (z)

(1.3.38)

=

L

o:"z" ,

Ftom the definition (1.3.37), we can deduce that

(b - aqz )/(qz ) = q{l - z) J (z) .

(1.3.39)

Observe that the Laurent expansion of f (qz) is valid in the annulus, Ib/(aq)1 < It ] < I/lql. and so the Lanrent expansions of both 1(:) and J (qz) ar Ib//I l. 8 restrict ion that we shall later remove by analytic continuation HelICe, by ( 1.3.3S) and ( 1.3.39), for Ib/(aq)1 < 1: 1 < 1, ~

(1.3.40)

L

~

(be" - ac.,._l)q" z" =

L

q(c n - c,, _ dz n .

Equating coefficients of z" on both sides of (1.3.40). we find that (ben -

ucn_ll~" =

<{(c" - c,,-d·

Solving for c,., we find that ( 1.3.41 )

1 _ aqn-l c,. = 1 bqn_l c,, _ l_

If n > 0, iterate (1.3.41 ) to conclude that ( 1.3.4 2) If n < 0, rewrite (1. 3.41 ) in the fonn ( 1.3.43)

4_.

1 _ bq"- L = I aq ._14.

Iterate ( 1.3.43), use ( 1.3_35) . and replace n by n + 1 to arrh'e at ( I.3A4 )

(a)n '" = -- ~. COPyrighlelJ'Material

SPIRIT OF RAMANU JAN

17

Putting (1.3.42) and (1.3.44) in (1.3.38) and using analytic continuation. we wnclude that

Iblal < 1=1 < 1.

(1.3045) [t

remains

to

e\1l[uat.e CO.

Write

-,

(1.3.46 ) J(z)=CO

L

. ~:~" :"+ Cil L ~:~"Z"=:CO(9(Z)+h(:»). ... . { I "

" __ 0 0 "

From the definition (1.3.37) of 1(:), ~'e see that 1(:) has a simple pole at ~ _ I. Sin"" 9 (~) i.!; "" pow", ..,..iell 3bou~ Z = 00 CO"""' ging for 1=1 > 16/01, it follows lhat gfz) is analytic at : = l. ilellC>e, by (1.3.46), if we let G(z) := CO( l - :)h(:), (1.3.47)

lim (l - :)/(:) = Cil ._1 lim (l - z)h(z) Jim G(:), ._1 ._1 =:

°

The function G(z) is analytic on < 1:1 < l/lql. because G(z) coo' "erges in a. neighborhood about z = 0, its simple pole at : = 1 is removable, and the next largest pole of both I (z) and h{z) is at z = l / q. Thus, from the definitinn of hI:) given in (1.3046) alld the definition of G(:) , ;.. . G() Z= ,lm Cil L...

N_ OQ

{(O l. (01.-, --

... 0 (6)..

(b)n _1

l. '.

with the understanding that (a )_1/(6)_ 1 = 0 , and so, by (1.3.47), G (1I =

N

.

"

J~"" Cil L...

.~

{ (O). (0). - , (b)n - (6)" _1

l

.

(1.3.48)

(a)N (a)"" = Cil J~"" (b) N '" Cil (6)",,'

On the other hand , by (1.3.37), ( \.3.49 )

(a)",, (ql a)""

lim (1 - :)/(:) '" () >_1 q 00 (' j a I 00 .

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

(b)..,(q/ a)oo ( 1.3.50)

copC;righlf1.'tJAJe/isiJ/'

B . C . BERNDT Putting (1.3.50) in (1 .3.45),

Wi!

CQmpl~te

the proof of ( \.3 .36).

Ex ercise 1.3.13. As a corolla'1l, convm Theorem LU :.! mto following $!f'Rmetri~ form not involving a bilat~ral sm ..... Co rollary 1.3.14. If l/Jql < Izl <

1/llIql,

0 th~

then

(1.3.51)

Exercise 1.3.15. Pro~e that h cobi'$ tripl~ produc/ identity, (1.3.11), can be dmucMfrom Ramanujan's l¢l .rummalion theorem. lfinl: fiNI let b = 0 and replace a bye. Now let:; = - ble and q = ab, and !he! /de _ O.

Exercise 1.3. 16. Prove that the q-binomiallhem-em, Th eorem \.3.1 , is a special ca6e 0/ Ramanujan" ,-WI summation theorem. We close this chapter with one further major identity, the quintu· pie product identity, which is enormo~ly useful in the theory 01 theta functions. We foHow our proof of the quintuple product identity with proofs of two beautiful corollaries tha t are analogues of Jacobi's identity in (1. 3.24). We use t hese tv.'O corollaries in Chapter 6 to help derive certain modular equations of d~ 3. The quintuple product identity can be formulated in se,-eral ways, and we gi\-e three of them below. We provide a proof of the identity in the first setling. Theorem 1.3.11 (T he Qu intuple Proouct Identity; First Version). Far z ",0,

L ~

(1.3. 52)

In' +n(:J" q- 3n _

z - 3.-1 q3'Hl )

S PIRlT OF RAMAN UJAN

19

Theorem 1.3.18 (The QUintuple Product Identity; Second Version ).

Far (I "# o. ( _aq; q)«>( _ I f a; q)co(a2~; q2)oo(q /a2; q2)",,( q; q)«>

=

(I _ I

(a 3q; tI )",, (q2/ a3; q3)",, (q3 ; til""

+ (a3q' ;q3)oo(q/ a3 ;q3l",, (q3;ql)"". Theorem 1.3.19 (The Quintuple Product Identity; Third Version ). We /wile 2 3 13 " ) f( _ x , >,x)/ ( _.u ) = f ( _ .\2X3 - >.x6 )+xf (- >. _ >.2X9 ) ( _. I ( X, Ax' ) , , To deduce (1.3.53) from (1.3.52), :-eplace q2 by q, set z = -aJq, and employ the J acobi triple product identity (1.3.10). Setting a = l / z. and a 3 = >.Jq in (1.3.53) and utilizing the Jacobi triple product identity again, we deduce (1.3_54). Proof. Let I (z ) denote the right side of (1.3.52). Then, for 0 < 00, we can express/ ( ~) as a Laurent a.eries

fh)m

the definition of

I, we find I ( l / ~) =

1::1<

that _ Z - 2 f( ~),

or, from (1.3.55), ~

L

a"z~ - 2

Equating constant terms, we deduce that ao = - (12 , and equating ooefficients of ~ - I, we find that (I, = O. From the definition of I, we also find that

Thus, from (1.3.55),

8 . C. BERNDT

20

Equating coefficients of zft on both sides. we find t hat, for each integer

E xercise 1.3.20. By itemtion, prove that, for every

(1.3.f>6)

(l3n

= q6 n_ 5 (l3n -

3

= ... = q3n' -

{1.3.57}

03n + l = q6n-3 aan _, = ...

(\.3.58)

(l3n+2 =

q6n -

1

(1Jn _!

=

;nttg~

n,

2n"A ....."

qa n >(I\,

= .. . - q3n'+2" (12·

By the Jacobi triple product idelllity, Theorem \.J.3,

(".; ,/)""f( ~)

=

I( - ~ 2 • -

'" L

( 1.3.59)

,,"

i <')/( - '1< , - '1./<)

(_ l yH q'j{j-Il+*' :2j+k.

j.*~-""

Equating constant coefficients on botb sides of (l. 3.59). we find that ~

(q.;q.)",<1Q =

L

( _1 )i qGl' - 7) = (q' ;q4)"".

) _ _ 00

by the pentagonal number theorem, Corollary 1-3.5. Hence, ao = I, and since a-, = -Oft. " 'c also deduce that ~ = -I. Recalling also that 01 = 0, we conclude from ( 1.3.56)-(1.3.~) that, for each integer n,

Putting these ''alues in (1.3.55), we wmplete the proof of (1.3.52), after replacing n by - n- I in tbesecond Sum arising from (1. 3.55). 0 Corollary 1.3.21. Rec4U /hat rp(q) and I( - q) all': defined in (1.2.2) and (1.2.4), Tr.s~cti""ly. Th en (1.3.60)

-

L

(6n

+ \ )q3n>H = ",,2( - l)/( _ q1 ).

Proof. Divide both side> of ( 1.3.:>2) by 1 - q/ :. By L'Hoopital's rule,

SPIRIT OF RAMANUJAN lienee, lett ing z tend (1.3. 15), "-e 6nd t hat

to q

21

in (1.3.52) and employin& {1.3.13} and

~

L

(6n

+ 1)q3"· ...,, =

(.r;q2 )!,(~ ; qO)!, .. ,. l( _q2)t( - ,I),

0

and thill complete!! the proof. Corollary 1.3.22. If >J!( q) and (1.2. 4), ,.upecLiv~llI, then

f(-~) (l~ d~fined

by (1.2.3) and

~

L

(1.3.6 1)

(3n

+ l )q3..• .. 1>o

_

\6(q2)J2 ( _q).

p roor. We again use the quintu ple product identity. Dj\'ide both sides of ( 1.3.52) by I - =2 and let : tend to 1. F"il"$l note that as z __ I, the left side of (1.3.52) tencb to

by (1.2.9). Thus, we apply L'HoopitaJ'. rule t.O deduee t hat

..3"'+" I"',-3n _ ,, _s.. _lqln+l)

,..""

lim L.. ... _ .., 'I 0_ 1

--~ C~""

1

3nq3n·- 2..

'" ~ Ct-"" 3~'+2'" _L

:2

+ ..too{3n + Ij,r· + ..

H

..... )

~oo(3m + 2W""+

lfoo )

~

. ..-..,

(3m

+ 1)q'''''.;.2""

.... here, in the antepenultimate line, "'"e set n '" - m in the forme!" sum and n '" - m - I in the latter sum . Ikmembering that ..."e have divided both sides ha\"e let z - 1, ""e find

B . C. BERNDT from the calculation above, (1 _3_14), !IIId (I.3.1&) that ~

L

{3m + l)qJ ... ·+2m

=

(q7; q2 ).,.,(q, q2)~{q4; q4);,

which completes the proof.

=

>/J(q7)f7{ _ q),

o

Although we have established only a few theorems about q-series a.nd theta functions, we have developed enough machinery to prove some beautiful, significant theorems aoout the arithmetical functions introduced at the beginning of this chApter.

1.4. N otes For an introduction to the element&y t heory of partitions that is eminently accessible to undergraduates, read Integer P(lrtitlo7l$ [23J, by G. E. Andrews and K. Eriks!lon. The most authoritative account on the theory of partitions is also by And rews [14], several portions of which are suitable for undergraduates. Hardy and Ramanujan [110J, [192, pp. 276- 309) IICtually provoo more than (1. 1.6); namely, they derived an infinite 8.Symptotic !leri... for p{n) in decrea.sing exponentiAl functions _ They suspectoo that their series did not converge, but it was IlQt until 1937 that D. H. Lehmer (140) shov.--.!d that it diverges_ In 1936, while preparing his lectures for a graduate course in analytic number theory at the UniveTl!ity of Pennsyh'B.nia, H. Rademacher [181], [18 2, pp. 108121] found an exact infinite series replesentation for p(n). At about the same time, A. Selberg [209, pp. 695-706], [52 , pp. 203-213] also discovered this same exllCt formu la. hut he never published his "'Ork. In their classic paper [110], [192, pp_ 276-3091, Hardy and Ra· ma.nujan int roduced their famous Mcircle method," which remains today as the primary tool of number theQrists using analytic techniques in studying problems in additive number theory. T he principal idea boehlnd th.. "circl.. metho-d" co.n be found in R1r.manujan '~ notebooka (193, pp _ 362- 363), although he did not rigorously develop his idellS. See Bernd t 's book {3 7, pp. 60-66] fOI a discussion of Ramanujan's first attempt at fonnulating the circle method. Despite its genesis in Ramanujan 's . the Hardy- Littlewood

S PIRIT OF RAMA NUJAN

23

circle method , because Hardy and J. E. Littlewood ext.ensi\"l~ly developed the method in a series of P O.~ Although not stated as a oonjecture, since that time the non~-anishing of T{n) h& been known as Lehmer'. <:Qnjecture. T heorem 1.3.1 , the q-binomial theorem , is due to A. Cauchy [66, p. 45J, while Corollary 1.3.2 W!lS first proved by L. Euler ]91 , Chap. 16J. The Jacobi triple product identity was pro,'e(\ by C. G. J. J acobi in his Fundamenta Nova [13( , the paper in which the theory of elliptic functions was founded and one of the most imp<:>rtant papers In the hIstory of mathematlCll. H owe~"er, the JacobI trIple product identity "'lIS first pro,'e(\ by C. F. Gauss (D7, p. 464]. The proof that ,,"e ha'"e gi'"en here is independently due to Andrews (12] and P. K. Menon [ISS]. F. Franklin devised a beautiful oombina_ torial proof of Corollary 1.3.5 (or Corollary 1.3.8). Sec Hardy and Wright's book ]112, pp. 286- 287J or Andrews's text [14, pp. lOl l] for Franklin's proof. J acobi's id~ntity, Theorem 1.3.9, has an elegant generalization found by S. BbargaVl\, C. Adiga. and D. O. Somashekara ]60]. Ramanujan·s ,1/1, summation theorem was first stated by Ramanujan in his notebooks 1193, Chap. 16, Entry 17J. In fact, he stated it in precisely the form gi'"en 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 [107 , pp. 222- 223J. Hov.'(!,-er, it was not until 1949 and 1950 that the first published proofs were giwn by W. Hahn (103] and M. Jackson [l30], respecth·ely. The proof that we ha'"e gi'"en here is due to K. Venkatachaliengar and was presented for the first time in the monograph by Adiga , Berndt, Bharga,.,., and G. N. Watson [5]; see also Berndt's book ]34, pp. 32-34]. As with the other theorems in this chapter, there are ]lOW many proofs of the 11/1, the<;>rem. Almost all proofs employ the q_binomial theorem at some stage. W. P. Johnson !132! has written an interesting semi.... xpository paper showing how Cauchy could have discovered Ramannjan's 11/11 SUmmation formula (but did not). Ramanujan left no clues about his proof Copyrighted Material

B.C. BERNDT

"

or proofs . However. i~ is quite possible that Ramanujan used the theory of partial fractions, which is a generalization of the method of partial fractions all calculus students learn to e\'3JuaU: integrals of rational functions. For a proof of Ramanujao 's I "'1 summation theorem employing partial fractions , see a paper by S. H. Chan [13J. but this proof also utilizes the q-binomial theorem.

Readeni might consider the following instructive exercise. E xercise 1.4.1. Consider the Laurent upansion

f1J

~

L

=~ "" :

1( : ) : =

..

ProceedIng Ilk we dId theorem. ded..a t1UJt

(1.4.. 1)

In

the proof of !Wm,,,nIJon'$ 1\iiI summa/wll

(, j di. c"

CR Z ~ .

~_ oo

= (q' /(bd» ..

(').

b

CQ,

- 00


<00.

It follows that (1.4.2)

1(: ) =

C(J ..

;c- (q'(,/{j di. (") " £:-"., M))" b .

Applying Ro"ulRujan's ,.", summa/;"" thwrem 10 the right nd~ of {1.4.2l . we find thot ( 1.4.3) (0':: jcc ( bj z l ee (a.q z I(M ))", (bd/( (1'::) loo ( q )",, (q'/(ab) )""

(dz )""( q/(dz

n"" =

C(I

(q~ / b)",, (q/(,,-z ))"" (q1/(bd))"" (qd/ a ),,,, .

HO weVf!T, aamining the ~eroJ ami poit:.. on wch side of ( 1.4.3), we sa that ( L4.3) ;., not /nre. Thu.s, (1.4.1) mu.st be wrong. What;" wrong with this argument ? If you nen! help. Jee 126]. The q-binomial theorem is a special case of a product represt'.nation of a basic hypergeometric series, whe~ here basic is not mea nt to be a synonym for fundam et!/al but instead refers to the q R'--"' t:h1y. in e"nprRl , ,. bask hypergeometric series J¢k h"" j q-products in the numerator and k + 1 q-products in the denomina· tor, where One of the q-products in tbe denominator is (q;q)n, if n is the ind(lx of summation . Clear and more extensive introductions to q-S
ha..,.

SPIRl T OF RAMANUJAN R _ Askey, and R. Roy 118, Chap. W]. By far, the moot comprehensive t reatise on q-.series is by C. Gasp-er and II I. Rahman (96]. An introduction to Ramanujan's exteJl.'iive oontributioJl.'i to theta functions CIIn be found in Berndt's lecture notes 1361. The interaction of q-series with number theory is best demonstrated in N. J_ Fine's beautiful but succinctly written mono~aph 194]. &nne illuminating historical remark'! on q-series are given in Roy 's review 1200]. The quintuple product identity has been discovered many times in the past century, and an account of all proofs known to the author in 1991 can be found in [34, p_ 83]. However, a mOre comprehensive survey discussing more than 2[' known proofs has been writlen hy R. (''''''I'''r [7!l]_ 1'hp formulMion given in ( 1.3_54) i~ that s:lwn by RamarlUjan in his lost notebook (1 94, p. 207). The proof of the quintuple product identity that we hal"e gil"en is a louiant of a proof due to Bharga"a [571, which in t urn is a combi nation of proofs of L . J _Mordell (1631 and L. Carlitz and ),1. V . Suhbarao [63]. A particularly simple and short proof was d isccl"ered \Vatsotl (218]. A similar proof was discxwered by R. J. Evans [921 and reproduced in the author 's lecture notes on theta functions 136, pp. 11-12]. For systematic approaches and »cwral new identiti
To the best of our knowledge, both ( 1_3.60) lind (1.3_61 ) are due to Ramanujan; they can be found in Entry 8(ix), (x) of Chapter 17 in his second notebook (193], 134, pp. 114- 115], and also in his paper [186, e<J - (65 )], (192, p. 1471. T be first published proofs are, llOVo"eI-ef, due to B. Gordon (100]. A proof of (1.3.60) is also gil1!n in Fine's book [94, p. 83J.

Copyrighted Material

Chapter 2

Congruences for p(n) and T(n)

2.1. Historical Background This chapter ill primarily devoted to prming some of Ramanujan '8 oongruences for p(n ) and T(n). We bep.n with II remark on notation. Throughout the cha pter, ,,~shall see congruences of the sort

(2.1.1 )

f (q) = g(q) (mod m),

where f (q) = LD" q" and g(q) = l:h"q" are po...;er series in q. The coug,ueu"" (2.1.1 ) i", <:qui ..alent to t.he "",,,dition
In 1919, Ramanujan [188], (192, pp. 210-213) announced thM he had found three simple congruences sa.tlsfied by p{n), na mely,

(2.1.2)

p(&n

+ 4) == 0 (modS),

(2.1.3)

p(7n

+ 1'»

(2.1.4)

= 0 (mod 7),

p{lln + 6) ==

o(mod 11).

He gave proofs of (2. 1.2) and (2.1.3) in [188] and later in II short one page note [1901. [192, p. 230) announced that he had also found II IJlovf of (2.\.4 ). He ,,1/10 remark.. in [1001 that ~It appears thM. thl'," life no eqUILlly simple properties for an~ moduli involving primes other than these three. ~ In a poothumously published paper [1 91[, [192, Copyrighted Material

27

B. C. BERNDT

28

pp. 232- 238], Hardy extracted diffcren; proofs of (2 . \.2)- {2.1.4) from an unpublished manuscript of Ramru.ujan 011 p{n) and 1"(n) [194 ,

pp. 133-177], [50). In [188], Ramanujan offered II more general conjecture. Let.; = S°:r> II C and let >. be an integer such t.'Jat 24). ;;; I (mod 6). Then (2.1.:;)

p(nO +,\,) ::;; 0 (mod 6).

In his unpublished manuscript [194, liP. 133-I77J. ISO), Ramanujan gave II proof of (2.1.5) for a.rbitrary II and b = c = O. He also began II proof of his conjecture for arbitrary band 11 = C = 0, but he did not complete it. If he had com pleted his proof, he would have notioo::l that hi~ CQnjectur<: in this """" nudod UI b
formulated his oonjoct ures after studying II table ohalnes of p(n), O :; n < 200, made by P. MacM ahon. After Ramanujan died, H. Gupta extended MacMahon 's table up to n = 300. Upon examining Gupta'. table in 1934, S. Chowla (75] found that P(243) is not divilsibJe by 73 , despite the fact that Z4 - 243 = I (mod 73). To correct Ramanujan·. conjectllN!, define 6' '" 5~7·'lIe, whue 1/ = 1>, if I> = 0,1,2, and II = ((I> + 2)/2), if I> > 2_ T hen

(2_1 .6)

p(n6 +~) ;: O{mod 6').

I" 1938, G . N. WlI\.IIOn [2 18) published II proof of (2.1.6) for a = C = 0 and gave a mQI"l detailed version of Ramanujan's proof Qf (2_1.6) in the case II _ C = O. It was not until 1%7 that A. O. L. At kin 128] proved (2. 1.6) for arbitrary c and a = ~ = O. The tau function r (n) was introduced by Ramanujao in his famous paper (186), (192, pp. 136-162). Although he proved little about r(n) in this paper, he did Formulate some fundament al conjectures about r (n). 10 (190), RamanujaD st ated ,,';thout proof congruences for r(n) modulo 5, 7, and 23. In his unpublished manuscript on p(n) and r (n) (194, pp. 133- 177], (50]. he proved these congruences and several Further results on r(n).

2.2. Ele m entary Con gruences for T(n } First, we show that

rtebPym)RPJ8 &f&ena l

SPIRI T OF RA MANUJAN Theorem 2.2. 1. odd equals

Th~

"umber of values QJn::;:tO Jor which r( .. )

(2.2.1)

[I

Ii'here [:to)

d~noles th~

1$

+,"'].

grealest integer leu tho .. or equal to:to.

Proo f. Observe that, for any positive inkger j, by the binomial the-orem,

Hence, (q, q)~ ;;; {q8, q8)"" (mod 2).

Th,.,re{ore, using the definition of r(n) in (\. \. 10), the congruence aoove, and Jacobi 's iMnlity, Theorem 1.3.9, we find that ~

L T(n)qR = q{q;q)~ ;;; q(q8 ; qS)!, ~

=

ThIl8 ,

l :(-1)R(2n + l)q(2n+1 I' (mod2 ).

"-.

r(nl is odd or even accordiog

a!:I

n is an odd squaro Or not.

Exercise 2.2.2. A s an elemenlary exercise, show thai the number of odd &quares less Ihan or equal 10:tO is prfl:iselll (2.2.1 ). So the proof of Theorem 2.2.1 is complete.

o

Theorem 2. 2.3. For ooch nonnegoli"" inleger n, (2.2. 2)

T( 7n ), T(7n

+ 3), T( 7.. + 5 ), T(7n + 6) ;;; 0 (mod 7).

Proof. Applying the binomial theorem, as we did in the previoll8 proof, we eMily deduce that (q; q)!o

!!

(q 1 ,
and so

L T(n)q" = q(q;q)~ = q(q;q)!, (ql; ql );!., (mod 7). ~

(2.2.3)

n~ l

Copyrighted Material

B. C. BERNDT

30

Since the powers in consider

(q7;q7)~

are all multiples of 7, we need only

~

(2. 2.4 )

q (q;q)~ =

L (- I)" (2n+ l )ql-l-.. (.. +I )/2 ,

"-.

by J acobi's identity, T heorem l.3.9. Observe t hat 1 + n(n + 1)/2 :; 0, 1, 2,4 (mod 7). Moreover, 1 + n {Ol + 1)/2 E 0 (mod 7) if and only if n = 3 (mod 7) or 2n + I = 0 (mod 7). It now follows from (2.2.3) and (2.2.4) that ;(7n) ;;; o(mod 7) . It aho follows that t here are no po'JI<eT'S of q on t he right side of (2.2.4 ) that are oongrueot to either 3, 5, or 6 modulo 7. The latler th~ oor.gruences in (2.2.2 ) thus follow &om (2.2.3 ) and (2.2.4). 0 For example, T(3) = 252 ;;; o(mod 7), T(5) = 4830 = O(mod 7), 1"(6) = - 6048 = 0 (mod 7), and ,.- (7) = -16744 :; 0 (mod 7). The reader will nilsen.., that 3, 5, and 6 ErrC the quadratic nonresidues modulo 7 and that indeed the proof (If Theorem 2.2.3 demonstrates t his.

Theorem 2.2.4. Let T , 0 < r < 23, denote any quadrntic resid\l.e modulo 23. Then, lor each positim: ifl/eger fI ,

(2.2.5 )

-r(23n - r ) :: 0 (moo 23).

P roof. By the binomial theorem,

Since t he PQ"'ers in (q2l; q2J)oo are all multiples of 23. ,,~ nero only consider ~

(2.2.6)

q(q; q)oo =

L

(_ I )·qL+"(3n +l)/~,

by Euler's pentagonal number theorem (1.3.18). Obser.e that (2.2.7) 1+ n{3n + 1) =

(6n+l ) ~ - 23f1(3; + 1)

::

(6f1+1 )~ (mod :l~).

2 T hus. -rem) will be a multiple of 23 when m is not congruent to a sqUanl modulo 23. In other words, if ..re set m = : 23,., + t = k 2,0 $ t < 23 , for some intes:,ers fI , t, aL)d ~ then t mUSt be a qUadr3t~ Copynglitea Mdlenal

SPlRlT OF RAMANUJAN nonresidue modulo 23. Recall tbat r is a quadratic residue modulo 23. Since 23 '" ~ I (mod 4), - r, r # 0, is a quadnltic IIOllresidue modulo 23. It follows that (2.2.8)

1"(23n - r) ;;; 0 {mod23),

O
o For example, 3 and 6 are quad ratic residues modulo 23, and so r(20) = - 7109160;;; 0 (mod 23) and r(11) = ~ 690ft934 - O{mod 23). We sballlater examillc COllgruences for r(o) modulo ft. However, to establisb tbese congruences, we first need to prove Ramanujan's famous congruence for p(n ) modulo ft.

2.3 . Ramanujan's Congruence p(Sn + 4) 0 (mod 5)

=

In this monograpb, we present four proofs of Ramanujan's oongruence for p(nj modulo 5. We offer tbree prods in tbis section. Tbe first and the second are more elementary tban tbe tbird , but tbe tbird gives more information. Tbeorem 2.3,1 . For eoch (2. 3.1)

non~ati~

p(5n + 4)

integern,

= 0 (mod ft).

First Proof of Theorem 2.3. 1. Our first proof is taken from Ramanujan 's paper [188J, [192, pp. 21G-213] and is reproduced in Hardy 's book [107, pp. 87--88J. We hegin by writing

By the binomial theorem, (2.3.3) (q;

q)~ ;;; (of; q~}"" (mod ft)

(f' q;~""

or '" 1 (mod ft). Copyrighted Material q,q ""

8 . C . BERNDT

32

Helloo, by (2.3.2) and (2.3.3),

q{q;q):" '" (q' ; q') ~

(:l.J.4)

-.., L

p(m)q..... ' (mod 5).

We now see from (2.3.4) that in order to allow tbat p(5n + 4) s: O{mod 5) ~ must show that 1M cotfficienl$ of ,f~ H on th~ left side of (2.3.4) are multiples of 5. By t he ]J.elltllgomd number theoreP! , Coronary 1.3.5, and Jacobi'.

identity, T heorem 1.3.9,

-

L- -

q(q; q)!, =q( q; q )",,(q; q )!, =q

L

(_ l YqI (3,i+ 11/l L (_l)l (2k+ 1)qk(HI )f 2

, __ ""

=

l aO

L (- I Y-+~ ( 2k + l )ql+1C 3J+ IJfHl (l+I ),7

Our ob,iecth-e ill to deteTmine ",ben tile exponents on the right Bide are muWple$ of 5. Obsen.'e that

20 + 1)2 +(2k+ I ) ~ '" 8 {1 + !i(3j

+ I ) + 111"(k+ I J} -

IOi' - 5.

Thus, 1 + !i(3j + I ) + ~k(k + I ) is a multi ple of:; if and only if

+ {2k + 1)2 . 0 (mod 5). It is easily checked t hat 2(j + I )' == 0,2, or 3 modulo:; and that (2k + 1)2.i1 0, 1, Or 4 modulo 5. We therefore see tbat (2.3.6) is ~ 2(j + 1)2

(2.3.6)

if and only if 2(j+ l )2~O( mod5)

and

(2k+l ) ' . O(mod5).

In particular, 2k + I ;: O(mod$), which , by (2.3.5), implies tbat the oocl!ident of tf nH , "
a

&ive a second simple proof due to G. E. Andre",. (15! lind based on the simple lemma give.. below. See aI.Io &n extensi'1l generaiiulion of t his lemma by Andrews &nd R. Roy (24J. In particular , taking" special case of t heir general theorem, Andre .... s and Roy establish the oongrne~ p(7" + ~l • O(mod 7). COpyrighted M;i'enat We

[lOW

SPIRJT OF RAMANUJAN

33

Lem m a 2.3.2. Let {a,,}, n > 0, ~ (lny UqUffltt of integef'$. Th.en the coefficient of q5n+3, n > 0, ill ~

L(q):= ( . I )2

(2 _3.7)

q,q

I >"qn'

00 •.-0

i.! divisible by 5 _

P roof, Write (2.3.7) in the form

L(q) = (q; q);,

L""

'(;f;n,q

m'

;: (q; q);,

E'"

m'

~~;):

(moo 5),

by tbe binomial theorem. Using Jacobi's identity, Theorem 1.3.9, ....e ~IJU:; ..,., d".l i~ ~uffi<= to ex .."""" tile coefficient of q$n ..., in ~

(2.3.8) (q; q)!,

~

~

L (lmq"" = ,., L {- I)j(2i + 1)t/U+ll/2 L a",q"".

m"

We .... ant those terms aoo''C for whicll j0 + 1)/2 + m 2 _ 5n + 3, where fI 2: O. It is ea.sy to see t hat this oondition is equivalent to t he oongruence (2.3.9) {2j + lj2 + 3m 2 ;: 0 (mod 5). Since (2j

+ 1)2 :;; 0, ± l (moo 5) and 3m2 :;; 0, 2, 3 (mod5), '\\'C lief! t hat

(2.3.9) ho ld. <mly wheD

(2.3.10)

m :;;

2j

+ I ;: O{moo5 ).

The coefficients of q5"H in (2.3.8) are then oomposed af terms of the sort (- I )j{2j + l)a m, which, by (2.3.10), are all multiples of $. 0 Exercise 2 .3.3. Use (1.3. 13) to 5hoIJ.' that (2.3.11)


{q;q)"" .

{ q:q)""

Second Proof of Theore m 2.3.1. Using (2.3.11), we find that

;;;;-;;;-;:0' :;:-".- = {q;q)""(-q:q)",,

1 ( q; q )(q;q)~ { q;q)""

B. C. BERNDT By Lemma 2.3.2, the coefficients p(k) on the left. side above are multiples of S whene\ff 21<: ;: Sj + 3 (modS), i.e., whenever I<: = Sn + 4. Thill then oompletes ou r S
of

p(Sn + 4)q" =

"..0

~ (.r; q~!~ . (q; q )""

Proof. We begin by writing (2.3.13)

(qI /$;q"~)QQ

(q5;
To see this, use the pentagonal number theorem (1.3.18) in the nu· merator on the left. side of (2.3.13) and, if n is the index of ~ummation, divide the terms into residue classes modulo S. If n _ 0, 2 (modS), then the pov.-enl are integral, and these. when d;"ided by (,f; ,f)oo, account for the terms in J ,{q). If n ;: 3,4 (modS), then the powers are of the form k + 2/S, where k is a positive integer. Le _, we obtain the terms J1(q)q2/5 . Exercise 2.3 .5. Lastly, when '1 = 1 (mod 5), use the pentagonal num~r thwrem, Corol/ary 1.3.5, to 8how that !hue lefTT18 in (qI/\ qI / 5)"" are equal to _ ql/5(,f;qS)"". lIence, we obtain the term _ql/S in (2.3. 13). Now cube both Bid"" of (2_3.13) to obt ai" (ql/S;ql/S)!,

",""-'..d" (qS;q~)~

(2.3.14 )

= (Jt - 3J~q) - ql/5(3J ,7 -

J,',J + 3J1, '''(1 + J,J1)

SPIRIT OF RAM ANUJAN

35

wbere for brevity we have deleted the argument q of J 1 and h. Next, we use Jacobi's identity. Thoorem 1.3.9. in the numerator on the left side of (2.3.14 ) and repeat the same kind of argument that we applied with tbe pentagonal number thoorem in (2.3.13 ). We subdivide the indices n of the sum in the numerator into residue classes modulo 5. As an exercise, readen! $hould use J llOObi 's identity ( J .3.24 ) to show that the oontributiollli of the terms with n 2 {mod5) are equal to 5 (q\qh )!'.r/~ . Thus, ,,~obtain &II equality of the sort.

=

(2.3. 15) where G, (q) ami G2 (V) are VV"..,r ....-i"" with iul."l,ntl

1-"''''''''''

I1ml

integral coefficient.!!. Hence, equating coefficients on tbe right sides of (2.3.14) and (2.3.15), we find that 1+ 6J,J2 = -5,

From allY one of these equalities, we deduce that (2.3.16) Our next task is to use (2.3.16), (2.3.13), and to show that

"rationalization~

(2.3.17)

(q';q'I_ (ql /~; 9"' /~ )""

J,

=

1 q' /5 + J 2q 2/5

(Jt + 3J2 9") + q' /5() ? + 2JJq) + q11$(2Jf + JM Jl 11q + q2Jf .r/~ (3J,

+

Jf

We oow demonstrate how to

+ Jtv ) + 5q4/$ llq+q2Jt pr()\~ th~

second equality in (2. 3.17). Return to (2.3.13) aod replace ql/' by wq' /5 , where 0.1 is any fifth root of unity. Thus, (2.3.18)

B. C. BERNDT

36

Let w run through all five fifth roots of unity and multiply all J"l.' su<;h equalities (2.3. ]8) to obtain

(2.3.19)

II....

(

1/5 )

1 /~.

""q( 5 ~ ~q)"" ""

=

q ,

II {J:(q) -

wql /~ + J~ (q )""lq115 }.

OJ

First examine the product on the left Jide of (2.3.19). Using the fact that the sum of the five fi fth roots of unity equals 0, we see that if n is not a multiple of 5, we obtain products of the form (2.3.20) ( l - q"/5)( 1 _ ",q"/5)( 1 _...,2 q"/S)( \ q,,/5)( 1 _ ",4q" /5) = I _ q".

_...,J

However, if n = 5m , then the corresponding terlllB are

inswad of I - qn that we obtained in (2.3.20). T hus, we find that

(2.3.21 )

IT...

(wqI/5, ,,,qI/ 5),,,,,

(q5 ;
=

We nOW examine the product on the right side of (2. 3.19). Since th~ are no fractional PO"l.'fS of q on the left side of (2.3.19) by (2.3.21), there are none on the right side 88 well. T hus, we only need to ex8.ll1i~ t ........, t~rm~ in t.h .. [>rOOnM, th~t give .is<' to int egral pm<~ of q. A brief inspectioll then convinces \18 thai (2.3.22)

II {),(q) -

-

wq"~ + )2(q )w2 q2/~ } = J~(q) - q+J~(q)q~ + C,q+C1q,

kL)

= 20 terms of the form -J, JzWI"-':lWJW~ = W'''-'2WJw~, and C, is the sum of th e b.~.2) '" 30 terms ofthe form - J? JJW1W?wj = -w,w~w~, since J,}z = -1. ilere, for each j, wJ is Ii fifth root of ullity, and in each product w; oF "'. if j oF k. First, examine the terms "'1"-':lWJ"'~ = ",.",;-'. Now , "'here C, is the sum of the

-;;-

L...

..........

_I

"'4w~

=

_ J

- w~w5

= - J,

where the SUIIl is O\"(!r the four fifth roots of unity W., except W$. Since there are fi\"(! possibilities fo~~. 'Vi:. IXmcluie that C, = - 5. Second, Copynghted Malena

SPIRlT OF RAMA NUJAN

37

--, ............ ,od

L

..... ,,"'.

(w?w~

+ w~w})

"" -<4w~ - w~w~ = - 2,

There are five jXl'iSibilitie1l for W;j, and 50 it would seem t hat ""e obtai n a oonuibution of - 10 to the val ue of C2 . Hov.'ever, beca~ of the symmetry of W2 and W;j, ""e have counted each contribution to C 2 t""ice, i.e., in fact, C 2 = -5. Using our values C I = -5 = C~ in (2.3.22), we deduce that (2_3.23)

IT {J, (q) - wq' f5 + J2(q)w'e /5 }

J N q) _ llq+J~(q)q2 .

=

• In summary, from (2.3.23), we have shown that

",,,,::-;;ri':;:-u;;w,, = D"'JH PI (q) J1(q)

ql /~

fL {J,(q)

+ J 2 (q)q2 /&

=

wql/~ + h(q)",2q2 / S}

+ J2(q)w2q2/~} fL ,,, {J 1(q) _ ",ql /5 + h(q)w 2 q2/&) Jf(q) llq + JN q)q2 ",ql/~

F(q) J Nq) - llq + J~(q)q2'

(2.3.24) where F

_

J~(q) - llq + JHq)q2

ql/5+J2(q)q2/S' If we consider the numerator and denominator above as polynomials in ql /~ and use long division, ...'l! find that F(q) indeed is t he numerator on the right side of (2.3.17). Thus, by (2.3.24), the proof of (2.3 _17) is complete. Recalling that on t he left side of (2. 3.17) 1/(qI /S; q\ /S)"" is the generating function for p(n), we select \hose terfDB 0 11 both sides where the p<)OI'l!TS of q are congruent to 4/ 5 modulo 1. We ~h"u t1ivitl" wtll sides by q'/s to fi nd that (q )- J1(q)

,

~

(2.3.25)

(q\!)""

L p(:>n + 4)q" = 'J~'(");-"ll~q~+-J"I7:(q"'W'

eo~yrfghled Matena\

q

B.C. BERNDT

38

However, from (2.3.19), (2.3.21 ), and (2.3.23), we 3lso know that (2.3.26)

(q:;; q~)~

(q; q )~

Utili.,.ing (2.3.26) in (2.3.25), we complete the proof of (2.3.\2).

0

Theorem 2.3.4 can be utilized to provide a proof of Ramanujao 's congruence for pin) modulo 25.

Theorem 2.3.6. For we')' (2 .3.27)

p(Z:in

nonntgoti~

integer n,

+ 24 ) ;: 0 (mod 25 ).

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

-

""" find that (2.3.28)

-

:~:::>(5n + 4 ) q ~ n_ O

= ,}

(")' q ;9 "" : 5{.r; q~ )t, L p(n)q~ (mod 25). (q; q )""

n_O

Fh>m Theorem 2.3.1 we know that the coefficients of q4 , q9 , ql' , ... , tf"H , ... On t he far right side of (2.3.28) an'! all multiples of 25. It follows that the coefficients of ~nH,n ;:: 0, on the far left. side of (2.3.28) are also multiples of 25, Le. , p(25n

+ 24) :: 0 (mod 25 ).

o

This completes the proof.

Ramanujan's oongruence in Theorem 2.3.1 yield!! a simple proof of Ramanujan 's congruence for tbe tall function modulo iI, as we next demonstrate. Theorem 2.3.7. For ooch (2.3.29)

nonn~ati~e ;nj~er

r("n)

E

n,

o(mod " ).

Proof. By the definition of r (n) and the binomial theorem , (2.3.30) ~ (q. q)2:> 00 "- r (n )q" = q(q; q) ~ = q (q: )eo = q( q~; q~)~ p(n)q" (mod,,). ,, _I

copyngn1e~ Malenal

L

" ..0

SPIRJ T OF RAMANUJAN

39

By Theorem 2.3.1, the coefficient of t ",n > I, on the fa.- tight side of (2.3.30) is a multiple of 5. Thus, the coefficient of ,;;" on the fa:r left side of (2.3_30) is a mUltiple of 5, i.e ., 1"(5n) = 0 (mod 5). 0

2.4. Ramanujan's Co ngruence p(7n + 5) _ 0 (mod 7) We consider next Ramanujan's congruence for p(n ) modulo 7. Theorem 2.4.1. For fW.ch nonnegalitte inlegl!T n ,

(2.4.1)

p(7n

+ 5) E

0 (mod 7).

pn....,r. Our pr ...... r i~ "-&,,iu tahu [wm Ra",,,"ujall'~ pa!",r 11881 1111u

was sketched by Hardy 1107, p. 88J. First, by the binomial theorem, q2 (q7 ;q7)",

f:

p(n )q" = q2 (q7 ; q7 )", = q2 (q; q)!, (q7; q7~"" .. ~n {q;q)"" (q;q)""

(2.4.2 )

E q'(q; q)!, (mod 7).

Hence, if we can show that the coeffici~nt of q7n+7, n > 0, in q2 (q; q )~ is a multiple of 7, it will follow from (2.4.2) that the coefficient of q7"+7 on the far left side is a multiple of 7, i,e. , p(7n + 5) = 0 (mod 7). Applying Jawbi's identi ty , Theorem 1.3.9, we find that q2 (q; q)~ = q2 {(q; q)!.,}2

(2.4.3 ) As we saw in t he previous parllgraph, we want to know when the exponents abo>." are multiples of 7 Now o~r\'e that

+ 1)2 + (2k + If = 8{2 + !JU + 1)+ !k(k+ In - 14, and so 2 + !iU + I ) + !k(k + I ) is a multiple of 7 if and only if (2j

(2.4.4)

(2j + 1)2 + (2k + 1)2 _ 0 (mod 7).

We easily see that (2j + If, (2k + 1)2 := 0, I, 2, 4 (mod 7), and so the only way (HA ) can hold is if both (2j + If, (2k + If - O(mod 7). In such cases, we trivit8'p~_fg}U~li~IDcients on the tight side of

B. C. B ERN DT (2.4.3) art:' multiples of 7. Hence, the cocffi~ient of q7 n+T,n ~ I, QIl the left side of (2.4.3) is a multiple of 7. As we demonstrated in t~ foregoing pal"agraph, this implies that p{7n + :J) ;;; 0 (mod 7). 0 We now consider the lUIalogue of T hoorem 2.3.4 for p{7n + 5), which was staled without proof by Ramanujan in his paper [188). In his unpublished manuscript on p{n) and T(n ), he gh"eli a '"Cry brief sketch of its proof [194, pp. 133- I77J, [50J. Then! are now several proofs of Theorem 2.4.2, but t~ detaiJ..; of RamlUlujan 's proof wen! worked out Oldy recently by Berndt, A. J . Vee, and J. Yi [54). Because the details are cumbersome. we provide only theoentral ideas and refer n!aders to [54J for a complete proof. Readers should not attempt to oompleu: missing details but merely try to gra.sp the idcillS behind Rarnanujan's proof. Theorem 2 .4.2 . lYe ha tJe

1= p{7n + 5)q" = 7 (q7;(q;qq7~!., + 49q (q7; q7~!.,. )"" (q;q)""

(2.4.5)

ncO

It is clear that T heorem 2.4.1 is an immediate cor(lllary of The0rem 2.4.2. P roof. Using (1.3.18) in both the numerator and denominator and then separati ng the indices of summation in the numerator into residueclasses modulo 7, we ...,adily 6nd that (q ' / 7.,"I 1/ 7) "'" - J +,1 /7J _ , '''+ ,5/1 J

(2.4.6)

(ql;q7)."

-

I

2

3·

where J It J2, and J 3 are power series in q ...·jth integral coefficients, and where the pentagonal number theorem was used to calculate the coefficient of "12/7 . Cubing both sides of (2.4.6), we lind that (qI /1 ;q' /7)!, (q' ;q7 )!, = (J~

+ 3J; JJq -

6J,JJq ) + q l/7(3J~ J 2

+ 3q2/ 7(J1.if - Jf + JW) + "13/7 (Jt -

-

6J, J aq + J~l )

6J,J2

+ 3J,J~q)

+3q~/7(J1 - J? + hJM +3q Sf7(J2 + J~J, - l1q) (2.4.7)

+ q617 (6J1J 2h - I). Copyrigllted Material

SPrRlT OF RAMANU JAN

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_1) into residue clas'leS modulo 1, we easily find that (2.4.8)

( q'I T., q 117)'00 = ,. ' )3

(q

,q

co

GI + ,1/7G 2 + , 317G3 _ 7,'" ,

where G"G 2, and G 3 are power series in q "'ilb integral ooefIicients, and where Jacobi 's identity, T heorem 1.3.9, wllS used to determine the coefficient of q6/7. Comparing coefficients in (2.4.7) and (2.4.8 ). ".., oondude that

(2.4.9)

J,)l - J~+J3q

= 0,

Jl-J~+hJlq

= 0,

J2

+./f)3 - )Iq = 0, .. - 1_

Replace ql /1 by wqlf7 in (2.4.6), where w is any seventh root of unity. Therefore, (2.4 _10)

(wqI/ 7; wql/1)"" = ) + ( ,. ') L

q ,q

00

", '/1 J2 _ w2,1/7 + w $J/T) . 'I 3

Taking the produc t.s of bot h sides of (2.4.10) over all seveR seventh roots of unity, W
(2.4.11 )

« ~: q~~

q , q ""

'"

II... (J, + wq'/7)2 _ wq2f7 + wqS/7 Jd·

Using the generating function for p(n), (2.4.6), and (2.4 .1 1), ".., find

"""

(UI2 )

n. ...

We only need to com pute the terms in 1 (J, +wqh - ,.,f +wqS )3 ) where the powers of q ~t1fflJ~Jlthta'i to complete the proof. In

B.C. BERNDT order to do this, we need to pro\1!~vera! identities using the ideutiti\l!i of (2.4.9). We do no~ give the details. Choosing only those ~rm!:l on each side of (2.4.12) where the powers of q are of t he form 7n +5 &nd using the omitted calculations, we find that ~

PIn )q~

L

=

~

1q.. ·, q..)'co (7 1" q , q )·

q (q7;q7 )~

0<>

(¢~ ; q"'g)!.,

+ 49q' )

,

"~

n.lii~ { mod7)

(2.4.13)

~ I . 9. '9 )' I ~~ . ~9 ) ' '"' p{7n+5},n = 7 Q , q ""'+49q7 Q , q "" . q (Q7;q7 ) ~ (q7 ; Q7 )~

!:o

Replacing q7 by q in (2.4.13), we complete t.he proof of (2. 4.5).

0

Recall that the identity of T heorem 2.3.4 yielded io T heorem 2.3.6 a congruence for pen) modulo 5~. Similarly, the identity in Theorem 2. 4. 2 yields a congruence for p{n) modulo 7', as we now demollSl.rate. T heorem 2 .4 .3. For each n.cmn(9<1tive integer n . (2.4.14)

p(49n

+ 47) =- 0 (mod49).

Proof. Write (2.4.5) in the form

I: P(7n + 5)q" = 7 (q7 j q7)~ (q j q)~ + 49q (q7 j q7 ):" n~(1 (q: q)~ (q: q)~ ~

(2.4.15)

= 7(q7 ;q7):;' L

(- I )"'(2m

+ l )q ", (m+ 1)/2 (mod 4.9),

m o'

by the binomial theorem a nd J acobi's identity (1.3.24 ). We ruw.· 6· amine the terms Orl tile right side of (2.4.1.5) where the poI'o-ers of q are of the fonn 7n + 6. Separating the summands into residue clas>eS modnlo 7, we see that the only terlllS yielding such exponents are when m = 3 (mod 7). But tllen 2m + I - O(mod 7). Thus. the tQefficient of t he JX"'o-er q7n+fi , n 2: I, on the right side of (2.4.15) is a multiple of 49. T he same must be true, of <:nurse, on the left side of (2.4.15), i.e., the coefficient P{ 4911 + 47) must be a multiple of 49. I.e., (2.4.l4.) has been established. 0 Copyrigllted Material

SPlRlT OF RA1\1ANUJAN

43

We shall return to congruences for p{n) afttt we introduce Eilllen· stein series in Chapter 4.

2.5. The Parity of p{n) In cont rast to Theorem 2.2.1 which provides a criterion for determining when T{n) is either e,-en or odd, "'-e know much less about the pa:rity of pen). It has long been oonjectured that p{n) is e--en approximately half of the time, or, more precisely.

(2.5.1)

#-{n < N: p(n) is e\-en} '" ~N,

as N - 00. T. R. Parkin and D. Shanks (178) undertook the first extensi\l: compllta~ionli, providing strong ev idence that indeed (2.5.1 ) is mO$t likely trll<'!. l)espite the venerability of the problem, it was Dot even known that p{n) lISSUmes either even or odd values infinitely often until 1959, when O . Kolberg [13S) established these facts. Other proofs of Kolberg's thoorem ""ere later found by J. Fabrykowski and t-.f. V. Subbarao [93) and by M. Newman [168). In 1983, L . Mirsky [159) established the first quantitative result by showing that (2 .5.2)

#- (n < N : p{n) is e\-en (odd)} >

loglogN

210g 2 .

An improvement was made by J.-L. Nicolas and A. SarkOzY [171!. who proved that

(2.5.3) for some pmiti\-e constant c. In the mOllt rettnt investig31ions, the methods for finding IO'o\-er bounds for the number of OCClIrrenoes of even values of p(nj h8\"e been somewhat different from those for odd values of p(n). Greatly improving on previolL'l result!:! , Nicolas, I. Z. Ruzsa, and Sarkmy 1170) in 1998 prm'ed that (2.5.4 )

#- (n < N : p{n) is even} >.,IN

and, for each { > O.

(2.5.5)

#- (n < N: p(n) is odd }:» .,INe-(q2+.) ...'"to:.~".

(We pause toexplain the notation >. We write F (N ) :> G(N), if and only if there exist.S a &pV~r'.m'~Je~:F that F (N ) > cG(N ),

44

B. C . BERNDT

for all N sufliciemJy large.) In an append ix to their paper 1170], J.- P. Serre used modular forms to pTO\~ that (2_5.6)

= 00.

At present, this is the best known result for even ,.",Iues of p(n). The lower bou nd (2.5.5) has been improved first by S. Ahlgren [&], who utilized modular forrns, and 5eOOnd by Nicolas 1169J, who used !DOre elementary methods, to prove that (2.5.7)

#{n
>

for some po!Iitiw number K . Ahlgren proved (2. 5.7) with K = 0_ An elegant, eleme ntary proof of (2.5. 7) when K = a was established by O. Ekhhorn [S6]. The lower bound (2.5.7) is currently the best kno"o"D result for odd ,-alues of p(n). Our gool in this ooction is to prove t he results (2_5_5) and (2.5.4) of Nicolas, Ruzsa.. and sarko~y (170) by rclali,-ely simple means. Owproofis a special instance of an argument devised by Berndt, Yee, and A. Zaharescu 1550), who PTO\·ed considerably mOre general theorems that are applicable 10 a wide ,,...iety of partition functions. ExCl'PI for one step, when "'e mUllt appeal 10 a theorem of S. Wiger! 1223) and Ramanujan /I 8S), our proof is elementary and self-rontalned. Theorem 2.5.1. Por eoch fixed c willi c > 21og2 ,md N sufjicilnlly large, (2_5 .8) Theorem 2.5.2. For eoch fired canstanl c ",ith c < 1/ .,{6, and for N su/Jictent/y /aryl, (2.5.9)

#In < N: PIn)

i3 e~ } ~

cVN.

Before we begin our proofs of Theorems 2.5.1 and 2_5_2. "-e ~ to establish some t(>rminology. Although ....-e shall use language from modern algebra, readers need not know any theorems from the subje.;t. [n fact. some of the information oolweyed in the next tWQ paragraphs will not be used in the sequel. bu t "-e think these facts &re interesting in the m~yrighted Material

S PIRIT OF RA M A NUJ AN

Let A

l!x lI

series in one variable

F2 be the ring of formal power X Over the field with t",'O elements F , = Z! 2Z , i.e. , :=;

A=

(2.5.10)

JI {X ) = f. an X " : a" E F" 0 < n < 00) . l n~1I

The ring A is an integral dQmain; n<>te that an element I {X ) = E ::':.oa" X " E A is in\'erti ble if and only if Olo =; 1. Since 0 and I are the only elements of F 1 , .",.., may write any element I{X ) E A in the form

f( X ): x n, +x", + · ··

(2.5.1I )

where the Sum may be finite or infinite and 0 any f( X ) E A, ob>olr....e that

:s n l < n, < .. .. For

(2.5.12)

In Qther words , if f {X ) is ginm by (2.:). 1l), then (2.:).1 3)

On A there exists a natural derivation which sends I (X ) E A to f'( X ) '" E A, i.e., if

1x

~

~

(2.5.14 )

I{ X ) ""

..L,a" X " ,

then

f' ( X ) '"

L na"X.. - I .

•••

Note thM for any fi X ) E A ,

r eX ) = 0.

(2.:).1:)

We also remark tbat for any I (X ) given in the fonn (2.5.11), the condition

J'(X ) = 0

(2.5.16)

is equivalent to the condition that all the exp<>oents n j are eveo num-

"'".

In QUT proof of Thoorem 2.&. 2, we need to koow the shape (2.&.1 1) of the series I (X) !( I - X ). F<>r any integel'!l 0 < II < b, "'.., see that in A

(2.5.17)

X· + X 6 = xa(! _ x~- a ) =X"+X • .;. I + ... + X ~I. I

X

Copyifglltlfd Material

B.C.BERNDT

"

We put together pairs of «.onseeutive terms X""" + X ""H to obtam the equality

(2.5.18) I (X ) I X

X'" + X '" X

I

=

X'"

+

+ X'"

I

X

(x n, + x .. ,+l + ... + X"' -')+ (x'" + . .. + X",-I) + .. . + (X"'HI + ... + X"""' - ' ) + ... .

If the sum on t he right side of (Z.:i-l l ) defining I (X ) is finiu, say j (X ) = X '" +X"'+···+X"· , then

{,XI = (x '" + X ",+i + ... + X ..,- I ) + ... + (X"'-' + x .. ·-· +I + ... + x .. ·- I ),

(2.5.19) if, is eWIl, and

I (X )

X

I

= (x "'+ ... + X .. ,-I) + ... ~

+ (x "·-'+··_+ X"·-,- I) +

(2.5.20)

if

~

LX",

is odd.

Before commencing our proofs of Thoorems 2.5.1 and 2.5.2, ~ introduce some standard notation in analytic number theory. We say that f eN ) = O(g(N )), as N tends to 00, if there exists a posit;'" constant A > 0 and a number No > 0 BUell that I/ (N )I S Alg(N )1for all N ~ No_ To emphasize that this positive constant A aoo"'e may depend upon another parameter c, "'e write f eN ) = Oc{g(N )), l1li N tends to 00. Proof of Theorem 2.5. 1. We begin with the pentagonal numoo theorem ~

(2_5_21 )

L ( ~I )n q" (3Q ~ I )/2 .. >
~

+ L ( ~ 1 ) nq"( Jn + I )12 = (q:q)"". "'"I

By reducing the coefficients mooulo 2 and replacing q by X in (2_5.21),

...-e lind that, if 1/ Fl!o~y~gRremU:fatfe infinite serie9 of (2.5.21 )

SPIRlT OF RAMANUJAN in .4, then

I ",

(2.5.22)

F (X )

(1 + ..f , (xn(3~-11I1 + X"(3~+ IJ.n) ) .

We write F ( X ) in the form (2.5_23)

F ( X ) "" I

+ X'" + X'" + ... + X n , +.

"'here, of ooucse, n"n1 ,'" are positi"e integers. Clearly, from t he generating function of the partition function p{n ) and (2.5.23 ),

#{I S n < N : p(n) il:Iodd},.. #{nJ < N}

(2_5_24)

wd

#( l < n

(2.5.25 )

~

N :p(n) iSe\len) = N - #{n, < N}.

We first establish a lo"'.. r bound for #(nj < N). Using (2_5_23). writ .. (2.5.22 ) in the form

(t. X"') (I L (x

+

~

(2.5.26)

=

m

{3 m -Il{2

m_'

~ (r

n (3"+I)/2) ) /J..-1 1/2+ X

+ x m (3 m +l){1 ).

.o\.'!ymptotically, there are ..,!2N / 3 terms of the form x m(3m- l l{2I ess than X N on the right side of (2.5.26). For II fixed positiw integer n j, we determine how many of these terms appear in a series of the form (2.5.27)

(I + _f.., (X"(3..

X "J

- 1)/0

+ x n{3n+ll(2))

,

arising from the left side of (2.5.26). Thus, for fixed nj < N, we eslimate the number of integral pairs (m , n) of solutions of the equation (2.5.28)

"i

+ !n(3n - I) .. !m(3m

- I ),

which we put in the form

(2.[>.29)

2nj = (m - n)(3m

+ 3n -

I).

By ... result of Wig..,rt 1223) and Ramanujan 1185), 1192, p. SO), the number of divisors <:of 2n j is no mOre than

c> log2.

O~ (N""'''-R)

for any fixed

Thus, ..,ach 0Cb~}~eNale~a"fd 3m+3n-1 OUIlll1'1ume

B. C. BERNDT

(N"''''P )

at moot O~ ,-alucs. Since the pair (m - 11,3"1 + 311 - I) uniquely determines the pai r (m , n), it follows thM the number of

Oc(N ...

solutions to (2.5.29) is 4 N) , where c is any constant such that c > 2 1ng 2. A similar argument can be made for the tenns in (2.5.26) of the form X"' (3"' +1 )/2. Returning to (2.5.26) and (2.5.27), we see that each series of the terms x",(3m - IJ/ 2 up to form (2.5.27) has at moost

Oc(N ... 41< )

X li that appear at least

0 11

the right side of (2.5.26). It follows that there an

O~ (N +- .... 4 N) numbers 11, < Nthat are needed to match

all the (l\Ilymptotieally J2N/ 3) terms X", (3", - I)/2 up to XN on the right side of (2.5.26). Agllill, 1111 analogous argumellt boJds for terms of the form x m {:Jm+1l11 We ha,-e therefore completed the proof of Theorem 2.5.1. 0 Proof of Theorem 2.5.2. Next, "'"e provide a Iov.-er bound for #(n < N : p en) is e,'eo ). Let (ml' "12,"') be the complement of the set {a, ,,], " 1," .} in the set of natural numbers (a, 1, 2, . . . ), and define (2.5.30)

G ( X ):= X""

+ X ...• + ... E A.

,

Then

(2 .5.31 ) G(X ) + F (X ) = I

+ X + X ' + .. . + xt + ... = ,,"""X

Since, by (2.5.25),

(2 .5.32) #{m; $ N} = N - -H{"j $ N} = (11 'S N: P(n ) is ",-en},

we need a Iov.'er bound for #{m, S N}. Using (2.5.31) in (2.5.Z2), we find that 1 + G (X ) (I

+~

1IX (I + f: .0' , =

(x n{3n- l l/2 + X" (3n +1 )/2)) ( X "(3"-I)/1 +

( I+X+X2+X ~+ XT

I

-'(;opyrigllted Material

X n

(3 .. +11/ 2) )

+ ... )

S PIRlT OF RA M ANU J AN _

I

I

X

(I+ X)+ (Xl+ X~)+'"

+ ( X (n- L)(3(n-L)H

){2

+ x .. (3n- L){":I)

+ ( X ~(3M L ){~ + X ("H)(3(,, -t L) -

(2.5.33)

"

L){":I)

+ ... ).

By (2.5.18), we see that the right side of (2.5.33) equaCs

(2.5.34) I

+ ( Xl + X3 + X 4 ) +. .+(x (n- I)(3(n_ L)HII2 +... +x ..(3n -LI/2- L) + ( x n (3n-t l I/2 + ... + x (nH)(3( nH I-LI{2- 1 ) + ....

Obsen'e that the gap betv.oeen

x n (ln- l )f2- 1

and

xn(3nH ){l

oontailllj:

n terms that are missing from the series (2.5.34 ). T his gap comes after a. segment of ~n (3n - I ) - 1 - ~(n - 1)(3(n - l)+ I )

+! =

2n- I

terms that do app-ea.r in (2.5.34). So "..., see that (2.5.34) contains asyrnptoticaUy

C"C~'"""-"I,,N '" n+ (2n

I)

I

2" - N _ ~N 3n I' 3

terlIlll up to X l' Now the sum in parentheses on the left side of (Z.5.33) has asymptotically ZJ2N/ 3 nonzero terms up to Xl<. Thus G( X) must have al least N / 6 uonzero terms up to X Ii in order fOr the left side of (2.5.33) to have at least 2N/3 terms up to X ,V to match thO$e on the right side of (2.5.33). We ha,-e lherefo~ completed the proof of Theorem 2.5.2. 0

J

2.6. Notes Theorem 2. 2.1 has been slightly refined by /11. R.Murty, V. K. Mutt)'. and T. N. Shorey (165), using a. more sophisticated lLTgllIDeot. T hey also obtain lll"'-et" bounds for the values of T{n) "'hen T(n) is odd. Another proof of Theorem 2.3.1 , rivalling Ramanujan's first proof in simplicity, has been given hy J. Drost 184]. See l\I. D. Hirschhorn's paper [120] for still another elementary proof. Many refereneal to further proofs of both T~Jfgmi}J MJ~;r~rem 2.3-4 can be found

B.C. BERNDT in the latest edition of Ramanujan 's C,,/Jedd Paper$ 1192. pp. 372375]. These pages also oontaln references t.o o~her proofs of Thoorelllll 2.4.1 and 2.4.2. Ramanujan himself summari:red the congruenoes he proved and the methods he uWized to prove them in a letter to Hardy wri tten from the nursing borne, Fitzroy liouse, in the summer of 1918. In partio;ular, he wrote (1)1 , pp. 192- 193], "Thus the divisibility by 5"7 6 11< when ... ., 0,1,2,3, b = 0, I, 2,3: c = 0,1 , 2 amounting 1.0 4 x 4 x 3 - I Or 47 cases of the conjectured theorem are prm1!
(2.1.4). Hirschhorn and O. C. Hunt [126] ga,,>'! an alternatiw proof al~ classical lines to thM of Ramanujan and Watson for Ramanujan's oongruence modulo 5". Those readers intrigued by our sketch of Ramanujan's T heorem 2.4.2 might consult F . G. Gan-an 's [95] proof of the more general congruence modulo 76 Rarnanujan appeared to conjecture that t he only congruences of the form p(ln + B );;; o(mod I), wh ..re I is a. prime, ace those wilen t "" 5,7, or II. T his was not prO\"ed until 2003, when Ahlgren and /II . Boylan [8) prO\-ed tha t indeed these are the only three such c0ngruences.

Copyrigllted Material

S P IRIT O F RAMANUJAN SupJ>Q:Se, h<m~ver, that Wi! drop the restriction that the moduli of the arithmetic progressions are the same as the moduli of the con· gruences. That is, are there congruences when the moduli are not the same or are n()t primes? The first theorems establishing lots of congruences for p{n) were found by Atkin [291. Then K. On() 1175J proved that, given any prime t ;:: 5, there exist infinitely many con· gruences ()f the type p{ An + 8) '" 0 (mod t). T his was extended by Ahlgren [7Jwh() established a similar re!u1t for any prime pow!.'T A consequence af their work is that if t ;:: 5 is prime, then for 1\ posith~ proparLion of positi,-e integers n , p{n) == 0 (mod t). Nonetheless, lind· ing roncrete examples iUustrating their theorems is not easy. Atkin and J. N. O'Brien \30Jhad earlier found II couple of such congruences; one is

r.

p(17303n + 237) = 0 (mud 13). R. Weaver [2191 devised an algorithm based on Ono's work and found

over 76,000 explicit examples, all with

",lin,

t

~

3\.

e-\

6 '-'4<' 1

:=

Except for a conple of sporadic examples, it turns out that f()r all of tbe congruences p{An + 8 ) == 0 (mod t) fOWld b)" Ramanujan, 000, Ahlgren, and others, 8 = -6t (mod f). In another hreakthrough. Ahlgren and Dno [10J, prm-oo that this residUO! class is only one of (1+ 1)/ 2 residue classes where p{n) po5 E 5 many sncb congrueooes. An informati\-e historical description of the quest for congruences for the partition function has been given by Ahlgren and Ono [9J. Suhbar80 [21 1 J first conjectured that in every arithmetic progression n == r (mod t) there are infinitely many wslues af n .ncb that P{n) ill e\-en and tbat there are infi nitely maIl)" values of n for whicb p{n) is odd . The most extensi,~ r..snlts pointing toward the truth of this conjecture have been found by Ono [173J. [174] and Ahlgren [6J. with a summary of previous results provided in these papers. The best lower bounds for the number of even and odd wslues of p{n) in arithmetic progressioTl.'i ar~ (2.5.6) \170Jand (2.5.7) {6), respecti>-ely. while the most gelleral theorems of this sort are found in [55] and

IS6J.

Copyrigllted Material

B.

c . BERNDT

In his unpubLisbed manuscript on p(1l) and T(Il), Ramanujan asserted and , in some cases, pro>-ed further congruences for T(n), rnOf such that p"ln but p<>+ l j n is called tbe order of n modulo p and is de noted by ordpn. Theore m 2.6.1. F()r any po!iti""

int~M"

n,

if tA~ msU a p ....me p such tAat (5) = -I and 2 1otdpn, T(n);;; 0 (mod 23), if tAe~ e:ri.!t.t a prim., p such lhQ/ O(mod23 ),

p

(-w

= 2,,2

n jJ

+ "'II + 3112 and ordpn ;;; 2 (mod3), ( I +otdpn) (mod 23), atAe,.,.,ue,

:= _2%' ·U~ ... 3~' OO"d. n_l(mo
\I.. he~

p run.! ()ver ,,1/ prime5 ~prQenltd. /)y 2;,:2

+ "'II + 311.

As mentioned in the Historical & ckground beginning this chapter, Ramanujan introduced his arithmetical fUn<:tion T(n) in [186J. [192, pp. 136-16], which was to become one of the most impo"ant papers in tbe history of number theory. In this paper, Ramannjan made three important conjectures about T(n). (a ) T(n) is multil)licative, i.e., T(m) T(n) = T{mn), wbelle\"," (m. n)= 1. Copyrigllted Material

S PIRIT OF RAMAN UJA N

53

(b) If p is any prime and n is any integer exceeding I , then

-r(pn+1) = -r(p)-r(pn) _ p"-r(pn- l ), (c) For each pr ime p ,

The 61'1:it two conject ures were 61'1:it pro-oed by L. J. Mordell [161 1 in 1917 and then greatly generalized by E. Heeke [113], beginning One of the most important chaptel'l:i in t he theory of modular forms. eo .... jecture (c) was also considerably generalized and was one of the most; famous unproved conjectures in number thoory until it was pr()\'ed by P. Deligne [8 1] in 1974. Readel'l:i ""8nting to learn more about -r( .. ) might begin by reading Iwo expository papers, one by R. A. Rankin ]1 97], and the other by M. R. Murty [164]. See also a paper by V. K. Murty ]166J.

Copyrighted Material

Chapter 3

Sums of Squares and Sums of Triangular Numbe rs

3.1. La mbe r t Series In this chapler, we shall derive formulas for ru{II ), for k = I , 2, 3, 4, and formulas for t1Jo (n}, for k = \,2. Obsen.., that '1* (n ) and tlk (n ) are generated by

..,2l (q) =

"-

L r1t (n)q~

IUld

.p2t(q) =

"L

t2.t (n lq" ,

respectively, where ..,(q) and w(q) are defined ;n (1.2.2) and (1.2.3), respect ively, and whel"'l, by oom...,ntioll, we define r n (Oj = 1 = In (O). Appearing in our proofs are Lambert series. Strictly speaking, II I..ambert series is II series of t he type

(3.l. 1) A generalized Lambert series allows mOre general exponents in both tile ourneratol'!! and denominatol'!! of (3.1.1), with tile minus sign in the denominator also possibly being replaced by a plus sign. Nonetoo. less, "''''' use the appellation, Lambert .series, for Ill! such series. Copyrigllted Material

8 . C. BERNDT

"

We also prove three further theorems on the representations of numbers by certain quadra~ic forms. For our purpose, a quadratic form is a function of k variables, J1 1, n~ , ... ,nk, and has the shape L: a;"n,nj, where a..j are nonnegative integers and fl" Il, .. _.• Ok are integenl. Thus, for example, rl (n ) denotes the number of represen· tations of J\ by the quadratic fonn nf + 7\~ + . _. + nt. In another example .....e examine the number of ways n can be representM by the quadratic form x' + xy + ,,'.

3.2 . Sums of Two Squares Theore m 3.2.1. For each po8i1i1H: integer n ,

.,(n) '" 4

(3.2.\ )

L

.--.

( _ 1 )(~-11/2_

We can st.ate Theorem 3.2.1 in the alternative formulation (3.2.2) where

d"k (n ) denotes the number of positive divisors

(3. 2.3)

d of n such that d ~ j ( mod k).

Immediately deducible from (3.2.2) is the "",II. known theorem that every prime p congruent to 1 modulo 4 can be represented as a sum of two squares 1172, p. 54]. Exercise 3.2.2. U6"'9 (3.2.2). pl"l)~ tho.t r2 (n) > 0 if and only if tlIef"1/ pnme p wngrnent to 3 modulo 4. in th e canonical factorizat'lm of n appears with an even exponent 1172. p. 55. Throrem 2.IS]. F irst P roof o r Theorem 3.2. 1. Using lhe Jacobi t riple product identity, Thoor<JIIl 1.3.3, we first deduce that

'"' L ~

(_ I )"a 2n -i-l q n(HI)!2

"- -Q<> Copyrigllled Material

S PIRIT O F RAM A NUJAN

57

C~,., + ~~J (- wa~+lq"("+1)/2 " .,...,.

L

n odd

~

=

a 4n + 1qn (1n+l )_

" _ _ 00

: a( _ a"q3; q' )",,( -a - . q; q4 )"", (q"; q. )"" (3.2.4)

-

,~ ( -a'q; q' )",, (_a - 41/; q' )_ (q' ;q' )oo ,

where "'e applied the J acobi triple product identity, T heorem 1.3.3, two additional times. We next Ulie logarithmic differentiation todifferentiate both sides of (3.2.4) with respect to a and then set a = I. Note that on the far lE'ft side of (3.2.4) the differenti ation of the infinite prod ucts is unnecessary, he<;:aUlie when a = 1, the factor a - I /o : O. We therefore obtain (3.2.~)

2(q; q)!o = 2(- 1/; q ' )",,( _ q; q4 ).",(q' ; q4)""

+ (- i\ q.)",,( _ q; q4 )..,(q4; q' ).", ~

xL

(olq4"_1

4q4" - 3

1 + q"n-' - 1+
0 "

= 2( _ q3: q' )oo( _q; q4 )oo( q4; q4 )""

Now divide both sides of (3.2.;') by 2 and by ( _ q; q )!, (q: q)"" = (_ q; q)",, (q2: q2 )oo

= (_ q; l )",, ( _ q2 ; q3)",, (q3 ; ql)"" - ( - q; l )",,{q'; q' )"" =

( - .r; q4 )",,( _ q ; q' )",, (q' ; q4 )oo

to dedUC
(q;q) ~ =1- 4~ 0.q·,,-" L- C+~CA ....' C _., ( q .q)' ,

OCCopyrigJ1/e(J

alerfal

D.C. BERNDT

" By (1.3.1 3) and Buler'li identity (1. 1.9),

-( q, q')'"'" (q' , q')"" -- ( (q;q)"" q; q)",,'

( ) 3.2.7

'P ( - q ) -

Using (3.2.7) in (3.2.6) and replacing q by - q , we conclude that


3 q4.. -L) L-, (q~~I q''' _~ - 1 q4n_1

..

1+ 4f: (f ", . 1

(3.2.8)

= 1+4

qf

. _1

f(

....01

f. 4m -Il.)

q (4m_3). -

r= I

L \- L din

~In

01 (moxI4)

4"'3 ( .."" 4)

Lastly, equate ooefficient.s of q",

tI ;::0:

I),..

I, on both sides of (3. 2.8) to

~hal

deduce

rAn) .. 4 (d". (n) - du {n)) , which, as y..e have seen, is an altemath'e formulation of (3.2.11.

D

We now give a serond proof of T heorem 3.2.1 that is qui te different and somewhat shorter t han the proof aOO1.-e. It requires, howe\'~r, a deeper theorem , namely, Ramanujan's 1 ¢l summa tion theorem.

Second Proof of Theorem 3 .2 .1. In Rama.Dujan 's lIP! summatioo theorem , Theorem 1.3.12, set a = {3 = - \ Uld % = I. After simplifying the product on the right-band side, we find t hat (3.2.9)

where in the next to 11\5~ nep we used (\ 1.9), and in the last step used (3.2.7). On the other hand,

L 0<>

"

1:

nm'

(3.2. 10)

1»

q

L I: (-\ )mqn t2..... = L: ( - 1)'" I: q(2m+l)n ""

=

0<>

.. _ I m-o

DO

00

m~a

n~1

..

= ~ (_ I)"'q(2"'tl) L \ q2m +! .

_

Copyrigllted Material

SPIRlT OF RAMANUJAN

"

Using (3.2.10) in (3.2. 9), ,,~ _ that WI! ha.-e nlaciJed th... first equality of (3.2.8). Th~ remaind ..... of th... proof th ... n follows as befor.... 0

3.3. Sums or Four Squares Theorem 3.3.1 . For

politiue

(GC/i

inl~er

n,

r~ (n )= 8 Ld.

(3.3. 1)

•• ••

Observ... that th ... sum in (3.3.1) is clearly never equal to O. Thus, as a corollary of Theorem 3.3.1 , ...-e deduce Lagrange's theorem tlult every positi1/(' integer n <;an he r... presented as a sum of 4 squares 1172, p.317]. First Proof of Theorem 3.3.1. We begin with Jacobi 's identity in the form (1 .3.28)

, (q;q);, = "2 L

~

(3.3.2)

( - I)~{2n+ I)q"("+llfl.

". - 00 Squaring both sides of (3.3.2), we ...asily find that

-

L

• 1 ) {q;qoo=;j

= (3.3.3)

( -I)"'·~ (2m + 1)(2n

~ (~,f:oo ~.t-oo) +

,v

,

"'.~.......

+ l )q(""··h"'.ft1/2

(- I).....· (2m+ 1)(2n + I)

"' . " oM

(m'.n·+ ... . n)/2 .

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

m =r +8,n = r - 8 or r = ~ (m+n) ,'= ~ ( m - n), and in the second on the right side of (3.3.3), set m= r

+', n = 8 - r C6Pyt'fQh/w,ljrnnar -

1)"

= Hm + n + 1).

8 . C. BERNDT

60

In both castS, as m &nd n run o\"u their respeCliY't range5, eacb of r and .. ruII!I through all of the integers. It follows rrom (3.3.3) that (3.3.4)

('1;'1 ):' .,

i C.f~"" -...f-.....,.,

(2r + 2. + 1){2r - 28 + 1)'1"·"+'

(2f+28 + 1)(2.s - 2r_ 1W· H '. r )

-,...L---

({2r+l)1_(2.s)1)qrZ. "+r

.. 2I(~ L

q'> (1 + 4'1 -d)~ L q"·<

I

~

' ''_<00

dq

''' _00

- f: ,.'•.• .,-"- f: ,.,) • _ _....

_

thj '''-00

~ ( _q;q~)!o (q2; '12)00 (I+ 4'1 ~) 2(_'1 1 ; q' l!,(q2; '12)00 - 2(-
wbere we used the product repreJelltation (1.3. 13) ror ,.,,('1), and where ""e applied the Jaoobi tri ple product identity (1.3.10) to

SPIRIT OF RAMANUJAN

, (S ~ (211 - Ilq2Q- l _ 4 ~ L.. 1 + q2n_1 L..i

"""'>q~n ,'",)

..- 1

"

n~ 1

(3.3.5)

Now divide both sides of (3.3.5) by

(_ q; q) ~ (q , q) ~ = ( _ q, q)~ (q2 , q3 ) ~ 10

=

(_q; ql ) ~ ( _ ql , q1 )!,(q2 ; q2);,

deduce that

(3.3.6)

7'''''''' '"", }·

(q;q)!, _ I _ S ~ { (211 _ 1)q2" - 1 L.. l+ q2n -1 I + qln I q,.q)4"" n a\

ThUll, substituting (3.2.7) into (3.3.6) and tben replacing q by - q , we find that

... -


{ (2n - I)
-

-

= l+sLnqn E .. ~ I

""

qn...

... ...0

= 1 + S L L dqdm ..... Id-l Copyriglf/~d Material

62

(3.3.7)

~I

B. C. BERNDT

..

+'I:I:"". ... 1 d j ..

where in the penultimate step we merely replaced n by d, and when in the l.a.st !>lep we set n =; dm and collected all coefficients of q". Now equate coefficients of q", OJ > I , on the extremal sides of (3.3.7) to deduce (3.3.1 ). 0

We now give a second proof of T heorem 3.3.1 that is

bp

I on

Our ,,"-ork in the second proof of T heorem 3.2.1.

Second Proof of T heorem 3.3. 1. In Ramanujan 's LTbI summation theorem, Theorem 1.3.12, first replace q by q2, and then set ~ = qeil, 0= - 1, and b = _ q2, where 8 is real.

Exercise 3.3.2. lj&ing (1.3.35), protO('! that for every integer n,

(3.3.8) Using (3.3.8) and replacing we find from (1.3.36) that

=

OJ

by - n for each negat""e integer n.

I + 4 ~ q"cos(n6) L., 1 + q2n

"-,

(3.3.9)

( - qe;'; q7)",,( -qe -;'; q')",,(ql; ql)~ = (~" ;q2)"" (qe - ";q2 ),,,, ( q';q2)!, .

Now replace () by If - 8 in (3.3.9). Mul~iply this new identity by (3.3.9) itself to deduce that (3.3.10)

SPIRIT O F RAMA NUJAN

63

by (3 _2.7). No ...· integrate both sides of (3_3_ 10) with respect to 0 o,-er !- "., ".). Using the orthogonality of {C03( nO)}, 1 < n < 00, on [- "., "'), we find that (3.3.11) Replaeing - q' by q in (3.3. 11 ), we dedu~ that 00

(3.3.12)

'P'(q) =

1+8~ (l+t

"

q)")"

Now ,

00

00

~ L (- i)m- 'm D

(3.3.13)

=

- 1I" (- qlm"

"" mqm (. L 1 + qlm m_'

Substitute (3.3.13) in (3.3.12) and obsen-e that we have now reached the first line of (3.3. 7). ThlL'S, the remainder of the proof is identical to that gh-en abo,,,,,. 0

3.4. Sums of Six Squares

:s

Of Jacobi 's formulas for r2 ~ ( n), 1 k :S 4, the most difficult to prove is perhaps that for "6(n). The proof ci S. H. Chan (72) that we gi,-e is possibly the simplest. Arithmetio;: proofs of the fonnula for r6 (11) are gi'-en in [167) and [154). Th eorem 3.4.1.

(3.4. 1)

+16( L

d' -

L "').

din

d\~

} ", l ( ,.,<>H )

1_3(mod ' )

Copyrighted Material

B.C. BERNDT P roof. Replacing a, b, and 2 by 1/, I/q, and x, respecti''l'ly, in Ramanujan's IV-oI ~ummation (1.3.36), wt obtain the useful corollary

(XII; q)oo( 9/(XII); q)".,( q; q)!, (11 ; q).,,(9/11; q)",, {x; q)"., ( q/ x; q)"" .

(3.4 .2)

where

Iql < Izl < 1.

E xercise 3 .4 .2. Observe thtJt the rigilt-hand metric in X and 11. Hence,

~de

of (3.4.2) U sym-

(3.4.3)

Give a direct proof of (3.4.3), i ndependent of (3.4.2).

Differentiating both sides of (3.4.2) witb respect plying both sides by x, we find tha~

\.0

X and multO-

(3.4.4) n .. _ oo

(X I/; q)"" (q/ (z .... ). q)",,(q: q);:" (y; q)"" (q/1I; q)",,(x; q )oo( q/ z; q)"",

Interchange z and !I in (3.4.4) and subtract the resulting identity from (3.4.4) to deduce that (3.4.5)

f;

nlln (:l:Y; q}oo( q/(x1l ); q)",,(q; q)!, n.. - oo I - z qn - (y; q)",,( q/ !I;q)oo (:I:; q).,.,(q/ :I:; q)oo

SPIRlT OF RAMANUJAN where Iql < lxi, )YI < L Le~~i ng Y ,,'" find that (3_4.6)

65

-x in (3.4.5) and simplifying,

L(q):= ~ ( nx~ _ n(- X)") = (-x~ ; q)00 {_q/X2;q)00 {q; q);, n::-oo I + xqn 1 xqn (:r: 2; q2)00 (q2/ :r: 2 ; q2 )00

Upon simplifying , we see

thl!.~

(3.4.7) R{q) = _ 2", (_ ",2; q ).,.,( _ q / ",2; q )oo{q; q);, (",2 9; 92 )"" (q/ ",2; q2 )00 (q2; q2);, (x2 ; q2 loo(q2/ x2 ; q2 )00 (xl ; q2 )00(q2/ ",2; q2)",, (q ; q2)~

= 2,;(1

+

where in t he second equality ""e applied (3.4_2), with "', y, and q replac:ed by q. x 2 , and q2 , respectively. Simplifying the left-hand side of (3.4.6) yields

~

..,

=L

+

nx"(i-{- I )") ~

l:

-

.. ~ I."'''O

+

« _ l)m+'nx-"- m-'qmn+,, + (_ I )"+I nx - .. - ... - l q",,, +,,) l: " " ' ,,,, ,,0 Copyrighted Material

B . C. BERNDT

66

~

L

+

({ _I J"' _ ( _ l )") n(xn+m+x- n-m ) qmn.

Noting tbat the summands disappear whenever m and n are of tbe same parity, we deduce that ~

L(q) = 2 L)Zn _ 1),.,2n - 1 ~

+2

L

(2n - I) (:I,2 .. -

1+2m

_ 2

L

2'1 (:l:2n +2m - 1

+ ", _ 2n _ 2", +1) q 2 n(2 m _ l )

~

X (X 2"-1+2'"

=:

21"( 1 + x 2 )

(I

+ X - 2'H I - z",) q {2n _ l) 2m

+ 2x( \ + :r

;1:')1

+ :1: - 2 .. +1 - 2", ) q (2n _ I)1 ..

:I

;:...

) L..- (2.. - 1 - 2m )x

_2n_2...

n ", .. I

(3.4.8) We nOw multiply both L(q) and .~(q) by 2/( %(1 x _ ). First, from (3. 4.7) we find that lim ~_,

let

2R(q) = %(1 + x 2 )

(3.4.9)

(q2;q2)~

=

= (_q2;q2 )~ '

where we have appealed to Euler's Seeond. from (3A .I\)

2L( )

+ x 2)) and

wi'

id~ntity

(1. 1.9) in the last step.

rI...-luce tha t

lim ( q2 )= 1 + 4 ~ (2n - I - 2m)(- w+m r _ .xl +% L... C&>

n ,mm \

tJ;JfHgfftetJ~~~/ I )2m

SPIRlT OF RAMANUJAN ~

_ 1+ 4

L

67

(2" _ 1)1 _ 4m 2) (_ I ) ~ (_.r )(~n-I ~

... ... .. 1 ~

"" I + 4

~

L (- W {2n - 1)2 _ L., ~

+ 16 L

~

'"~

L (- I t+ l ( _ q2) (2,,_ I )'"

~

= 1 + 4. L (_ q~)k

L

(- I)"(2n _ 1)2

(~"_ I ) l k

l ml ~

(5.4.1O)

{_q2)(ln - I )...

+ 16L{-,l)·

L

( 1)"+1,,,2.

(2" _ Il .... t

t_l

Now IUbstitute (3.4.9) and (3.4.10) illio (3.4.6) to deduce Ihat

~2;~~~

( q , q )""

= I + 4 f)-q~)k_ l ~

+ 16L{- q')t_1

L I, ..

(- I)~ (2n - Il

- I )!-t

L

(_ I}"+i m 2.

(In _ I) ... . t

Equat; ,,& t he ooeffidento; of (_ql)" on hnth ~itlell abo\-e and using

(3.2.1), ",-e complete the proof of Jacohi', six squarellheon:m.

0

3.5. Sums of Eigh t Squares Our proof of Jacobi '. formula for r. (n) ...mfocus on the function (3. ~. 1 )

F(z ) _ F (q;: ):= ( I + :)( Zq; q)""(- qlz; q),,,,{q;q~!... (1 z)(,q;q )""(q/ : ;q),,,,( q;q)"",

Lemma 3.5. 1. For Iql <

Iz! < l /Iql end

t"

1,

(3.5.2) P roof. In R1unanujan'B 1"'1 $\Immation formula, Corollary 1.3.14, Ie\. Q ' " (3 _ - 1. replace ~,by~ ~l_ z by .jq:. We then

B. C. BERNDT

68 rooily find ~bat, for I

< 1: 1< Ill'll.

I + 2~ q"~ ~ + 2~ : _n _ {q; q)~ (-:q; q) oo(- l /z ; q)"" .f;j I +q" 1+,," ( ' :'l )!.(: , :, )"" (l! :;q)",,

=:

(1 + l)(q ; q)~ ( zq:q)oo( .,/ z;'l )"" _ - F ( I , I -, (1 z)( '1:')00(:":' ... (" / z :'l)",,

(u .a) nu~ , for 1 <

1' 1.

= 1+ 2

:-1 "" .,~ . '~_~. - 2 ~ -;: 1 :- 1 L.., I +q"

.-,

=z, -+', _,f"ncn , ,I +q"

(3.5.4)

.-

Pu tting (3.1>.4) in (3.5.3), multiplying bot h sides by -1, and usill! analytic conti nuation. "'-e complete the proof of (3.5.2) for 1,,1< 1:1< 1/ 1,,1 and z", I. 0

Theore m 3.5.2. For (3.5.5)

FI (, )_

1.,1< 1:1 < 1/1 '11 and z ", 1,

(,+,)2 +S :[!-. (- l)"-I nq" +4~ nq" (z"+z- "). 1 : L...l q" L l q" .. _ I

.. . \

P roof. We begin by squaring both sides of (3.5.2), and in doing so, ",-e \lie the elementary identiti
(3.5.6)

C

+ : ) (z-n - :") = (z" + :-") + 2 ~ (z~ + z- t ) +2.

,-,

(3.5.7) (: _n _ z" )( z- '" _ zm ) "" (z"+'" + z- n- ",) _ (z.. - m + :",_n). Cop~fed Malerial

S PlRlT OF RAMA NUJAN

69

It follows from (3.5.2), (3.5.6). and (3..>. 7) tha~

(3.5.8)

1+ ~ ( I z

+2

f

".,

q"(z-" 1 +q" =

Equating: coefficienu of (z " (3.5.9 ), ...-e find that "

CIO

Zn»)'

c+:r

..,

+Co + fc,, (z" +

+ z-" ) On

the right s ide'! of (3.5.8) and

"+'"

"_I

..

C = 4 q +8 " q + 4" q " I + q" L. 1 + q"+'" L. (1 + q"' )(l ..... 1 .. .. I

.. 00

(3.5.10)

z-") ,

+ '1" - "' )

q... q" ......

- 8L ,"(1"+::,"'.")(" 1+"',;;;''''' ''r

For the purpose of simpliScatlon, re....rite each of t he three sums in (3.5. 10) and ILITive at

(3.5. 11)

8 . C. BERNDT

70

Observe that the first portio!! of the lint sum on the right side cancels with the second portion of the third sum on the right side above. After still further cancellation, we find that (l .5.11) reduces simply to

C~ =4 q"

l+q"

(3.5. 12)

+ 4(n - l) q" I

qn

+8,.-"'"~ I

,"

qnl+q"

" 4n-',-:;; 1 q"

=

Observe that the left side of (3.5.8) \'3.IIishes at z = - 1. Thus, setting z == - I in (3.5.8), we readily find that "" (- I)"-'nq"

{3.5.13}

Co- BE

q"

I

"-,

.

Putting (3.5.12) and (3.5.13) in (3.a.8), we complete the proof of (3.5.5). o

Theorem 3 .5 .3 . lYe h4 ve

(l.5.14 ) Proof. Rewrite (3.5.5) in the fonn

(3.5.15)

F 2(Z) =

(,+,)2 +4~ z L .q I

"-,

Divide both sides of (3.5.15) by ( I L'H66pital 's rule, we find that (3_5 .16)

nq" qn

+ z )2

(:" +:-"_ 2(_ 1)").

and let z tend to - 1. Using

'" urn

:"+:-"_ 2(_ 1)" = ( _ ', )"' n . • --\ ( I +:p

Letting: ..... - \ in (3.5.15), using (3.5.16), Ilsing the definition (3.5.1) on the left side of (3.5.15), and lastly tropJoying (3.2.7), we conclude that

(3.5.17) Replacing q by - q in (3.5.17), we deduce (3.5.14) to complete the proof. 0 Copyrighted Material

SPIRIT OF RAMANUJAN Theorem 3 .5.4. For each

po,giti~e

71

inleger n,

r8 ( n ) = 16{- I)n L: ( - I )Ad".

(3.5.18)

'I· Proof. From (3.5.14),

""

16L L d.,

'l (q ) = 1+16 2: I = I

+

d 3 q4 (

q

)<1

~

d.

~

,tZqd(_q)d... ! m_O

=

I + 16 ~) - qn L ( _ I)ddJq", n.. ' din

",hert! we put" = d(m + I ). The desired result now follows by equating CQ('fficients of q", n > I, on both sides above. U

3.6. S ums of 'friangular Numbers In this short section we provide easy proofs of formulas for t-;, (n) and t. { n }. We begin witb some elementary identities for theta functiollS. Exer cise 3 .6 .1. Pro1H!l that (3.6.1)


(3.6.2)


(3.6.3)


~' (q).

Theorelll 3.6.2. Rec411 tAat dj.k(n) is defined in (3.2.3). Then for each positi1H!l integer n,

(3.6.4)

t, (n ) = d! ,~ ( 4n + I) - d3 ,~ {4n + I ).

P roof , Using (3 .6. 1)- (3.6.3 ), we easily find that

(3.6.a)


1

= 8q w

{q').

Employing (3.2.2) and using the generating {u nction (1.2.a) for tk (n), we see that

4

-

L

m_1

(d, .• (m ) - d3.• (m )) qm - 4

-

L

(d".(m) - d3.• (m )) (- q)'"

"' .,\ ~

= 8 Copyrighted Material

L t2 (n) q~n +l . "'

.

B. C. BERNDT

72

Equating coefficients of q, .. -t-l, n O:! 0, on both sides above, "'e COD· clude that

1I1 .• (4n + 1) - d3,. (4)> +

which is what we wanted

to

\) =

12(11),

o

prove.

Theorem 3.6.3. Fc.r each positive

int~ger

n,

(3.66) uM~ a{n} = Ldl.. d.

Exercise 3 .6.4. I n ana.l<19l1 with (3.6.5 ), prove /hut <j02(q) + ","( _ q) = 2
(3.6.7)

Proof. Multiply (3.6.5) and (3.6.7) together and use (3.6.3) to de. due<'! t hat (3.6.8)

Now invoke Theorem 3.3.1 and employ the generating function (1.2.5) for t. (n) 1.0 arrive at ~

8

~

L L dqn - 8 L 2:> n ooil din t Id

~

{- q)" = ]6 2:> . (n)q2 n+l.

n~O din

" KG

"Id

Equating coefficients of q2 n-H on both sides above,

L

.,.

"''e

conclude that

d "" t. {n ).

dl(2 .. +I)

Noting that the condition 4 I d ;S 8'.lpoeriluous for divisors of odd integers, We oomplele t he proof. 0

3.7. Representations of Integers by x 2 + 3y2, and x 2 +xy+y2

Z2

+ 2y2,

In Our next theorems, we derive analogues of Theorem 3.2.1 for rep-resent&tions of pOSiti,-e integers by th~ quadratic forms ,,' + 2y2 a.nd ;,:' + 3y'. Howe\-er. first we need a leuma which is a beautiful identity by itself. Copyrighted Material

SPIRlT OF RAMANUJAN

73

Lem ma 3.7.1. For \al, Ibl < 1, (3.7.1)

I

+2~

~

= ( - a,ub)oco( - b; ab)",,(ab, ab )!, L.. . . (.b)" (•.,ob) ~_ l , , ,(h"') , , e(e.1,.,"')' 'ee un

(,n

Proof. In Corollary 1.3.14 , "",t a = (J '" - I, q = Vab, and ~ = Va/b. Then (3.7. 1) readily follows upon observing that

(-I;ab)..

2

(ab;ab)~

l+ (aW'

o

for each positive integer n. Theorelll 3 .7.2. We !uJUt ""

(3.7.2)

",(q)'f'(q1) = 1 - 2

L "-,

(_I)n(n+l)/2q1~ - 1

I

q1n -l

.

Proof. Set a = q and b = ql in Ltmma 3.7.1. Using (3. 2.7) Or (1.3.13), (1.3.1 4), (1. 3.32), and (1.3.30) .....e find tha.t 1 + 2 ee q" + tin = ( -q;q~),.,( _ q3; q'}QO(q~ ; q4)!., 1 + q'n (q;q4)oo(qJ;q4)..,( q4;q4);,

~

= (

q; q2)", ""l( _ q4) (q; q~) oo

¢(q)

2) ( ')

= ",,(q2)

(;(q) V"'(

¢( q)'f(-q "" q

V~

q)lI'(q)

= ""(q),,,,(ql).

(3.7.3)

Expanding 1/( 1 + q4n ) in a geometric serie> on the left side of (3.7.3), ....e arrive at the following a.rray or po"'ers:

,

q' q'

"" "

-,' _ qlO _ql~

- q' _ q"

- l'

..

q' q"

q" q q" q»

. Copyrighted Material

B.C. BERNDT U v."e sum this array by succe:ssi~"e columns and rows, we find that. 00

(3.7.'\)

L

n_ 1

q"+,f"

"" (_ I )"( n+I J/2+l q'n- 1

1+q'n=L

n_ 1

1

q2 n -l

.

If we use (3.7.'1 ) ill (3.7.3), we complete the proof of (3.7. 2).

0

Theorem 3.7.3. ut rl.2(n) denote tM number of ~cntanon.t of the positive integer n (J.S th e sum 01 a square and twice a squa~. Then, fOTn > I , rl .2(n) = 2(dl~(n) + d3 •S (n) - du(n) - dr.s{n)),

where d).a(n), j ., 1, 3,5.7, U defined by (3.2.3).

Proof. In (3.7.2), expand l /{ l - q2 .._ I) in 3 geometric series, and the desired result folLows. 0

Proof. In Lemma J .7.1 , set a = _ q and b = _q2 to deduce that 1 + 2 ~ (_q)n + (_q2)" L1 + ,f"

"-.

( q;.r )oo( q1;q3).",{ ,f;q3)&,

• (3.7.6)

{q;q)",,(q3;q3) oo ( q;f)",,( q3 ;q3)""

=~{ _ q !~( _ q3},

by (32.7). Next , on the left s ide of (3. 7.6), expand 1/ {1 + q3n) in a geomet. ric series, invert the order of s ummation, and sum the two resulting geometric series. Th is elementary task then gives us (3.7.7) ~ (_q)n + (_ q2 )" '" I m+1 (q3m+I q3m ... L1 + q3" L- ( ) 1+
.;p., _

2)

SP(RIT OF RAM A NUJAN

"

Inserting (3.7.7) in (3.7.6), replacing q by - q, and employing, for bl'evity, t he Legendre symbol (j), we find that

~ (

<j:I(q)<j:I(q3) - I + 2 ~ (3.7.8)

~

11'..,+2)

1/3"'+1 1 + ( 1/), ... +1 - I + ( q)3......1

(") 1+(," ,)"

-I+'~ 3

~('") ,~ .. +2]; ~ ('"+1) 0"' .. "3 l+q, 3 I q' ..... 1

-1+2f:i

_ I+'~('")( ,'" ~ 3 l_q"

,'''') I

.. - I

+, l:~ (2n+l) qh+1 +,,, (~) , + ,,~

3

1_1/2nH

n

00

_ 1

~

.. _ I

3

...." 'r

I

""

qn

4"

~

"_ I

. ..

(~) --'-':;;;; 3

l _ q4n

o T heorem 3.7.4 hl.8 the follnwing arithmetical inU'rprel.alion. Theorem 3.7.5. Let r l.3 (n ) d....oJe tAe number oj repruentatiOft$ oj the po!itilJ/! int~tr n by %' + 3y' , where % lind y lire integer., lind u.err ftpru.... tllh'on.t ....jth % and 1/ hlWing different.igru a.., counted II.J dilti""L /kaJ./1 that diJ«n ) .. d(fined in (3.2.3). Th .... Jor tad! po.itiVl! integer n, (3.7.9)

q ..J( n) - 4 (d..u(n) - d •. n (n )) + 2 (tfu(n ) - du (n )).

proor. If ...-e expand each of t he quotients 1/( I - q"' ) on the right side of (3.7.5) into geometric series. colletl the ooefficieots of all common exponents in each of l~~~~,,~~1e series, and then equate

D.C.BERNDT

76

coefficients of like pOwers of 11 all each side, we deduce (3.7.9) to

o

COml)let.e the proof.

We prove one furt her t heorem and its arit hmetical oonsequenoe in t his <;hapter, but it is closely relatEd to Theorem 3.7.4. To effect the proof, Wf! first need a beautifulana.logue of Lemma 3.7. 1.

Lemma (3.7.10)

3.7.6.

101. Ibl <

FrJr

I,

P roof. We apply (;Qrollary 1.3.14 with q = ab, Z = - 1/ a' , and P '" l /(ab) to deduce that

0: =

lib,

(3.7. 11 )

= I

~ l - I /{ab) 2.. (ab):!" I b

+~ I n~ l

I

= ( l-ab)

(

I

~

+ L.. "~ l

(I~ ..-

DO

ab

+ ..L

~2

I

J - ab 2" (00)2"+1a

I l

(Ob)'''- I -

AI 1.I1 tiplying the extremal sides of (3. 7.11 ) by

CIO

L1

.. _ I

al P -

the proof of (3.7.1O).

b2n-'/a ) (aWn- 1 . (lb), "'ll complete

0

Theorem 3.1.7. 1/ 1/J(q);" definw by (1.2.3), Wm (3.7. 12)

Exercise 3.7.8. Use Throrr.m 3.7.7 to find a formula /rtf' the number of repre! ffl/atiVfl$ of a JIOsitj~ integer by the Sum oj a lriongulllr o u mbi.r and th ree times a t';angular n~mr..er. Copyrighted Material

SPIRIT OF RAMANUJAN

77

Proof. In Lemma 3.7.6, put a = q and b = q2n _ ' _ ,,5(2~ - ' )

00

" ~

1

q4(1 n

I)

i'

to find tbat

'2);' -- q." i'''''''i';i''~'h'\#, '(q2;'ll~ )""('llt ;'l11)",,(q4;'l12)~ ( ,,' , ,,'?) oo(,t . q '?)",,(q ' \ q

_ (q.;q4 )""(ql1 ,'l12)",, _ .11 '1 '1'1 - q (q2; q')""(q4;'l12)",, - 'lv' q 11' q .

(3.7.13)

[t remairu; to simplify the left side of (3.7.13). Expand I/( l _ q
1fl _ l

'\" q t.....

1

5.(2,. - 1)

- q

q4{2n _ l )

n_ 1

""

""

= ~ '\" (q(Gm+l}(2n _l) _ 'l(Gm+~)(2"_I ) )

L. L.

m _ O .. _I

(3.1.14) Substituting (3.7. 14) in (3.7. 13), complete the proof.

""e

immediately deduce (3.7.12) to 0

Theorem 3.7.9. Recall !.hal ",,(q) and f(q) are definal b!l (1.2.2) and (1.2.3), rupectiwl!l. Then (3.7.1:)


"'""

L~ (1 _ "~ q

J,,-+-I

Proof. Using (3. 7. 12) aDd the trivial identity

1I>'e

easily find that

~ ( ' l4(JHI )

(3.7.16)

-L

1 q4(J,,+l ) - 1 t"lfpyrighled Material

q'(3n+2»

q4(J"H)

)

.

B. C. BERNDT

78

Multiply the extremal sides of (3.7.16) by
(3.7.17)

.

L-

}.t __ oo

, = I +6L- (i'n+ L ql"H) 1 rjln+1 - 1 ,fnH .

<1' +1 1+*

nd

-

Proof. By T heorem 3.7.9, it suffices 10 show that (3.1.18)

L

qi'+J Ht ' = \p(q);o(q3 ) + 4qtf.o{q2)1/J(l ).

} ,k __ o<>

We begin by observing that

; 2 + jk + k1 =

Ch + k)2

+ 3(h)2 ,

sothatif j = 2n + l ,

(h + k)2 + 3 U; )1

-

= (n +k)(n + k

+ 1} + 3n(n + I) + I.

Thus,

L

qi'+i1 +*'

'"

;. k ~_ ""

c.~_ + ,JJq(!H'l'H(!'l' j ..... n

, <>
L•

-

q ( nH)( n + HL )Hn ('Hl )+l

.. .t _ _ oo

L

Q..,( m +l) + 3 n (n +l )+ 1

where we have used the biia(.eraJ repr~ntation of / (q, qJ) = 2\11(q) in (1.2.3). Now employ Theorem 3.7.9 on the rar right side abov". This therefore completes t he proar. 0 Theorem 3.7.10 h"" the followins immediate and beautiful a:rit b-

metie interpretalion.

Theorem 3.7.11. Let r (n) denote 1M .. umber of represe..tatioM of the posjti~e integern by the quadrotic form j ' + jk+k1 , when: different Copyrighted Material

SP IRIT OF RAMANUJ AN

signs of j and k yitld distinct po!itivt int~ff n.

79

~pre~entaliofM

of II. Thttl for roch

(3.7.19)

Proof. Return to (3.7.17) and expand tbe sUIIlIIl3.ndB On the right side into geometric series. Hence, ~

(3.7.W)

L .i.h. - oo

~

qi'+.iHk' = 1 + 6

~

:L L (q{3m.+ll r _ q(3m+2Ir) . mooOr .. '

Now equate coeffidents of qn , II > 1, on both sides of (3.7.W ), and 0 the desired formula (3.7.19) follows.

3.8. N otes The formulas for r2 (n ), r4 (n) , r6(n ), and rs (n ) gi\= in Theorems 3.2.1,3.3.1,3.4.1 , and 3.5.4, respectively, are due to Jacobi [131]. All four theorems can be proved arithmetically. For example, see Hardy and Wright's book [112, pp. 241 - 242] for an elementary proof of Th"" .... m 3 2.1. In f::h"pt.er 6. we return to these problems and ii'"C completely difJerent proof$ of Jacobi's formula!; for r4 (1I) and ra (n) arising out of RamanujlUl's theory of elliptic functiollll. T he first proof of Theorem 3.2.1 that ....e have given is due to M. D. Hirschhorn [l 15J. In a later paper [119], he gave another simple proof b&$ed on Jacobi's identity, Theorem 1.3.9. One year later , Hirschhorn (122J showed that one can derive further theorems about r~ ( n) from T heorem 3.2.1. The first proof of Theorem 3.3.1 that ""C have gi '"Cn is also due to Hirschhorn (116J, who earlier [114J had given another proof. S. Bhargava and C. Adiga {58J have also utilized the ,1/J, sum mation formula to give simple proofs of Theorems 3.2.1 and :S.:S. L. Starting with Tb.,..,,,,w 3.2.1 fOJ tWO 3qu",~, B. K. Spc
B.C. BERNDT

so taken from

II

also establish

(3.8.1)

paper by S H. Chan [72]. Chan used the same idea to 8.

formula for 16(n), namely,

16(n) =

iL

(_l)(d+l){1d~.

4 14"+3

We are uncertain who first prO\w (3.& I ), but an equivalent formulation can be found in an unpublished manuscript of Ramanujan [194. p. 3::.6], j19, pp. 398-40IJ. T he first published proof known to us is by K. 000, S. RobiJl.'i, and P. T. Wahl (177, Thoorem 4]; this proof uses the thoory of modular forms. An arithmetic proof is gh~n in j128, p. 262, Theorem 11 [. S. Cooper and H. Y. Lam (801 al'lO employed t he ]w] summation formula to prove Theorem 3.4.1. The proof of T heorem 3.5.4 that we have given is due to J.- F. Lin 1144J, who in ano ther pllp
L

,"

='",,;:;,"'"

;-1

n~_""

(3.8.2)

F (a, t) F (b, t ) = t

•

at F (ab. t) + F (ab, t)(PI {a) + PI (!I».

Cooper (76] offers several formul as for t.he number nf int"t".... ' ''pr... sented by an even number of squares or an even number of triangular numbers. Cooper and Lam in (80] UOO Ramanujan's IWL sum mation formula and Venklltl
SPIRlT OF RAMANUJAN

"

numberli, for k = 2, 4,6,8. For completene!Oi, the formula for ta{n ) il; gi,~n by

(3.8.3) A. M. Legendre (139, p. 133) was the first to prove (3.8.3). An analytic equi,.,uent of (3.8.3) can be found in Ramanujan 's lost notebook 11941, (1 9, p. 401 ). Ono, Robill.'J, and WahL (177, p. 82) ha,·e established (3.8.3) using t he tlleory of modular fOrIllll. An arithmetic proof of (3.8.3) can be found in [128]. We prO\-ide a proof of (3.8.3) in Chapter 6; see Theorem 6.2.12. The connection bet"'«!n fOTllmLas for sums of squares and sums of triangular numbers was further solidified by P. Barruca.nd, Cooper. and Hirschhorn [32].... ho pro,-ed that rl (8n+k) = clfk (n ),

... here

Cl = z"(l + ~(:))'

l
Thus, the study of Ik (n ), for 1 S" k S" 1, is reduced to t he study of the subsequence rl (8n + k ) of rk (n) . Another elegant approach was gi>'en by L. Carlitz (62J. lie employed a beautiful formula due to W . N. Bailey (31), namely, (3.8.4)

xq" yqn} xqn) 2 - ( I yq,, )2 -

(xyq)oo(q/( xy))", {xq/ y)",, (yq/ x )",, (q):" (xq) ~ (q/x );,(,-/q );,(q/ ,-);,

to give proofs of formulas for r. (n ) and r6 (n ). An equh-alent for_ mulation of (3.8.4) can be found in N. J. Fine's book (94 , p. 22, eq. (18.85)]. Bailey'S proof 131) of (3.8.4) employs the Weierstrass p-function from t he theory of elliptic functio!l!l. Shortly thereafter . J. M. Dobbie 183) gave a shorter, more elemenlary proof of (3.8.4). Williams gave an arithmetic proof of Theorem 3.::'.4 based on an extension of an identity of J . Liouville [225). PI'. 353-3::.5] contains II. fragRlImanujan 's 1000t notebook P"'= on Lambert series. ment providing

B.C. BERNDT

82

This

fragmen~

h88 been examined by Berndt [3 9), with the arith-

metical coIl5equences of Ramanujan's Lambert series identities also discussed by him. See also Berndt 's book with Andrews [19, Chapter 17).

Andre,,'S (13) used the theory of basic bypergoomelric serie5 to give a uniform approach to proving Jaoobi's formulll5 for ru (n), 1 5 k < 4. Formulas for rk(n), when k is odd, haw an entirely different flavor. Gauss found a formula for r3(0), which was put in a more concrete form by G. EiS('nstein (88), 190 , p. 505J. who also glI'1! aLI analogous formula for r~(n). Chapter 4 in E. Grosswald'$ book [l Oll is d evoted to the study of . , (0 ), while pagffi 121< "'Dr! 12!1 in th~ AAmP text provide information about r~(n) . For recent work on formulas for rk (n ), when k is odd, see papers b:- Cooper [711. [78]. Theorem 3.6.2 is due to J acobi. '!'be first proof of Theorem 3.6.3 was found by Legendre [139, p. 133). Further proofs "'ere giwm by Cl'uchy [65, p. 572J. [66, p. 64) and Plana [179. p. 147J. For an elementary proof based on an extens;Cln of I'n identity of Liouvi\k>, see a paper by J. G. Huard, Z. M. Ou, B. K. Spearman, a.nd Williams [128). Jacobi [131J claimed that V. Y. Bouniakowsky fi rst proved Theorem 3.6.3, but he did not give a reference. Lin [143[ derived a formula for the number of rep ........ nt~tio"" of " pooitj'"e intcg<:r by 8 triangular numbers. The analogue of Lagrange's theorem is the theorem of Gauss [98, p. 497) stating that every p<:>Sitive integer can be represented as a sum of three triangular numbers. An elegant proof via q·series has been given by Andre",!! (16). Williams [226) derived an elegant formula for t he number of representatioIL'! of a pnsiti''e integer as a Sum of two triangular numbers plus twice the sum of two triangular numhcrs. Liouville (146), [147) determined the p<:>$i t ive integers n, b, and c, \ < a :$ b :S c, such that the polynomial Oll. l + 1>00.2 + cil, J, where ll.l, il,2 , and ll.3 denote triangular numbers, represents all p<:>Sitive integers. They are

((I , b,c) = ( \ , 1, I) , (1, 1,2), ( I, I , 4) , (i , 1, 5),

(1,2,2), ( 1,2,3), (1, 2, 4). Copyrighted Material

SPIRIT OF RAMANUJAN

8J

For an exposition of Liouville's methods giving formulas for ru (n ). 1'5 k < 5, see Chapter 14 in M. B. Nathanson's text (167). Although not explicitly stated by him, Theorom 3.7.5 is due to P. G. L. Dirichlet [82). The first explicit statement of Theorem 3.7.5, Ilowever, ill due to L. wrenz in 1871 [152]. The proof of Theorem 3.7.5 that we have given was independent.ly given by &rodl [35} and by Shargava, Adiga, and. D. D. Somashekara [59). R. A. Askey (27) used Ramanujan '$ I ~ summation theorem to derh~ a formula for the oumber of ways a positive integer can be represeoted by a square and twice a square. Although not stated explicitly by him , Theorem 3.7.11 is also due to Dirichlet (8 2], who proved a general theorem for representations of integers by binary quadratic forll15. Thus, every theorem in this chapter concerning representations of integers as sums of squares is, in fact, contained in Dirichlet'. general theorem. T heorom 3.7.4 has been enormously generalized by Williarru; (224], who derived a representation for certain sums of the products 4?" Formulas for k > 4 are more complicated tban those for k < 4. Fot: those who b!\\'e some familiarity witb modular fOrIIlll, "'e remark that the generating function ".,n (q) for r2k (n) is a modular form of "''light k. For k:::; 4, the dimension of the spaee of modular forms in which Copyrigllted Material

B . C. BERNDT o;?k(q) lies is equal to l. For k > 4, the di_nsion exceeds (1111':, and so other modular f<>rIIIS playa role in formulas for ~k ( n), m&ki~ the.ie fOTIIlulas m Ore complicated. In the latler part of the nineteenth oentury and the early part of thnl in Ramanuja n's fonnul ... for 1"21 (R) gh-en in hiB Tllble VI. A new cl""" of formulas in..,I>1ng products of certain Eisenstein 8erieo; hM '-n oonject.nrnd by H. H. Chan and K. S. Chua (69). The th~1llii of Jacobi '-n enormously generalized by S. C. Milne [156)- [158), ""ho also provides the most extensi>-e li t.erllt ure sun"y kllOWn on formulas for snrm; of squares. K . Ono [176) and l.. Long and Y. Y&ng [150) h8,-e gi,-en furtl>e-r proofs of IiOIDe o f Milne's results.

lu.,,,

E1ementllt)' texts in number theory <:(Intaining m ...!eri&! re\ ... ted to that gi,-rn in this chapter inclnde t h06e of HlITdy IIIld Wright [112, Chapter 20] and 1... K. Hua [127, pp. 115-120, 2(l.j,-21O, 301-309]. The moot e:o:lelll~ text On trum8 of squares is by GTO:!8>I-ald [101).

Chapter 4

Eisenstein Series

4.1. Berno ulli N umbe rs and Eisenste in SeriC5 in the definition of the c~ical EiSlCnsuin serWla are Bernoulli nwubeni, ..·hith we define first. Ap~

Definition 4.1.1. TM

8 .. , n > 0, "re /kfint:l1 l>y

B~nu>tl1Ii n~~.

tlu: gmera!ing /unctiDn

z

(4.1.1)

e'

" " :" B ",,! ,

, -:L

1:1 < 2"..

"~

(Note t bat the function :!(e' - I ) is analytic at : "" 0, and i~ neareIIt singularities to 0 are ±2,.-j: thus. the series On the riy,t side 01 (4.1.1) O:llwerges for 1%1 < 2>1". ) Tn particular,

, , 8"= 0,

B, __ "

85

0:.

0,

~ = 42 '

T

B~

- 0, ~ = O.

that 8 .. ., 0 when II is odd aM greater than I. This is easy to prct\-e. It

appelUll

T heorem 4. 1.2. For n?: I ,

B2ft +' '" O.

-

85

B . C. BERND T p roor. Define (4.l.2)

*' + I)I) '" ","",'c,+ _,I z_1 + EBft z:. e' I n.

I(z}''''

"-.

2(~

.

It is easily checked t hat.

I(-z) '" f(z), i.e. , 1(%) is an even function _ Hence, from (4 .1.2), _ oonclude thllt Bn '" O. when n is an odd integer at least equal to 3. 0 T heorem 4 .1.3. For 1%1< 2,,-,

= ='" ~ 2 cot 2 ~

{- l ).. B:tn~

(2n)!

.

P roof. Using (4.Ll ) with z '" -lz, ..... find that

-cot-,,--:r :r ;:r ~""'--, O 2 2 2 e'" I ;% ~ 8 {-.-:rr

---, +L.

m

m o'

Eioonstein series function

Il/'e

special

CI\OOI8

m.!

of Ramanujan's

IIlOJ'e

geDf'nl1

~

(4.1.3)

~~",(q}:-

E k~n'q"", ~ ... _

l

where r and if are nonnegative integers. Tri,~lLIIy, ~ •.• (q) '" ~ .... (q) _ For our PurpoIleII, tbe Il'IOO>I. important special CII.'ie8 are when r '" (I and 6 is an odd p06itive integer, and when r _ 1 and , is an e,= posith." integer. D efi nition 4.1.4 . For tach poMlIt: inUger T, rkfiruc (4.1.4)

c Br + I () "T:"-2(T +l)+ ~D.~q '"

SPIRlT OF RAM A NUJAN

87

Definition 4.1 .5. The c~col Ei3em1t:in a pori""," integer, an: Ikfined I>JI

~me.

E;,.(q) , wIi..,.., n i3

(4.1.5)

Tb""

'",-

E;~(q) '" -8- S~n_ h

(4.1.6)

n ::! l.

(Our ootation here is sligbtly different from the classical notatioo; oormally, E.t,,(T ) is defined by the right side of (4.1.,,) when q '"' eO.;, and lm T > 0.) In Ramaoujan'~ notation, Ihe three most important special cases of (4 _1.,,) are

P .. P(q) , .. E;(q) = I - 24

(4.1.1)

L-

1 '0'q" - - 248"

' 0' Q ~ Q(q) :- E.(q ) ~ I

(4 .1.8)

' 0'

R "" R(q):= &!(q) = I - 504

(4.1.9)

k~ t

""

+ 240 L

""

L

I

qqk" 240S~,

Ik~" qk" - 504S. _

b.

Our fint goal is to derive two re.;urnmCEl relations for fint of which .. Iso iDvo1""" 4',,'n _

82~+1 '

the

4.2. Trigo nome tric Series Our goal in tbis section is to present Ramanujan's elementary method 1186J, 11 9 2, pp. 136-162) for
stein serie!. We begin with an

exe~

Exercise 4 .2.1. Verify ~trnl)

(4.2.1 )

UOI"

u
the dtmento'l' idfntity

I +2

mo'

L

_0'

oos("O) + cos(m9) '" cot (!O) .in(m8),

valid Jor tm;h posilM inl"ger m.

C()I: yr

IXl

B. C. BERNDT

88

We use (4.2.1) when we sqUlU"e the left >
(1

, ' ,~ t/sin(kB)) , "" ('4"001 ( 400t '2 +L I
"J' + ;C-c (") L

~.1

~C<>'< ,w,

q -C

wbere the Fourier ooefficients en depend upon q but not UpOn 9. Squaring the left side of (4.2.2), Ullin,; (4.2.1) for the lermJI ilM,)/"i", cot( !6) &io(1;9), ""d using the identity sin l (k6) = ~ ( I - oos(zkO)), ,,"I: find that

. .-.

,-

= '2 2: (1 =

--

~L L


""1-

~. l"''''

I

mq'"

00

-"2 L

{4.2.a}

1

q ....

mo '

To e&.lculate C ... n 2: 1, ..,e need to be careful. FiNt oJ;",er.."!: that we obtain contributions from

.

,"

qi I q


.. 1 qI qi 1

r/'rI' ~(.,....(j -

k)8 - <Xl6(j + k )6),

for j, k > I. We ahIogain oontributions from the ten\UI cot(!8)sin{m8), fOf m

2: n. TillIS, we find !.hat, for n 2: I , I

q"

'"

._1

C"=2 1 qn + Ll I n_1

{4.2.4}

.-.

qk

q,,+k

""" qnH q"+k + L J 1/' 1 qn+l

._1

qn_k

-z L1 rtlqn.· Copyngh!ed I

a~al

SPIRIT OF RAMANUJAN

89

Using tho! easily verified elementary identities 1

g"H

q~ +k

•

(.

+ ,. ) . q"

2(1

1

q"

1

qk

q" )

+(

(

q"

1

,.

_ --,--q"'.'. . ) l - q"+~

qk

(n - l )q" 2( 1 q" )

q" ) ' I q"

g" ( . ") =lq"lq" - 2'

(4.2.5)

Substituting (4.2.3) and (4.2.5) in (4.2.2), we find that

(

~ oot ~8

4

f!..
2+ L..J h .1

1

(~ 001 ~8)2 ~qkOO5(k9) 4

qk

1

+ 2" L

(4.2.6)

+ L.., (I ,t V

2

t~ L

k,f

DO

I

k_ 1

k

q

( I - oos{k8)) .

This is the first of the two primary trigonometric series identities that we need. Using (4.2.6), we establish a recurrence formula for tbe functions S •. Appearing in the recurreTlOl n!lation are the funct ions from (4.1.3) 00

(4.2.7)

• 1/1" (q)

_

-

~

L.., m ... .k .. 1

Theorem 4.2.2. For n

(4.2.8)

> I,

00

k2"mk _ ~

q

k2 .. k

- L.., {I h ,1

q

'1" )1 '

B. C. BERNDT

90

P roof. From the eieme ntNy

iden ~ ity

(4.2.9)

and T heorem 4.1.3, "'" find thllt

Using Theorem 4. 1.3, (4.2.10), and the Maclaurin series fo£ and 00II(/<8) in (4.2.6), ,.," deduoe that

sin(~)

Collecting coefficients of like pa>."e11I of q in (4.2.1l ) and ffCIIlling the repreeenc.&tiono of S. and 4>r•• (q) gh"en in (4. \.4) and (4.2.7), respectively, ,,~ find that

+ S'q _ ( 1.. 'If! l!

S3eJ + S5 ~ _ ... 31

51

J'

~ _l_+ S, _ " 1 .2 ~+ 4> ,.411' _ . .. 41P 21 41

+'

(5311' _ S~8" + s,fI' _ .. .

221

41 6 1

J.

SPIRIT OF RAMANUJAN

91

Equating coefficientl! of (/h' , n 2: I , on both

(_1),,-1 Sz,,-+- l 2 (2n)1

~ ( 1)"-' - -

!idel

above, we lind ~ha\

(-1) ~

+ (2n)! "' ,,2~

(SI1!(2nSz,, _11)'+ 5,3! (2nSz,,_,3)! + ... + (2nSz,, _1I )! 51) I!

(- I)"Sz,, -+-l

+

(2n+l)! '

or, upon simplification,

.

(2n+3) '" ) S.. _ISz,,_>k-+-I , 2(2n + I)S2n-+-l- 4> 1 ,>n= ~ ~ ( 2k-1

,

o

which ill (4.2.8).

Our next tflSk iI! to Il.'le Theorem 4.2.2 to deri"" the three impor_ lant first order differential equations satisfied by P, Q, and R. From

(4.L5).

(4.2.12) by (4.2.7). In partieular, from Our table of &moulli numhllrft at tbe oopmung of Ulil! chapter lUld the detinitiOTlll (4.1 .7)--(4. 1.9),

(4.2.13) (4.2.14) (4.2.15)

Y'

m

B. C. BERNDT

92

Putting II = 1 in (4.2.8), ~'" find that (4.2.16)

"(4_ 2_17)

288~ 1.2 - Q - p' ;

putting II _ 2 in (4.2.8) . "" find that

7 lOSs - 4>1 ,_ = SS,S3. 7'204>1,_ '" PQ - R ;

(4.2.18)

IWd putting n

~

3 in (4.2.8), we lind that 9

1451 - 4> 1,6 -

12SIS~ + 2051.

or, witb the """ of the identity Q2 _ 4805" which "'" shall p ......... below in (4.2.38),

100841 ,,&,., Q2 _ PR.

(4.2.19)

He'll<''', putting (4 '2.17)--(4.2_1 9) in (4 _2.13)-(4.2 .15), respectively. we deduce the follo1'ling important theorem .

Thoorem 4.2.3.

W~

Iwvc

(4.2.20)

dP

P' - Q

qd<j =

12

(4. 2.21)

PQ - R q dq = 3

(4.2.22)


dQ

Ii'

The next theorem is also of enormOUll in,port.ance. Theorem 4.2.4. We "" vc

{4.2.Z3}

~(q) - ~(q)". 1728q(q; q)~.

Proof. By a .trD.igbtforward U8e of logarithmic differentiation, "" easily find t hat d q dq log (q{q;q): ) - P(q).

Copyngh!ed

a~al

SPIRlT OF RAM A NUJAN

93

On t.bo= other hand, using ~traightfo""''III"d d ifferentiation along "'-ith Thec:w~ 4.2.3, '""" find ~hat d

qdq log (ct(q) - ~ (q)) = P (q).

The la5t

tM;>

equal itie!l imply that

d

d

dq log (q(q; q)~) '" dq log (Q' (q) - ~(q),

"(4.2.U )

q(q; q):: .. c (ct(q) - R"(q) ,

for 8OtDe COWllant c. Equating ooefIicientsof q on each side of (4.2.U), "'., deduce that 1 ,. '-(3· U O + 2·504) = 17Z&,

and 80 .nth tile Vl\lue c .. 1/ 1728, Theorem 4.2.4 folm..'lI from (4. 2.24).

o Th obtain a SCC()nd rocurrence relation for 5" "'., begin ""ith ana_ logues of (4.2.1) and (4.'2.6).

Exerebe 4.2. 5. Fir,t,

(U 2S)

cot2 H8)

p~

that

(I - 00!I(n8))

.., "-,

_ (2n - I ) + 4 ~)n - ,I;; ) oos(k9) + COII(nD). Exercise 4 .2 .S. ( TIl., ., a t>ef1l difJieult aen;i.!e_) S«<md, using (4.'2.2S), ~ /lUll

(

1

(4.2.26)

L(9) ,.. 1

1

Ii cot

2

29 + 1'2 +

1 )2

= ( gcot2 29 + 12

f

11<»

,1;;

1 "'", (I - 00II(109»

.,

'

)

I "" Jilq'< + 121:: q'«5+00!I(ke))_: R(9) . 1

Theorem 4 . 2 .7 . For- etlCh po';Ii<.,

i"tefJ~

01 ,

(4.2.27)

ngntrd

I

ale

B. C. BERNDT

proor. Using (4.2.10), .."
(4.2.28)

1

1

,I

ii cot "2 8 + 12 '"

1 2(fl

l~

+"2 ~,

-'

Using (4.2.28) and the Maclaurin ""'ries ror 00fI(k8), ..."!: find that the left side or (4.2.2tl) has the rarm

(4.2.29)

£(8) '"

. .( L--, 2fJl +"2

~ k
+~I ct~ For n

> I , the wclIicient or 8""'"

8"")'

(2n)!

within pareotheses

abm"
is equal t.o

by (4.1.4). Thus, the left side or (4.2.26), i.e., (4.2.29), is equal t.o

(4.2.30)

1

L(8) -

"" (_ 1)_'

(201 + _., L

{2n )1 s",+11f"

'.

)

Returning to (4.2.9) , we lind thM

,p

dIP <:018 '" 2 rot /l a;ci/l,

'"

diP cot 8 '" - 2 -

8cot2 8 - 6cot 4 8 d

- 6 +8 <18 cot 0 - 6cot" 0,

(4.2.31)

C

j1 !!d I alenal

•

SPUUT OF RAl\lANUJAN

95

ThUll, by (4.2.31), (4.2.9), and Thoorem 4.1.3, (4.2.32)

1

4

-00< 64

B+

1

2

- M 48

I

B+ 144

'" ..!.. (I + ~3~~ «>to _ 6 .!. J3 CQto) + "- (-1 _.!!... 001 9) + _,_ ~ . 48 ~ 144

,

,'"

-26. 32

00t8 27 .3 d03

•

_,_ ~ ( - l j"/J.rn2 2"(2n -1)(2 .. 27 .3;::' (2.. )1

' 26·:J2

,

'( 6

=

26,.

I

'"'

I'"

..

+ '15-3-5 - 2~.3 L,

3)02"- '

27nH

)

(2" + 4)(;,,)!

- rr - ls + L

=26. 32 - 27.3 1

"" ( l )"a,..

:2

2)(2,.

8'''

-So-

where in the penultimate line, we used the [act that 8. = Using (4.2.32) with /I rep~ by !9 ""d the Maclaurin!!eries for OOII(kS), we find that the right .ide of (4.2.26) is eqU&! 10 ( 4.2.33)

R (B)

1 - 484

I

+ ZS . 3.:;

I ~ (-I).. Ba.. .... ..a.. - z3. 3 L.. (2" + 4)(2,,),"-

..,

'"" k't/' ( "" ( _ ll" t2R n) +I1..!.. L, ql 6+L (2n)! £i1 . k_ 1

The C!lnst.nl

term

in (4.2.33) equals

I

2'. 3.5 +

(4.2.3-1)

.. _,

..,

I ~

k"t/

I

2: L.. I ql - 25 .,

by (4. 1.4). For n 2: I, the coefIicient of (J2n in (4.2.33) equals (

(4.2.35)

I )n+l ~......

24 (2,,+4)(2,,)!

+

( I )" "" k""+'q1" 12(2n)! f, I
-

( 1)" 12{2n )!s",+,!,

by (4.1.4). Hence. (4.2.33) can be ",,,Titten in the shape

_

(4.2.36)

1

R(O) - 48"

I

1 "" (- l )~ ~ (2.. )1 S:!ft+lg2 .

+ 283 + 12 L

..,

'

.. "' ,.

8. C. BERNDT

"

Hence, by (4.2.30) and (4.2.36), 11o-e deduce that

Equating coefficients of (fln , n

> 1, on both sides abo>~, """ find that

(4.2.37)

which completes the proof of Theorem 4.2. 7.

For example, putting n

=

o

2 in (4.2.27) and using (4.1.6) and

(4.1.8) yields

(4.2.38)

0,

P utting n = 3 in (4 .2.27) yields

(4.2.39)

Using (4.2.27). tbe definitions (4.1.8) and (4. 1.9). and induction on r , we show that, for each nonnegati'"
where the numbers c.... n an constants. It is dear that (4.2.40) is ,1I.Iid for r = 1, 2 by the definition (4.1.6) of 52T+ 10 800 for r = 3, -1 by (4.2.38) and (4.2.39), respectively. Assume that (4.2.40) holds. We prove (4.2.40) witll r repl~ by r + I. ,By Theorem 4.2.7 and the CopyrigJJted Malena'

SPIRIT OF R.A.?lANUJAN induction hypothesis, c...'.~' Q

.'

R",

.. ' ... '2:0

~_ l

.",' t6 .. ' . 7k+ l

...","~2:0 4 ... ~ +,,", ~ _2. _ 2k 4 '

m_ m' + m" ,n_ ,,' 4 " " 2:0 4_,,", _2r+4

Exercise 4.2.8. Sim@ri,l, u.tin9 (4_2_8), aJ.mg with 1M
t:.{,m, ..

P'Q'" 1(',

' ."',.. >0. /<1

UH", ..1m_:fr+2

""'ere tIu numben ct''''' ~ 0'" """"'t4nl.o. (4.2.16).

For 1M C
4.3. A C lass of Series from Ramanujan's Lost Notebook Expressible in Terms of P, Q,

and R On pages IllS and 369 orhis lost ~book, in the pagination of [194), Ramanujan examines the J!elies defined for 1, 1< I and e&eh nonnegati,." integer It: by (4.3.1) T zt := T 2II (q) ~

:.. 1 +

L (- I )" { (6n -

.-,

1)ztq"(3n-1)12 + (6n + J)U q"(3ftHln } _

Note that the exp<Jnentll n(3n ± 1)/2 >U"e the generalized pem&gOnalnumben and that when It: .. 0, (4.3.1 ) redlJCe8 to .he Beries in t he pentagunal number t heQrem , Corollary 1.3_5. Ramanujan records formw.. for T2l , It: .. 1, 2, .. . , 6, in termli of the Eiflen8tein !!tries P, Q, ILOd R , d/!fined by (4.\.1)- (4.1.9). Ram .... ujan ·s formulations of

COl

ed mall'

Ia)

B. C. BERNDT these formuJ3!I are cryptic. The first is gi\'en by Ram&.lluj&.ll in tbe form C '

.:'1c';;'q-=c',,',,';:'C+...c·_·. = P. q~+

In succeeding formulas, only the first t\\'O terms of the numerator ~ given, and in two ifL'ltances the denominator is replaced by a dash - . At the bottom of the page, he gives the first five terms of a general formula for T:u .

In this se<;tion, we indicate how to prove these se_-en formulas and one corollary. Keys to our proofs are the pentagonal number theorem ( 1.3.18), (4.3.2) (q; q)""

-

= I + 2) _I )n {q"(3"_1)/2 + q"(3"+I )/2} = To(q),

"-.

where Iql < I , and Ramanujan 's famoWl differential cquatioll!l (4.2.20}-(4.2.22). We now state Ramanujan 's sUe formulas for corollary and his general formula. Theorem 4 .3. 1. IjT1_ ..,

d~finw

T,_ followed by a

bll (4.3. 1) and P,Q, and R Of"f

definw by (4.1.1)-(4.1.9), then

(i) T, (q) = P, (q;q)""

(ii)

~~ (q) =3p 2 _ 2Q, (q, q}""

(iii ) Te (q) = 1:;p3 _ lVPQ+ 16R, (q;q)oo

(iv) (v)

rs(r) q;q

= 105P' - 420P'Q + 448PR - 132Q2 ,

00

r1 0 \q ) =94 ;,P; - 6300plQ + l00s0p2R- ;,940PQ' + 1216QR, q,q""

,wl (\'i )

rn\q }= 10395p6 - 103950P6PQR -

2712Q3 - 9T28Rl . Copyrigllted Material

196020p2Ql

•

SPIRIT OF RAMANUJAN

99

The first fonnulll bM 1111 interesting arithmetical interpretation.

Corollary 4.3.2. For n 2: 1, hI a (n) .. l:4!Rd,
-24

(- I )t oUI '"

Since .,-U) is nmltiplicat;''e,

(- 1)'(6r-lF, ifn;r(3r-l )/2, (- I r(6r + I?, ifn=r(3r + 1)/2, 0, note thllt .,U) is ever> except when

,,1)

j is II "'I.1lIL«' or '''-;ce II "'l."&Ce. Thl1.8, from CornUill)' 4.3.2, " ...,

!lee

that, unless n = r(3r ± \)/2, tbe !lumber of representations of n!lS .. s um of II square or twice II square and .. generalized pentagonal number .I:(3k ± 1)/2 ;. "'...... For
5= 4+1. In geneno.I, as Rllmanujan indicated, for cenain CQll.'llanl.$

Ct ...... ,

Tn ('ll = ('I)""

Important in OUt proofs are the simple identities (4.3.4)

(6n ± I?_ U n (3n± 1) +1. 2

We abo use the notation

(4.3.5)

lv,{P,Q, R) := Tn (q)_ ('Ii 'I)""

Proof of 'Thc:>,em 4.3.1. Obaen.,-, tbat

_.

d -

P('l ) '" 1+24qdqLIog{1 - 'I") d

= 1+ 24'1 dq Iog(q; 'I)""

= 1 + 24q £('1; 'I )"". ('I; 'I )"" COl r

ed malarial

B. C. BERNDT

' 00 Thus, using (4.3.2) &nd (4.3.4),

""l

find

~hat

(4.3.6)

(q; q).,.,P(q) - (1/; 9)"" ""

24q~

'"

(I + f:(-I)"

{q,,(3n-l)/2

+ 'I"(""+I)12})

-, '" Z
e

2

-

I )-I)" { (6n -

-,

2

I)' - t) q,,(3.o_1)/2 + «00 + If _ 1) q'"(3ntll/l}

00

_ 2:(-1)" { (6n - 1)''I,,(J.,.-I)/2 + (00 + J)2q";3n+l 1f2 }

.0'

- (q; q)"" + I - T,(q) - (q;q)ooThis completeS the proof of (il. In tile proofs of the .... msjning identities ofTboorem 4.3.1, in each case, we apply the operator to the preceding identity. in each proof we use the identici....

Z4Q£

k ;:>: 0,

(4.3.7)

which foLl",,' from differentiation and the ust of (4.3.4) in exactly the same manner that "re employed aoo.-e in (4.3. 6) to pM\"f: the special

(·U8) We l\OYo' PfO\.", (ti). Applying the operator 24q£ to (4.3 .6) and using (4.3.7) and (4.3.8), "'" deduce t hat d

P (q) (To(ql - (q; q)oo) + (q; q),.,24q dq P (q) "" T. (q) - T 2 (q) .

Employing (il tosimplify and using the differential equation in (4.2.20), we ani,-e at

pl(q)(q;q)"" + 2 ( pl(q) - Q (q)) (q;q)oo '" T~ (q),

"' ,.

'"

SPllUT OF RAMANUJAN

"(4.3.9) a/:I

T4 (q) = (3p2(q) - 2Q(q»(q; 'I )"" ,

desired.

To prt)\"I! (iii), we apply the operator 24q~ to (4.3.9) and

Wle

(4.3.7) and (4.3.8) to deduce that

TG -

T~ "" 24 (6Pq ~ _ 2q: ) ('1:'1 )"" + (3p1 -

2Q) (T2 - ('1;'1)",,)

'"' ( 12P(P 2 _ Q) _ 16(PQ - R » ('I; 'I )""

+ (3pl -

2Q)(P - 1)('1:'1 )"",

where..., used (4.2.20), (4..2.21), ILIId (i). If "'"I! and simplify. ""I! conclude thai-

now

employ (4.3.9)

TG - (151"'- 30PQ + l(lR) ('1 ,'1 )"". In g~al, by applying the o!"'rator 24q~ to Tn and using (4.3.5) . (4.3.7), and (4.3.8). ".., Snd that d

T""u-T"" .. Uq dijfu (P, Q, R) ('I; '1 )",,+ h.{P ,Q, R )( P - l )(q; 'I )"" , where.".., ha'"I! \1900 the notation (4.3.5). Then J>roc:eooing by induotion while using the fonnula (4.3.5) for T2•• "'.., find that T:UH d ( . ) - 24Q .lj2k(P,Q,R) +Ph.(P.Q, R). '1,'1"" ....

Thus, in the ootation (4.3.5),

(4.3.10)

d

hH1( P,Q, R ) '" 24q-h. (P, Q , R ) + P fu (P, Q , R). d,

Exerd.... 4 . :J .:J . \Vith 1M. ...... 0/ (4.:J. IO) ..nd

th ~

dii!eren./iol o:qua-

lions (4.2.20)-{4.2.22), prot>e the nm1t1ining ;denlilia, (i")-{\i).

o Proof of Corollary 4.3.2. By expanding the summands of P(q) in (4.1.7) in geometric serifii and ooLlecting the ooefficif:nUi of 'I" for each posit""I! integer n , ""I! find that ~

P(q) = I - 24

~

L 07 (n)q" = - 24 L:IT(n )q" , .., ••• r:

ngntrd

t

ale

""

B . C. BERNDT

- ii.

upon using the definition 17(0) '" ThllS, by (4.3.1 ) !IIld (4.3.2), Th..."..,m 4.3.1(i) can be written in the form (4.3.11)

_24I:,u(J)r/ . (I + t <-l)k ,.e

{t/' (lk- I)/3

+ qk(1t+ lln})

k MI ~

..,

.. 1+ L (-l)n {(61'\ _ I)V·(:t..- l )/2 + (6 .. + 1)'q"(""H)n } .

Equllting coefficients of q", n ;:: I, on both sides of (4.3.11), ...-e complete the proof. 0

4.4 . Proofs of tbe Congruences p(5n + 4) = O(mod 5) and p(7n + 5) _

o(mod 7)

In his mWluscript [188]. [192, pp. 21(}-Z13], in addition to proving lhe oo~ (2.1 .2) - (2.L4) , Ramanujan prtr.-es furthN 00ItgnreD
(q" ;q'J)""

~

L p(13n -

..,

7)qH

=<

II

-

L

..,

,.-(n) qn + 13J.

He then writes, ~ It ill not nOO Pry to know all tm. detAils abo\-e in order to prxn-e (4.4.1). T he proof can be ''UY much simplified .... follows; using Theorem 4.2.3 and ... we can show that

(4.4.2) where J '" J {q) ill a JXW...... $
manujan realized this simplification at precisely this juncture while "'Tit ing hill paper, for he did not return to his proofs of (2_1.2)--{2.1.4 ) to utilize this observation...oo thereby simplify them. Instead of using further (more complicated) Eisenstein ileries identities, the differential equations of Theorem 4.2.3 /Ire employed. In his doctorAl dissertalion, J. M. RlllIhforth j201] used this ideo. to simplify Ramanuj&n 's proof of (2_\.4 ), lIS extracted by Hardy for (191]. In this section , we

Cor

o"d

I

SPIRlT OF RAMANU JA N

"3

use Ftamanujan's observation abo>-e along with (4.2.23) and (4_2_20)(4.2.22) to give simplified proof5 of (2. L2) and (2.1.3). The proof of (2.1. 4) is more tedious, and we refer to Berndt's paper [40] for the proof of (2.1.4), which is precisely that of Rushforth [201]. Theore m 4.4.1. For each IWIlnegaliff integer n, p(5n + 4) ;;; 0 (mod S).

(4.4.3)

Proof. From the definitions (4.1.8) and (4.1.9),

(4.4.4 )

~i\"ely,

R= P + SJ,

Q = I+5)

,ince II~ = II (mod S) by Fennat's little theorem. It follo~"S from (4.4_4) and (4.2.20) that

Q'J _ R2

:z

Q(1 +SJ )2 _

( P +S) ~

= Q - p2 + SJ

(4.4.S) But, by (4_2.23) and the binomial thoon'Dl,

(4.4.6) ..-.3 _ R2 =I72S (. )'l4= '" q q, q DO

3q (q; q)~+SJ = 3q (rt;rt)~+ SJ. (q;q)""

(q;q)",

Combining (4.4.S) and (4.4.6) and using the generating fUllCtion for "'II), "'e find that (4.4.7)

We now equate tbo5e terms on both sides of (4.4.7) whose J>O'"''eT5 are of the form qlm to find that (4.4.8) Rarnanujan's congruence (4.4_3) follows immediately from (4_4.8). T heor e m 4 .4 .2. Fur each 1l0llnegal;ff integer n ,

(4.4.9)

0

,..

B. C. BERNDT

p roor. The Ii..- two Sl~ of ow- proof are tbe -.me "" lboee of Ramanujan and Hanly \1911. 1192, p. ~l. From tbe definition of

R. ;t ill olwioull

that

R _ l+7J.

(4 .4.10)

Ueil1& one of the identities from (4.2.38), namely,

fermAt'. little theorem, aDd the definit ion (4 .1.7), _ abo tiD
(4.4.11 ) !felice, from (4.4..10), (4 ....11), (4.2.20), and (4..2.21 ), (~

_ R2)2 ... (PQ _ I ..-7) "

= P2q'l _ 2PQ + l + 7J "" P(Pl -

QJ - PQ + R + 7)

_ 12Pq dP _ 3qdQ + 7J (4.4 .12)

.. d

..

dQ dJ =6qdq~ - 3qdq + 7J _ 'dq +7J.

O n t he oI.ber hand, by (4.2.23) and the binomial theorem ,

"

() ('. ' )' (Q'l - R' )' = ,Pq;q"" + U - tl"q "'+ 7J

(q;q)...

('I;' ).,.

~

-

_ (q';q' )!., Lp(n)q"+' + 7J.

(4.4 .13)

We now equate the right &ldo! of (4..4. 12) and (4.4 .13) and then trllCl. \.ho8e ~rntlI irl\'olving 'IT. Equa.lilli them , "''e find that

e>;-

~

('I'; q7)!.

L )l{7n + 5),'''+? .. 7J, "".

ffOlQ

whwoe (4.4.9) irnUJO't'l;!Ite1y fo/lQlll'll.

o

SPIRlT OF RAMANUJ A N

lOS

4.5. N ot es Eisenstein series were first introduced and st udied extensively by G. Eisenstein {ST]. {S9 , pp. 357--478] in 1847_ In contemporary 1>0talion, the Eiseru!tein series G~j(T ) tuld E-.j(T) , where j ie a positive illteger exceeding one, are defined for 1m T :> 0 by

..........

( m." ),,{,l,O)

"d

(4 .~.1)

where (z) = E ::". l ,,-', Re z :> I, de""t"'! the Riemann zeta function, B", n 2: 0, denotes the nth Bernoulli number, and .,.~ (n) .. Edl" d". The serie« G~j( T) dual not oonveTge if j _ 1. ""d >0;) the firnt equality in (4_5.1 ) does not hold if j _ 1, although the series on the right side converges fnr j = I. We have defined Eiselllltein &«ries by the latter two repretientMions in (4_5_1)_ These u n be elie. For example, _ T. M_ A])/>I!tQl's text ]2:>, pp. 18-20, 24J or B. Schoeneberg's book !20:>, pp. 51-03J. The Eigenstein ~ ~j (T ) is a modular fonn of weight 2j on the full modular group. Along with the discriminant function 6 (T) := q(O; O): . q = .,h;~ , thEse functions form the building blocks for all analytic modular forms on the full modular group. For basic properties of Eil;enBtein series from the point of view of modular forms, see, for example, the aforementioned texts w; well w; R. A. RAnkin 's text [196, Chap. 6]. Ramanujall made many original and fundamental oontributions to the theory of Eisenstein oories, mOf;t notably in his lo!it notebook and letter!! to Hardy 11941_ In particuh,,· , in letteno he ",rote to Hardy from nursing homes near the end of his OOjollfO in England, he derh-ed formul88 for the ooefficient~ of oertain Quotients of Ei'lffll.tein series.

B.C. BERNDT

H"

These formulas are an8.logous t.o the lI.'iymptotic formula for p(n) of

Hardy and Ramanujan that was discll ved in (1.1.6) and the IIOtes for Chapter 1. Howe>-er, their forms and proofs are much different. For accounts of this work, ~ papers of Hardy and Ramanujllll (I l l ), [192, pp. 3W- 321J, Berndt and P. Bialek [42], and Berndt, Bialek, and A. J. Vee [431. For an account of aU of Ramanujan'B theorems OIl

Eisenstein series from his lost notebook, other unpublished papers, and letters, see the book [20J by G. E. Andrews and Berndl. For an expository survey of many of Ramanujan 's theorems on Eisenstein ser;!)5 from his lost notebook , see the paper [53] by Berndt and Vee.

In his notebooks (193), Ramanujan made further claims about Eisenstein series. In particular, in Entry 14 of ChapleT 15 of his 8eCond notebook, Ramanujan recorded another recurrence relation for Eisenstein series; see Berndt's book [33, p . 332). Also, Cbapters 17 and 21 in Ramanujan 's second notebook contain many claims about Eisenstein series. See Berndt's book [34] for proofs of all these theor rems.

An important function which "'e ha\'e not discussed in this book is the modular j ·invariant defined by

. Ql (q) '(T):'" (. }'" q q,q "" T he values of j(T) and its Fourier or po",er series coefficients are of great import ance in the theory of modular ro~. At about the same time that Ramanujan discm'ered the differelltial equation (4.2.20) satisfied by P(q), in 1911, J . Chazy [74] c0nsidered the third order differential equation

(4.5.2)

y'" '" 2!1/1" - 3(vf.

where y = y(t) and sh()';l.·ed that one solution is ,.;P(q), wbere Q = e l .". If ,,'e eliminate Q and R from the system (4.2.20)--(4.2.22). "1l deduce the differential equation (4.~.2) for P In fa<;t, the system of equatioM (4.2.20)-(4.2 _22) is equivalent to a system of three differential equatioM first studied by G. Ha.lphen [104J in ISS\. The equation (4.~. 2) is of fundamel\ta1 importance in tbe tbeory of integrable systems [1), [2J. [3J. [41. Copyrigll/ed Material

SPIRIT OF RAMANUJAN

107

The results in Sections 4.1 and 4.2 paper [186], [19 2, pp. 136-162].

an:!

tllken from Ramanujllll 's

Exen:;i.se 4.5. 1. SMW that &manuftm'. key identity (4.2.6) oon be deduttd from Vt:nkatachaliengar'6 ruult (3.8. 2). In her thesis [1.831, V. Ramamani extended Ramanujan's use of trigonometric series 10 study t ","O analogues of +•.• (q), namely,

-

L

II'r .• (q):=

-

L

Fr .. (q):=

(_I) .. -lm r n·q ....•

(2m _ 1)r n _q\2m-l)n/2,

...."--00 where Iql < I. Ramamani employed the classical theory of ellipti<: functions in her wurk . In extending Ramamani's study of II' r,.(q), in ber tbesis [102), n. Habn chiefly emp)~ ideas from RaIDanujan's viewpoint in tbe t heory of ellipt;': functions. H. H. Chan [68] derived an analogue of (4.2.2) with cot replaced by esc and used it to give new proofs of Ramanujan's famous formulas

-

L

"-,

(")

~

(.. ,,')~

q"

(I

q")1

=

q (q;q)""

;c.(")

" l + q" ( )' ( ' . 'I' 8q2 (qT;q7)!., ~"1 q (1 q"P "" qq ,q ""q , q 00+ (q,q)"" where (i) denote!! the Legendre symbol. These formulas lead, respectively, to proofs of Ramanujan 's congruenoes p(~n + 4) - 0 (mod~) and p(7n+~) = O(mod7). Z.-G. Liu [148) used the theory of elliptic functions and 1L!<'!OCiate.:! complex analysis to deri..e a trigonometric series ideutity invol\"iI1& tbeta functions that is analogous to R.am.a.nujan's trigouomet ric series identities. Whereas Ramanujao used his results to study sums of squares, Liu used his identity to obtain representations for tu (n). The contellt or Section 4.3 is takell from a paper by Berndt and Vee [~3) . K. Vellbt~31iYn'jftJg a!fr;atl-32) hll..'l gi,-en a similar

".

B. C. BERNDT

b ut mOre abbnwia.ted acnrunt of this work. On p&ge 369 in his \oI;t notebook [194J , RamanujlW offers another infinite class of series

(4.5.3)

-

F2~(q):'" 2:)- I )k(2k + 1)2n+!qk(k +l l {2

•••

that can be represented in tenD!< of polynomials in P , Q, IWd R. Obsene that ,,·hen,., = 0, Fo(q) = {q;q);:", by Jacobi'. ~tity, T heorem 1.3.9 . Thus, (q;q);, !ll1d Jacobi'6 ident ity play the roles iII the .. rorcmentioned ident;t;"" that (q; qj"" and the p
""".,.s

are

The proofs in Section 4.4 can be found in Berndt's paper 14.0J.

Rushforth's proof of the oonpuence p( lI n + 6) - 0 (mod 11) can be found in his thesis [201] IUld was reproduood in Berndt'. paper [40j. M indicated earlier, the key ideft behind th ..... proofs &rises from fut,. manujan's previOWliy unpublished manWlCript 011 tbe partition &lid

tau functions [194[, for whieb Berndt and 0 00 [50]

ga'" addition&!

details and references for several proofs. The latter ))II))l'r "''lIS extended and annotated st.iLl further by Andrtnro's &lid Berndt in their hook [21 ).

Copyngh!ed I

a~al

Chapter 5

The Connection Between Hypergeometric Functions and Theta Functions

In thi'l chapter, we prove one of the oonlral theorems in the theory of elliptic functions and, without a doubt, the primary theorem for Ramanl\ian in the development of his highly original approaclt to the theory of elliptic functions.

5.1. Definitions of Hypergcometric Series and Elliptic Integrals Firet, _review tho notation for a rising fac/orial Or II givw in (Ll.3). o."fine (a)Q = 1 and, for n;:>: 1,

(:;.1.1)

(a) .. :'=' a{a

.hift~Jacu,ritJl

+ 1)(0 +2)·· '(0 + " - I).

UnfQrtunately, the Il()tation (5.l.l) is the !!alDOl as that in ( 1.1.1).

However, the notations (1.1.1) and (&.1.1) are rarely used in the same context, and!lO no confusion ~ho"ld .... L~. Note that ( I ).. = n!. The ootllOOn (S.U) ill also called the Pochammcr 6ymbol in ~pecial.ly

-

H19

m

B. C. BERNDT

110

the older literature, but tbe notation is misnllffied, not suggesti\-e, and should be laid 1O mit. Definition 5.1.1.

ut II,

II, lind c be arllilnlry comp/n nwnbe ....

rwnJXlsitit'( inUgeJ". Then, for 1,1 < 1, /Ji( GIl .........n or ordinary h~_m:.e function , 1", (II, II; e; ~ ) .. dtfimd tzctpf liIat C CUrlJlOt /)(: a

b, ~

,F, (Il,b; c;z):-

(S.U)

L "~

Readers can easily apply the rali<;> test. to show that the series in

(5. 1.2) inde
Izi <

I.

Certain speci.al cases of hypergoometric funct.ions with elliptic inlegral$.

ILI"
oonnected

Definition 5. 1.2. Tht: compktt dlipfiI < I by (5.1.3) The number k i.! oolled the moduJlII, and 1M numba k' ., viI called the rompknumtary modul ..... Lem ma 5.1.3. For

It 1<

,p

it

I,

(5.1.4)

Proof. In our proof, we ~press the value of M integral in ternul of the daasical Gamma fun<:tlon r (z), and UlIe the well.known ,,,jue

r Hl

"',fii. Readen! unfamiliar with the GlLIDmll function might

e•...J.UlIte the integral below. Using the binomial series in the integrand of K (k) , inverting the order of

,,-ant to

Uge

indlKtion on n

10

"""mat;on and integnr.1.lon by .. bsolute ooo'wg"""" , and lL'ling the aforementioned ,...J.ue of the integral of sin2K Q m-er [0, .-/2], "'"/: find

"'

•

SPlRlT OF RAMANUJAN

that

/.'" ,j,

o Allhough not nee< saT)' for the dew~lopment of tIN: theo:wy in tbis chapter, we next define elliptic integJals of the 8OCOnd kind_

Definition 5.1 .4. TIw romp/de elliptic illlegml oj 1M M:anld kind it Ikfinl'lll fo r 1.1:1 < I by (5.1.5)

£ (k) "..

l~/l VI - Fsin~ ,pdJ.

Re&denI ,,·ill be familiar wit h complete integrals of the second kind from elementary calculus. Recall that if an elli(l'le is given by the pNlUlletric equat ions :.. =" 006 6, y = II sin o, then the length L of the per1nleter of this eJli(l
L _4a

/.'" ,Ii II

where "denotes the OCCC1lt ricity of the ellipse. (Not.\! that L can be e>:pl ~ by 3l\ elliptic integral of the SOlCOIld kind.) An llllIllogue of Lem.m.s 5.1 .3 holds for elliptic integrals of the second kind.

Exercise 5.1.5. Prove UUlt (5.1.6)

0

-'2''2,1; 01:, ) .

. (1 Ek=2~11 ("

We Del
COl r

ed malarial

B. C. BERNDT

112

Theorem 5 .1.6 (Landen's Transformation). For 0 < x < I, K

('"Ii) _(1 + x)K(xj, 1 +%

or, equiwlent/y. by Lemma ,s.1.3, (5. 1.7)

2 Fl

11 4.) =( l+xh F) (12' 2"1:I:x') . (2'2:I'(I+;[)2

Corollary 5.1.7. ForO

(a.LS) 2FJ Proof.

O. ~; I ; 1 -

O~""'ing

< z < I, (:

~

:Y)

= (I

+ xhF[

G. ~; I;X 2).

t hat

( I :XX)2 =1 - C~;)2 we see that (S. I_8) follows immooiawly from (5.1 .7). Copyrighted Material

o

SPIRIT OF RA M A NUJA N

113

The version of Landen 's transform~tion lhat ",-e have ginm is actually aspecial case of a mOre general Landen transformation. namely. (&.1.9) 2Fl (a,b;2b; ( I ':-x)2 ) = ( I + X)2g 2Fl

(01,0 -b+ ~;b + ~ ; Xl) .

By Lemma &.1.3 and Theorem 5.1.6, the special case of (&.1.9) that ,,-e ha\"e proved is for a = b = ~ . There ill still one further version of Landen's t ransformation. To describe this transfonnation, ",-e need to define incomplete elliptic integrals. Defin it ion 5 .1.8. 110 < a th~ first kind is defined by

~

1
~J/iplv.

int,grrd ()I

We now state Landen's transformation for inoomplete elliptic integrals of the first kind. Exercise 5.1.9 .

Il lxl < I and

" sin II = sin(2,8 - a) ,

0$a<1I",OS,8<1
prove that

(&.1.10)

(1+")1" o

Vi

When (> = 1< and {3 = 1
Lemm a 5. 1. 10 . A8X - 0+ , ($.1.11 ) ",herr. C i.f

0

ronstonl-copyrfghled Material

B.C. BERNDT

11< P roof. By Lemma 5.1.3, it rruffices

to

prove that

2K(,Ji) '" - locO - z) + C.

(~. 1.I2)

In the I'(!presentation (5.1.3), Jet I _ li n ~, 110 that COlI 6 = ';1

,2.

Thus, (5. 1.13)

Note thM, for 0 <

%

< I,

J.

(5.1.14)

1

o

Now examine. for 0

xdt ""''0 .. - q() I xt

::!') .

< :z: < I , I

.

..

}~.

Ob&er,'e that R(:r ) is continuous at r,. 1-. [n fact .

(5. U S)

R(l )_

Thus, as

% -

y mlltoti<:

w

2(2

I}

-I - I

1-, the two integrals

i~

,, _

- - '" log2. J.'~ o l +t

(5. 1.1 3) and (t:>. L14 ) are

air

e!\Ch o the r, since their difference apprOllches a oonstant ,

namely, log:'!,

(5. 1.1 5).

fo' {I

\Ill

x - 1- , Hence, (5. 1.12) follows from (5. 1.14) and 0

5.2 . T he Main Theore m Briefly, our primary tOO is to

p~

that

where q is a certain function of x. We shall establish t his relation lifter P series of &ncillary lemmas.

First define, for 0 < (5.2. 1)

X

< I,

11'

SPIRIT OF RAMANUJAN

i;

By eontilluity, since 2F\ (t,~; 1;0) = I and ,F\ (t, I ;~) ~ 00, as l' ..... 1-. by Lemma 5.1.10. "'e C&rI extelld the definitioll of F (%) to z: '" 0 and z: = I by setting

F(O) = 0 Lemma $.2.1. ForO

F (I) = I.

< z < I,

F(~') .. F'

(5.2.2)

Proof. In (5. 1.7) replace

Cl :z),)·

by ( I - %)/( 1 + ~). Obsenll' that

~

..

,

% ~ 1 -( 1+%)l'

'x

,

(1+%),- 1-:2: , 2 1 +z:-c7C 1+<

Hence,

"'''l!

arrive at

(5.2.3) I ! x

IF\

G,~; I ; 1 -

(I

:zX)2) '" 2F\

G,~ ; I j 1 -

z2) .

Now divide (6.2.3) by (S. U ) to d<>duou t h.t ; I j l -%')

(5.'2.4)

, ,' I ,'~') .

Multiplying both side. of (5. 2.4) by - ,.. , exponentiating both sidea, and invoking the definition of F(z ) !rom (5.2. 1), "'''l! com plete the proof of (5. '2. 2). 0 Lemma 5.2.2 . 1/% , 0 < % < I,;" dtJintd by (52.5)

(5.2.6)

: : : ..

~~~) .., A .. A(Il).

B . C. BERNDT

116

Proof. By ( L3.32), (3.6.7), and the definition of >. in (5. 2.5 ),

>.2 2 _ rp4(_q') _ { (q ) -

rp4(ql ) ,

=

4"

2rp(q)rp( - q ) } ' rp1(q) + rp'( q) 4 !-" i+i

=

(1+1=)' ,..

(1+>.)7

= I _xl

. o

T his concludes the proof. Lemma 5.2.3 . If n

=

2 m,

wh~re

m u any nonntg
inttg~r,

(5.2.7) Proof. Replacing x by ( I - x)/( 1 + :::) in Lemma 5.2.1, "'-e readily find that (5.2.8) Applyi ng Lemma 5.2.2 to (5.2.8), we find that

(5.2.9) Iterate (5. 2.9 ) tQ deduce that

F(),'(q)) = F ' (>.2(q2J) = FO' (),2(qf'») = . .. = F'~ (),2(qr J) , that is to say,

o Le mma 5.2.4. If n = 2m . where m U <1"11 nonneg
(5.2.10) P roof. From the defini tion of F{x ) in (5.2.1 ) and Lemm a 5.2.3, we see that if rp4(-f~ )

COP~hle!MZfJri~/

SPIRIT OF RAMAN UJAN

'hm

'"

,FI(!.tj l ; l - xI) _n2Fl(\~; I ; I - % .. ) 2Fd!.P;X,) ,FI i,4;I;x .. ) .

We rewrite the IMt equality in the form

2FI ( , ; 1;xI) t F\ (~.~; I ; %") n 2F,( . ; I; I -x.) = ,H(!.t;I;1 x .. )· But thils implies that F"(1 - %1) =

Fn - %. ) ...·hich ill the same Ill!

0

~'I. Theorem 5.2.5. We haw

(~.2. 1I)

F

Proof. Write Lemma

(1- ¥')~~») = q.

~.I.IO

in the form

(~.2.12)

wbere 0(1) denotes a function which tmds to 0 as %

0". not l"IflC:IMaarily tbe same with each appearance below. Using (5. 2.12) and the definition (5. 2.1), \\~ lee that (5.2. 13) 11$

-

F(x) '" e""~-C"o(l) = Axeo(l) = Ax{ 1 + 0( 1»,

X _ 0". where A =e- C .

L<, (~.2.1 4 )

Note that Xn tend$ to 1 as n - 00. W~ now take the nth I"O()I. of both sides of (5. 2.10) &Ild let n tend to 00. We need to t ake care on the right side when letting n _ 00. TIll"', in t he notation (5.2.14) and with the use of (5.2. 13), we find that

F ( 1-

¥') ~~») = nl~'!, vr.,F-(~I-,~.I

.--

= lim y'A(l

% ..

)( 1 +0(1))

B.C. BERNDT

1I8

(::'.2 _15)

=

q,

where in the pen ultimate line we used (3.6.S). T his completes the proof of Theorem ::'.2.5. o

P roof. From (5.2.&) a nd (5.2.6),

(l + xf = 1+"'(1 _ ",2) = ",,' ('1) ",,4(_'1') 1 J: ~( q)
Le mma 5.2.7. We have

,

(5.2_18)

'P (q) = 2Fl

(I2' '1 2;

1; I -

",,-(-ql)
.

Proof. Iterate (5.2.16) m times to obtain the equalities ,F1 (

'

1

2' '2: 1; 1 -

(5.2.19)

10"4( _ '1 ) ) ,.l'(q) 'P' (q) = r (q4}

,F,

(' I

2' '2; I ; I -

(1 1 - "'=r (qo~ ) 2 Fl ",,'('1 )

",,4{_ q4) ) ",,' (tt )

-»).

2

""' (_'1 2'2 ; 1;1 -
Now let In _ 00 in (5.2.19). ObserviJ:g that
SPIRI T OF RAMA NUJAN

119

Theore m 5 .2.8. Recall that F(x ) i3

defillw loy (S.2.1 ). For 0 < x <

L

(S.2.2O) Of,

ill o/h.,.. words, if

(;'.2.21 )

(;'.2.22 )

P roof. W~ begin by sllmm,...i ...ing fi".t.I",mmil S.2.7 and second Theorem ;'.2.S. If
Fix) )

1.1:=
,,'

F (r) = F ( I - 1.1 ).

(".2.24 )

The last equality (".2.24) implies that (".2.25)

IF, U , ! ; J; I 2F,

x) _

(! '2;' I ; r ) -

2F, U, ~ ; I ; u)

("

2F' I . I ;I ; 1

u

).

Suppose that we can show that the denominators in (5.2.2S) ace equal. Le., that

(". 2.26) Then it will follow from (5.2.23) that
(! ,!; I ; r ),

i.e., (S.2.20) holds. We show that (:'.2.26) easily follows from the rnOllQtonid ty of 2F, ( i, 1; x ) o n (0,1). SUPp<>6
I;

of 2F"

B. C.BERNDT

120

It follows from (S.2.2S) that

,Fl

(i.!: 1;1(0) <

, F L (~,!: I: l - xo),

< I - 3:0, which is inoompat_ ihle with the previous condition I -Ito < Zo_ Hence, our assumption that there exists a j)Oint 3:(1 such that (5.2.26) does not hold has prowhich implies, by mOllotonicity, that Uo

duced a contradiction.

0

We place Theorem :>.2.8 in the context of the classical theory of elliptic functions and summarize some of Our principal results in this chapter. Set x = Ie', where, as abm-e, Ie ili the modulus. Also set (5.2.27)

so that the complementary modulus k' is

gi"'~n

by

;0'1 q)

(5.2.28)

r(q) .

T hen , by Theorem 5.2.8, (5.2.29) w here

(5.2.30)

»)

2 ~. (2Hti,!: 1 ;1 le '1 =~ -=<'Xp - ". F (i 1.}'1<2) 21

- cxp

(K') K ' -:If

2'2 "

where K = K (k) and K' = K (k' ).

5.3. Princi ples of Duplication and Dirnidiatioll In this section y.." show how a formula involving x (or k 2 ) , !I (defined by (5.2.30», and z (defined by (5. 2.29 )) can be converted tQ an equation involving 2y or 1//2.

Define, for 0 < (.~ .3. 1 )

~'

< 1, %

from which it follOW!! that (5.3.2)

<)' 1 + .l1-:r:

, = (1 /1

SPIRJT OF RAMANUJAN

121

FunherlDOre, define (~. 3.3)

By

e-·' ''''' Jo'(1")

and

(~. 2. 1 ), ( ~.3. 2), Lemma ~.2. I ,

(U4 )

%' -

and

2Jo'l (i. I: 1;1").

(~.3.3),

e-'_ F(X) =F( 4~ ) _JF(r)=e -,'f2 (1+

r )l

Hence, (~. 3.~)

11 -

FurtherlllOrf.!, by : = , F]

(~.2.29l,

(5.3.2),

(~. 1.7) ,

(~.3.3),

and

(~' ~;I;X) = ,F (~,~; I;( I :~)2 ) 1

tUG) ThIUl, !iOlving (5.3.6) for (~.3. 7)

iv'·

r

= (I

+ 1?)2 F I

- (I

+ .;;})I.

and lI!Iing

(~.3. 1 ),

.+

G ,~; I:X')

we find tbat

• 1 + ';1 - 2(

xl:·

Theorem 5.3.1 ( Pri nciple of Duplication). Suppose thaI IWO setA of pGrurnel erl, ::<, y, and z and r', y' and:', are re/aud by the tqUalwlUl (~.2.27), ( ~.2 .29). and (~.2.3O) with ::< . 11. lind % rep/aeal by r'. "'. and :'. ruptc.llw/y, Suppose /hey salasfy an equation of the form

fI(r',"" I) - 0, and z as rewltd 10 x' by ( ~.3.2). Then , ~ (5.3. 1). It'e obtain an equalion of the form

(~.3.~),

and (5.3.7).

Theore m 5.3.2 (Principle of Dilnidilition). A s III the previol/S thw · rem, s uppose that fw<:I sets of porornd erl, x, 11, and : alld r'. y'. and :', are rewted by the ~$Il&iilYeJh/f .29), and (5.2.30) ""th x,

B. C. BERNDT y, and z rt:placm. by x', an equation of /he form

11'.

and z', rt:$p8:tively. SIlPPOU they ..ati.!fy

ll{x',!I', l) = 0,

and we reuerse the role.! r1f x, !I, and z wi/h those r1f r', 11', ami z', =
n

((l :~P' ~y,

(1

+ .,ti):::)

= O.

5.4. A C atalogue of Formulas for Theta Functions and Eisen st e itl Series Using (5. 2.29), (5.2. 27), the principles of duplication and dimidiation from tbe previous sect.ion, and elementary thel-a function identities, such as thCl'le given in Thoorem 1.3.10, we can derive a plet hora of eva luations of the functions "', >/J. f . and X at di fferent powers of t he argument q in terms of z, x. and q. Using the tools mentioned above, readers should be able to easily derive each of the formulas below. Proofs of all the results in Theorems 5.4.1-5.4.4 can be found in 13 3 . pp_ l:!:!- I:t:>, Entries IU-I:!J. Alter we state the formulas, we offer a few proofs in iUustra t ion. ThCQre m 5.4.1. Ifx, q, and z art: related by (5.2. 27), (5.2.29), and (5. 2.30), tkn (i)

",(q) '" .,fi.,

(ii)

",(-q) '" .,(i.'(I-x)!/~ ,

(iii)

",(_q2) = .,fi.(1 _ x)!/8,

(iv)

",(l) '"

(v)

",(q4) =

(v i) (vii)

.fZ-Vi (1 + ,)1 x), ~.,fi. (I +(1 _ X)l/') ,

"'( v"1l = .,fi. (\


=

+ ,.Ix) !/2 ,

0(1- ,.Ix)! / 2.

Copyrighted Material

SPIRIT OF RAMANUJAN Proof. We offer proofs of (ii)-(iv ). To prO\"e (ii)- (iv), employ (i) a.nd use. re;;pe<:tively, (5.2.27), ( " 3.32), am! (3.6.7). 0 Theorem 5 .4.2. 11;t, q, and .: ore related by (5. 2.27), (5.2.29), and (5.2.30), then

(i)

~(q)

(ii)

~(-q) =

= jf"z(;t/q)I/S,

Jfz (;t(1 -

x)/ q)' /8,

(iii)

~(l) = !';:(;t/q)' /t ,

(iv)

~(qt) =

! Jfz {(1 _ ,,",-=-.") /q} 1/2 ,

(v)

~(q') =

i,fi{\ - ( I - z)"t}fq,

(vi) (vii)

>/i(,jii) lb(- ,jii)

= .;: H( I =

+ vTxl} 1/4 (z/q)l/lfi,

';:H(1 - .fi)}I/·(z/q)'/1~.

P roof. We prove ollly (i}--{iii). To prove (iii). use (3.6.8) along with Theorem 5A.l (i), (ii). To prove (i) , use (iii) and the principle of dimid iation. Part (ii) follows from ( J.3.30) and Theorems a. 4.1(i), (ii) and 5.4.2{i). 0 Theorem 5.4.3. lIz, q, and z are rdoted by (5. 2.27), (a.2.29), and (5.2.30), then

(iii)

I(q) = ';:2 -1/6 (z (l- z)/q}l/l4 , I (-q) = .h2- 1/ 6( 1 _ zi / 6(z / q)l/lt, I (- q' ) = ';:2- 1/ 3 {z(l - x )/q} lIn,

(iv)

I (- q' ) = ,fir I / 3(1 _ z)l /l t (X/ q)1/6.

(i) (ii)

Proof. We prove (i)- {iii). Bot h (ii) and (iii) follow from (1.3.34) and 0 previously proved results. Next employ (1.3.30) w prove (i). T heorem 5 .4.4. lIz. q , and z are ..dated by (5.2.27), (5. 2.29 ), and (r..2.30), t!.en

(i) (ii) (iii)

B.C. BERNDT

124 Exer cise 5.4.5. Provt $Ornc further \\'~

restd~

in Throret'll3

5.4.1~5.4 _4 .

conclude thi:; chapter by deriving repr
p {q2 ), Q(q), Q(q2), R(q), and R(q') in terms of x and z. However, first,,~ need the following two important lhooreDlS, the first of which readers (:aD easily prove.

Theorem 5.4.6. The function z = IF1(l,!; l ox) sotisfiu liLt differentia/. equation

vrz dz I :t( I - z).u 7 + (I - h)dx-;jz=O.

(5.4.1)

Exercise 5.4.7. The diilCllmtinl equction (5 A.I ) ho.s a ffgIllar singular po;>int ot X = O. Sollie (S.4..1 ) by Ihe method ()/ Frol>enitl.! to find that 1m!! of liLt two linearly independell,j soluti01M U ~Fl (~,

!; 1;%).

Those reade,." unfamiliar with sol.jug ordinary differential equations wi l h regular singular points may use the definition of ; = 2F.(!,~;1;:t) to easily check that z = 7Fl(~,~;I;:r:) is indeed a solution of (5.4. 1). Theorem 5.4.8. If y is defined by (5.2.30) and z is defintxl

(5. 2.29 ), then

~y

,

(5. 4.2)

:.::(1

X)Zl'

Proof. Using (5.2.27), (5.2.29), (1.2.4), and Thoorem 5.4.3(ii), (iv), we can readily establish the identity :.::

(5.4.3)

1

f8( - q' ) x = 16'1 fS( q)

(q4;q4)8

=]6'1 (q;q)~<>O.

'Thke the logarithm of both sides of (5.4.3). differentiate "";th respect to q, and multiply both sides by q to deduce that d.r 00 <>On 4 ,,,

="'--c:.::(1 x) d
(5.4.4)

=

1- 8"

~ I .. _ 1 00

=1+8 L l

.,"

n= l

nq

q'"

+8 "

~ I .. _ 1

nq

,"

n~n

n = .p4(q) = Z2.

q

Copyrighted Material

SP IRIT OF RAMA NUJAN

125

where ...-e ha.\-e employed the fifth line ill the displa.y (3.3.7), and (5_2.29). Using the trivial fact dq

.u. =

dy -qc/.x

o

in (5.4.4), we complete the proof. T henrem 5.4. 9 . \vith z defined by (~.2.29) ,

(5.4.5) (5.4.6)

P(q) = (1 - 5:r:)z1 + 12:r:( 1 - :r:)z

p (ql ) = (1- 2x)z2

+ 6J:(i

-

d::r,

:r:)Z ~ .

Proof. We prove ollly (5.4.6); the proof of (5.4.5) is similar. Applying the definition of P (q) , Theorem 5.U(iii ), the definition of I{ - q) from ( \.2.4 ), and l3Sl1y the chain rule along with Theorem 5.4.8, we find that

=

l - 12~ dy

f:

10&(1 -

e-2n~)

n .. '

= I _ 12 :y log( e _1~; e-2~)"" =

- ~ log{e - ~ l2(-e-~~)J

=

_!!.. Jog{e -~z62 -4.:z: ( I _ .:z:)e~}

d,

d, d

= - -log{ z6.:z:(1 -.:z:)} d, = .:z:{l _ .:z: )z1 Jog{z 6.:z:( I-.:z:)}

!

l'

d2

= (1 - .:z:) z - .:z:z + 6.:z:( 1- .:z:) z dx

, d, =( 1 - 2.:z:): + 6:J(1 - .:z: jzdx· Copyrighted Material

o

B. C. BERNDT

126

Exercise 5 .4 .10. AIJematiliely, applv the proce.s of dimidiation io (5.4.6) hi deril!e (5.4.5 ). T heorem 5 .4 . 11. We have (5.4.7)

(5.4.8)

Q(q ) = z4(1 + 14:1: + %~ ), Q(q2) = :4(1 -:I: + x' ).

P roo f. We 6rst Pr(lVC (5.4.8). Secondy, we establish (5.4.7) by applying the process of dimidiation to (5.4.8). From the diJlerential equation (4.2.20) and the chain rule, we find

lhol (5.4.9) From t he definition of q, i.e ., (5.2.30 ), Theorem 5.4.8, and the chain

rule, we next find that

(5.4.10)

dq dx

dll

- ~- -q -

dx

q

z (1

X) :2 "

It follows from (5.4.9) and (5.4.10) that (5.4.11)

x(l _ X1 z2dPJ:2) = P2(q' ) ; Q{ql) .

We next use (5.4.11 ) and Theorem 5.4.9 to establish (5.4.8). Differentiating (5.4.6) with respeo;t to :t, applying the prod u~ rule several times, ILIld 5irnplifying, ""E! deduce t hat (5.4.1 2)

dP {ql)

'" ,

d, (d')' +6x( l - x )z d'dJ;2, =-2z (z - 4{1- 2x )d') - + 6x( I - x ) (d')' +6z{ I - :r:)z-d' , dx dx d2;2 = - 2: + 8z( I - 2x) d:i; + 6x( l - z ) tU

d' , + 6x ( I - x ) (")~ d'z = - 2z .4x( l -x)d2;2 dx +6x(l-x )zdx2

~v. '

= - 2zx( l -x) dx d' ,2 + 6x ( l - x ) dx Copyrighted erial

SPIRlT OF RAMANUJAN :-2%(1 - ", )

127

,p, 3 (") tb; ' ) ( ztb;2-

•

"'here ie the antepenultimate uep we ~mployed the differential equation (~_4.1 ) . Using (5.4.12) and (5.4.6) in (5.4.1 I), "'C 6nd that Q(q2 ): ((1 _2%):2 + 00:(I _ ",)Z ::.)2

-6,{l - , ),' ( - 2x{l- ,) ( , =

(1 - 2Z)2 z·

+ 12x( 1 _

~: - 3 (~n

z)(1 _ 2z):3 ~

)

+ 12z'(1 _ x)2 z3 ~~

: (1 - 2x f z 4 + 12x(1 - x)( I - 2X)ZJ !;

+ 12x( 1 _ x): 3 ( -( 1-2X):; + ~z) = (I - ", +",')z' , where in the penultimate !iDe we utaiW
differen~ial

equation

We now use the Prieciple of Dimidiation to establish (5. 4.7). Ac-

cordingly, from (5. 4.8). Q(q) '" ( I

4Vi (1 + ViP

16X ) ~ .. z + (I + ,fi)' (I + v ) :

'" ( I + 14x + x' )z\

o

after simplification. Theorem 5 .4.12. W", hoOlt (5. 4.13) (5.4.1 4)

+ x)(1 - 34x + ",2), :6(1 + x)( 1 - ,! z ){ l - 2x).

R (q) = :6(1

R(l ) =

(5.4.14) and pr""""-'
(5.4.15)

3,{1 _ %lZ2~Q(ql) = 2P(q2) Q{, ') _ 2R(q'). CopyngIOd Matenal

8. C. BERNDT

128

Solving (aA.IS ) for R(q~) and employing (5 .4.6) lind (5 .4 .8) (twice), "'e find that

R (q2)

=

P(q~JQ(q~) _ ~Z( l _ X)Z2 d~2)

.., { (1 - 2x)z2 +6X( I -Z)Z:!; }

~x(l -

_ = 26

{(1 -

%)z2

{4Z~:(1

z4(1 _ z + x~)

_ z+ x 2) + : 4( _ 1 + 2% )}

2:1: )( 1 - x + ",2 ) + ~Z( l

= :6(1 - 2x)( I + %)

- x){ l -

h )}

(1- ~J:) ,

which completes tbe proof of (5.4.14). Applying the Principle of Dimidiation to (5 .4.14), we find that

• =. (1+ (1 '.fi) 8.fi) + ,/%)1 (1 - {I +.fi)1

R(q) = z( l + v z }

X(1 - (l!~2) '" 26 (I + 6.,.tX +x) (I - 6"fi +x) (l + x ) = 26( 1 + x)( 1 - 34x + 1?),

o

which complete!! the proof of (5A. 13).

Exercise 5.4.13. Pro1H: that P {q4) = 22 ( l - ~x) + 3x( l Q(q' ) = z' ( I - x+ R(q4 ) =

,,8 ( I _

~x)

d-

xl. ~ ,

h:r:2) , (1- X -

J~ X2) .

5.5. Notes We have fo!lov.-ed Rarnanujan 's presentation in Chapler 17 of his second notebook [193], as pre!lented by the author in 134, pp. 91-102, Sections 2-6J. in our devdopment of Thoorem ;'2.8. However. the clever proof of Thoorem ;'A .8 that we have given was communicated to US by H. II. Chan and is much simphlr than the proof in (34. p. 120J; Copyrighted Material

SPIRJT OF RAMANUJAN

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_ Ilia paper with Y. L. Ong (71]. T he Principles of Duplication and Dimidia~ion are due 1.0 Jacobi [131 1. Ramanujan states the Principlp of Duplication in Entry 13(vii) of Chapter 17 in his second notebook [1931. [34 , p. 127]. It is dear that Ramanujan frequently used the Principles of Duplication and Dimidiation in his wo rk on elliptic func· tions. Our presentation of these principles is similar 1.0 t hat given in the author 's book [34, pp. 125-126]. Por Ramanujan , it was natural t.O ask whether there "'ere other theories of elliptic fWlctions in wbich the hypergeometric fnnc~ion ?F, (~, 1; x ) is T(>placed by other hypergeometric fnnctions. In his famous paper, Modular equatwns on~ approximations to 1r [184), [192, pp. 23- 391, Ramanujan Wl"Qt.e, ""There are <»r~pondi n& ~heo­ ries in which q is replaced by one Or other of the functions

i;

where

K, = 2Fl(l,1;I;x),

K2 =

2 Fl(!,i,I;x),

K 3 = 2F,(1,

1.l ;x).M

Pag"" 257-262 in Ramanujan's second no •.,book ","". .-1"""1....-1 .n.-1.... ''eloping these theories. The many claims 00 these six pages were first prowd by Berndt, S. Bhargava, and F. G. Garvan [41 ]. See also Chapter 33 in Berndt 's book [38]. Ho".ever, Raroarmja n's theories are by no means <»mplete, and since the appearance of [41 ), many other papers have been "'ritten on Ramanujan's alternati,'e theories, with still much remaioing 1.0 be accomplished. The fonnula (5.1.6 ) is due 1.0 the Scottish mathematician C. Maclaurin [153] in 1742. Our proof of Landen 's transformation, Theorem 5.1.6, is almost identical to the proof that the English astronomer J . l vory [129) ga\'e in 1796 for (5. 1.6). When h'Ory aubmit· ted tllS paper to editor John Piayr.. ir, he nat urally j",;luueU .........ver letter, but, rather surprisingly (at lealt 1.0 us), the cover letter was published along ",ith the paper. In this letter, he relat.es how he dis-<;overed his formula . ~ Having , Il8 you know , bestowed a good deaJ of time and attention o~~~Ql'Mm~IIl~~ of physical astronomy

B.C. BER NDT

'30

which relate!! to the mutual disturban<es of the planet.5, I have, naturally, been led wooDsider the various methods of resolving the formula (4~ + b~ - 2ab cos(,6)n into infinite series of t he form A + B 008 (,6 + C cos2¢ + &c. In the course of t hese investigations, a series for the rectification of the ellipsis occurred to me, remarkable for its simplicity, as well as its rapid convergency. As I believe it to be new, I send it to you .... "

Landen's transformation was introduced by J. Landen in a pa_ per written in 1771 {U T] but developed more com pletely in his paper 113S] published in 1775. This transformation was crucial in OUT proof of the fundamental Thoorem 5.2.8. The importance of Lan· den's transformation is conveyed by G. MittaJ:-Leffier, who. in his survey article (160J on elliptic li.mctions written in 1923, emphasizes, "Euler's addition theorem and the traz:sformation theor~m of Landen and Lagrange were the two fundamental idellS of which the tbeory of elliptic functions was in possession when this theory WlIs brough~ up for renewed consideration by Legendre in 1786.~ Born in 1719, Landen WlIs appointtd as t he land·agent to the Earl Fitzwilliam, a post he held until his retiremeut t"""O yean! before his death in 1790. According to an edition of Encyclopedia Britanniw published in 1882, ~H e [LandenJ lived a '~ry retired life, and saw little or nothing of sociely; when he did mingle in il , his dogmatism and pugnacity caused him to be generally sllunned. ~ Landen made several contributions to the Ladie5 Dillry , which Wlls published in England from 1704 to 1816 and "designed principally for the amusement and instruction of tbe fair sex.n As WlIs COmmOn with other contributors, Landen frequently used pseudonyms, !uch as Si r Stately Stiff, Peter Walton, Waltoniensis, C. Bumpkin, and Peter Puzzl~m, for problems he proposed and .'IOh"ed. The largest portion of each issue ,,"as de'"Oted to the presenta tion of mathematical problems and their s0lutions. Despite its name, of the 913 contributors of mathematical problems and solutions Over the years of its publication, only 32 were womcn. For additional informMion ahout Landen 8nd the Lo.di~. Diary, see a paper by G. Almkvist lind the lIutbar [llJ. Readers are also recommended to read G. N. Watton's article, Th~ marquu and th ~ land- ag~nt; a tale 0/ the eighteenth " nt'llT'll [2 17J. (You know the Copyrighted Material

SPIRIT OF RAMANUJA N

\31

identity of the land-agent; to augment your curiO/iity, we refrain from tellina: you the identity of th" m,.,.r",;~) Exercise 5.1.9 is just one of many beautiful transformation fOT_ (Iluias for elliptic integrals, many of which are due to Jacobi and / or Ramanujan [34, pp. 104-113J. We offer a few of these transformatiou formulas as exercises. Exercise 5 .5. 1. I/O < o,p <

ill" and tann

= .II

:z: tan,8, show

au" T he uext exercise is a form of the of elliptic functions.

~ddition

theorem in the theory

Exercise 5.5.2 . I/O < n, (J < ~1r ar.d ooto ootfJ

.,'

=

.II

:t, prow:

The following exercise gives the cbssical duplication formula for elliptic integrals.

Exe rcise 5.5.3. I/O < 0. , fJ <

!,,- and ooto.tan(.B/2) = JI

xsin 2 0.,

prove that

2[

' JI d:sin2 1/J =

t

JI d:ain2 1/J'

Exercise 5 .5.4. I/O < 0.,,8 < ~1f and (I (I +%)siuo, prow: that

+ :tsin2 0.)sin{J

=

T he transformation above is called Gauss's transformation and is similar in form to Landen 's transformation given in Exercise 5.1.9. Historically, the theory of elliptic :Unctions arose from the problem of inverting inco~~JH~~, such as the incomplete

B . C . BERNDT elliptic integral of the first kind given in Definition 5.L8. These in· version problems were motivated by the welJ-knOl"n inversion of tbe t rigonometric integral arcsinz =

f.' ". in ..,II ~2'

O
I.

Landen 's transformation for inoomple>e elliptic integrals is only one example of many beautiful relations that exist between elliptic integrals. many of which are due to Jacobi (131] and Ramanujan [193. Chapter 17, 5e<:tion 8], (34. pp. 1M- 1 13). T he text [180, C bapters 24) by V. P rasolov and Y . Solovyev is a superb introduction to elliptic integrals describing t he fundamental ,heorems of Legendre. Jacobi, ann N. H. AI,.,!. Lemma 5.1.10 is a primitive, special case of a much more general asymptotic formula for G aussian hypergeometric series; for example, see [33, pp. 77-18]. The catalogue of formulas for theta functions given in 5e<:tion 5.4 is taken from Sections 10-12 of Chapter 11 in Ramanujan's second notebook [1931. [33, pp. 122- 124J. T he formulas for Q (q) and R (q) in Theorems 5.4.1 1 and 5.4.12 can be fou1d in Section 13 of Chapter 11 in Ramanujan's second notebook (1931. [34, pp. 126- 121]. Although the formulas for P (q ) in Theorem 5.4.9 are not found in Section 13, from the appearances of other formulas for P (q) in that section, it is clear that Ramanujan knew Theorem 5.4.9. Ramanujan used his formulas for P (q), Q (q) , and R(q) to derive a multitude of elegant evaluations of infinite series in 5e<:tions 13-11 of Chapter 17 [34, pp. 126-139]. For example, I

~

1 + 22: ~_ l

cos

h(I ) = z y

"d

Copyrighted Material

Chapter 6

Applications of the Primary Theorem of Chapter 5

6. 1. Int ro duction 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 terms of z and 1: that ~ from T heorem 5.2.8. We first demoll5trate in Section 6.2 how the formula.!l we deri,~ for Ei!;
elegant formula for ta (n ). In ~ion 6.3 "'I) define one of the most important concepts in t ho! theory of elliptic fuoctioll.5. namely, a modular equation _ Landen's transformation, Theorem 5.1.6, can be thought of as II. modular equation of degnl
easy. There is no single method that one can use W produce modular equations. In Soction 6.3, we shall derive !;Orne modular equations of degree 3.

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->3,

H.C. BERNDT

'" 6.2. S ums of Squares and 'l)iangular Numbers

"'e firot offer ""me corolla.o. i"" of T heo. ~1l1 ;;.4.9. Recall tllIot 0 <.: '" <.: I

and q and z are defined by (5.2.30) and (5.2.29), respectively. Corollary 6.2.1. We have ~

(6.2 .1)

..,

2P(q2 ) _ P {q) = 1 + 24 '("""'

nq

"

t..... l + q ..

= z2( 1

+ x ).

Proof. Applying Theorem 5.4.9, we immediately deduce that (6.2.2) 0" t he o t her hand ,

(6.2.3)

~(" 2P(q1 )_ p (q) = 1+24'" _ nq 2" ~ I q" n~ l

")

+ I TU[ qn

=

1+24 '~, " nq . L.. l + q~

.. _ 1

Combining (6. 2.2) and (6.2.3). we deduce (6.2.1 ). Corollary 6.2.2. lVe

0

oo~e

2P(q4) - P(q2)=1 +24 f nq2:

n~ ll + qn

= Z2(1 _ ~%). 2

Proof. Using the process of duplication along with Corollary 6.2.1, we find that

o Corollary 6.2.3. lYe ha'l1e

1 + 8~ (

,. ,

1)"'Iq" :%2( I _ z).

I +q" Copyrighted Material

L..

SPIRIT OF RAMANUJAN

135

Proof. Using Corollaries 6.2.2 and 6.2.1 , we find that

o

from which Corollary 6 .2.3 is immediate.

Theorem 6 .2.4. Far (6.2.4 )

~ach

p<)5ilivt:

;n~egu

n,

r~(n ) = S:2.~d.

'1"

"'

Fi rst Proo f of Theorem 6.2.4. From Theorem 5.4.I(ii) and Corollary 6.2.3,

Iha~ IS,

w.th q replaced by - q,

which arises fTom (3.3.12) and (3.3.13) in OW" second proof of Thoorem 3.3.1 , i.e., Theorem 6.2.4 . T hus. the remainder of the proof of Theorem 6.2.4 is identical to that of the aforementioned proof. 0 We now give a second proof of Theorem 6.2.4 , which is a variation of Our first. proof above.

Second Proof of Theorem 6 .2.4. Recall that o(n) = L dln d. We extend the definit ion of d en) by sett ing den) = 0, if n is not an integer. With this ill ~ j ~ ~m 5.4.1, T heorem 5.4.9,

8 . C. BERNDT

136

and Exercise 5.4.13, we fi nd t hat 4 I ~ r. (n)q'· = ",'(q) = z~ = '3 P {q') - j"P(q)

= ;

(1 - 24f"("/4)q~) - ~ (I - 24 f: 0'(n)9") n~'

n~l

= 1+ L (8o'(nl - 32o{n/4)) q".

""'

Equating coefficients of q" , n > I , on the extremal sides above, we find that r ,(n} = So{") - 32q(n/4) = 8 d,

L

.,"."

o

which completes our second proof of Theorem 6. 2.4.

To re-deri\"e J acobi 's formula for r8 (n), ....e need the following consequence of T heorem 5.4.11. Corollary 6.2.:>. We (6.2_5)

haUl:

(

- ( I)""'" "q )

16O(q2) - Q(q)= 15 1+162: -] " _I

=15z'(I - :tf.

q

Proof. Dy a direct calculation, 16O(q2) - Q (q) = 15 + 16,240

00

32n

L In qq

2.. -

00

240

.. _ 1

= 15 ( I + 16

3"

L 1" q n .. q _ I

f: (-?'":q").

.. _ 1

q

On the other hand, using Theorem 5.4.11 , we find t hat 16O(q2) - Q (q) = 16z'(1 - x + ,,2) _ z'( I + 14% + xl ) = \:>z' (\ -

The desired result now follows from the last twO identities.

Theorem 6.2.6. For each pasiti"" intqer n, (6.2.6)

."

Copyrighted Material

:rf. 0

SPIRlT OF RAMA NUJAN Proof. From Theorem 5.4.I (ii ) and Corollary 6.2.5,

- (..,8( _q)=Z 4(l _ x )2 ,. 1+16'"

"-. L-

Thus , replacing q by -q , we deduce that

' J" "'I '" q"

I

which is preei.'lely (3.5.14) in our previous proof of Theorem 3.5.4, or Theorem 6.2.6. T he remainder of the proof is ~hen the same as 0 before. Exercise ti.:>.. 7. Gwe anotller proof 01 Tlleorem 6.2.6 in tile foUowing mannU. First, using formu lru for Q(q), Q(q2 ), ond Q (q4) from Theorem 5.4.11 and Exercise 5.4.13, re$pedively, proue that

-"-.

L r!(n )q"

= z4 = I

-

+L

( 1603(n ) - 3203(n/ 2)

+ 2560 3(n/ 4 )) q" ,

wh ~re

0"3(m ) = 0, ifm is not an intefJ". Second, by considering the cases n even and n odd sepamtely, shew that

1603(n ) - 320"3(n/2) + 25603 (n/~) = 16{-

W :~:) _I )d,f.

""

We DOW ~urn our a~t.en~ion \.0 sums of uiangular numbers. First, we give a proof of Theorem 3.6.3 along the lines of the previous proofs in tms section. The following simple exercise will be used in our proof below. Exerdse 6.2.8. If m u on even poritive integu,

th~

O" (m) - 3O" (m/ 2) + 2o (m/ 4) = O. Theorem 6.2.9. For each

nonnegati~e

integer n,

P roof. By Theorem 5.4.2(iii ), (6.2.7)

-

L !4 (n )q2n+1 = 'l'/>4 (q2 ) =

.

_ x z2. n_O Copyrighted Material 16

B.C. BERNDT

138

T hen, using (6.2.7) in conjunction with Corollaries 6.2.1 and 6.2.2 and Exercise 6.2.8, we deduce that 1 L"" t. (n)q2 n -l- l .. - 241 P(q) + gP(q2) "~

1 12 P(q4 )

~

.., L

=

(o(m)-Su(m/ 2) + 2o(m / 4)) q'"

~

L (J(2n + 1 ).2 ~+1.

=

Equating the coefficient.$ of CQmplete the proof.

q2n+l

on the extremal sides above, ....e

o

In the notes to Chapter 3, we mentioned a formula for ts(n), which we now prove. We first need another corollary_ C orollary 6. 2 .10.

IV~ ha,,~ 00

L ,.,

1..:1

1 sinh(ky) = gZ4 x .

P roo£. Using the definition (&.2.30) a.r.d T heorem 5.4.11, "..., find tbat (6.2.S)

"" L

,,3

co

smh(ky) = 2

k_ l

L

k_ l

=

J!l

ek~ e-k~

""

= 2

L

k3e- k~

I

e- >kv

k_ l

I • 8: x.

o Corollary 6 .2.11 . We

h
SPIRIT OF RAMANUJAN

139

proor. From ThooreDl M.2(i), I

q.,r.."(q) .. 16~'z. Employing Corollary 6.2.10 and (6.2.8), "'l' complete the proof.

0

Theorem 6.2. 12. For ach po.rihl/e illieger n .

I,{n) = 4J(fH-l)

('HI )I" _

Proof. By CoroUary 6.2.11,

..,

.... \

oil" ,,' " odd

Equaling coefficients of q"+1 on both sides aoo-.'l', "'l' complete the

0

proof.

R£call that in T heorem 4.2. 4 we established the following: fundamental theorem.

Theorem 6 .2. 13. lYe hove (6.2.9) Our repreaentations I'row Theorems 5.4.11 and 5.4. 12 enable to give a very sim ple proof of Theore m 6.2.13.

Wi

Proof. It will be simpler to use the argument 'I' instead of q. Thus, from Theorems 5.4.11 and 5.4.12, "'" find that

Q'(.r) - R2(q2) = Z12(1 _ Z + Z2)' _ Z12(1 + z)2(1 _ (6.2.10)

~;r)2 ( 1 _ 2z)2

= l3'zI2;r2(1 - z )',

titer a dOllage of eleDlentary algebra. On the other Iwld, by Theorem 5. 4.3(iii), 172&t (~ ; ~)!!

(6.2. 1I )

= ITZ8q' {"{-q') .. :r . 3''1' ~122-·:r:2{ 1

:r:)2/ .,2

_ ~3':12z2 ( 1 _ Z)2.

Combining (6.2.10) and (6_2.11) and repla.clng: 'I' by 'I, we complete

the proof.

Capyr;ghted Material

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140

B . C . BERNDT

6.3. M odular Equatio ns Defi nit ion 6. 3 .1. LeI K. K', L, a"d f} denote complete ellip/ic integrals of the first kind a830ciated wilh the moduli k, k' : = ';J kl , t , and := ';1 (l , re3p«tively, where 0 < k . t < 1. Suppose that

e

K' £' "- ~ K L for s(lme positive integer n. A ..,lation between k and ( induced (6.3.1) is aJlted a modular e.quati(ln of degru n. (6 _3 _1)

b~

Following Ramanujan , set () _ k 2

,,'

We often say that jJ has degree n over (). Using Lemma ". 1.3, we may replace the defining relation (6.3.1) for a. modular equation by the equivalent relation

i;

(6.3.2)

n2F1(i, i, I ; 1 - Q) "" lF1(i, I ; 1 - jJ) 2F,(i, ~; 1, 0- ) lF1 ( i ,! : l;jJ)

U~i ng

(:>_2_29) and the formulas from Section ".4, we see that a modular equation can be considered as an identity amongst theta fun<;t.iOI15 with arguments q and theta functions with arguments q". In fact, moot often One ""t"bli.h""" moduJ...- o:quation of deg>ee" IIi first proving the requisite theta funct.ion identity. Then ,,~ use the formulas from Se.::tion :;.4 to e1Cpress theta functions with argument q in terms of Q , % = Zl, and (possibly) q. and the theta functions wi th argument qn in tcnru; of P. z.. , and (pa;sibly) q", where (6.3.3)

T he mult iplier m of degree n is defined by

(6.3.4 1

m = ",,2{q) .r{q")

=:!. : ..

We note that the method of establishing m<>
SPIRIT OF RAMANUJAN

'"

discover or construct modular equations. One need! to be resourceful and use II variety of tools. Generally, as the degree of the modular equation increases, the difficulty of establishing modular equations rises sharply. The foHowing simple, bu\ enormously useful, device ellabIes WI to produce new modular equations from previously derived modular equations. Theorem 6.3.2 (Method of Reciprocation). If wt rep/act a by 1- 13, P by J - Ct . and m by '111m in a moddar equation of dey""" n , thtn lOt oblain a new modular equation of lilt 3ame degru. Proof. Jfwe make the indicated substitutions in the definition (6.3.4) u( th" wultiplier, w
~'" 2F I(j,1;1;I-Pl m 2Fl(i,~;1;1 0 )

(6.3.~)

Rearranging

(6.3 .~ )

and using (6.3.4 ), we deduce that 2Fl
'

i.e., I!."Cohtain the defining relation (6.3.2) for a modular equation.

0

We now discuss modular equations of degree 3. We need several identities for Lambert series. Theorem 6.3.3. If (1) denQte~ the kgendre symbol, then

(ii) '0'(')

~(q3)

(iii)

= 1+

- Wr"+l (_ I )",jIn+2 ) 1 +( q)Jn+1 + 1 +( q)3nH '

("'"

qGn +l "" rfn ...S ) 1 tf"+1 - ~ 1 IIn+$

fi~

~(q) ¢(q3) = I +3

(iv)

~ (

~

Part (il is identical to Theorem 3.7.7, and part (ii) can be found in (3.7.8).

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B. C. BERNDT

142

Proof of (iii). Replacing q~ by q in (1.3.60) and employing the Ja.cobi triple product identit y, equation (1.3.11 ), we find that (6.3.6) M

't'2 ( ~ q) / ( _q ) =

L

n __ oo

=

(611 + l )q(3n' +n)/~

d

dz {Z/ (q2 :~ , q/~6l} 1 ' _ 1

d : / (q, q2 ) dz (log {zf (q2z6, q/ zG) }) I•• •

: I (q, qZ ) (1 + :. (log ( _ q/ 2 6 ; qJ) ",,(_ q226 ; t/ )",, (q\ q3)",, ) 1•• ,) . Using the /l. laclaurin series fOr Jog( l + z), inverting the order of sum· malion, and then summing the resulting geometric series, we find that , for lal < 1, M

(6.3.7)

log e- 0 ; q3 )"" =

..,

L log (l + aq3

ft )

Using (6.3.7) in (6.3.6), we find that

r( - q)/ (- q) ( d "" ( - qz -~t d "" (_qz z6) n) J (q, qZ ) = 1 - dz n (1 orn) - dz 0 (1 qJn)

L

L

n_ ]

n~ l

(6.3.8)

""

= 1- 6

L

n -o


<XI

L

,. \

q"n 'U I +q3n +2 '

n -fl

where we expanded the summands in geometric series and then inverted t he order of summation. Now, by Theorem 1.3.9, (1.3. 15), lUld Copyrighted Material

SPIRIT O F RAMA NUJAN

143

(3.2.7) , ",,2(q )f(q)

",,2 (q)( _ q ; _q )""

f ( q, q2 ) - (q; =

ql)",,( q2 ; ql ).,,( ql;

",,2 (q ) (- q;-q)." (q; q).., (q;

q3 )""

(q: q)oo
J( ) (tt; _ q3)"" = op q ( q3; qa)"" opl(q) = op(q3) '

(6.3.9)

Replacing q by - q in (6.3.8) and then using (6. 3.9), we complete the _~ ( ,q. 0

P roof of (iv) . The proof is similar to the previous proof. Using (1.3.61), the J acobi triple product identi~y in equation {1.3. 1I}, and (6.3.7), we find that (6.3.10) ~

~(q2 ) f2(_q) = ~ (3n + l )q3n'+2"

d

= dz {zf(q~:3,q/Z3)} 1'''1

d == / (q, q~) dz (log {z/(q5 .. 3, q/ z3)}lI •• , =

J(q, q~)

:z

(log { z(

- l ..3; q6 )",,(- q/ zl; q6 ).,,(q6; q6) oo} )1 •• 1

,(1 - 3~~ {q'"" t".'}) . 1 +q6n+I - I+q6n+S

= f {q, q )

Using ( \.3.11 ), (1.3.14), and ( \.3.15 ), we fiud that ~(ql1 f2{q)

_ f { q, qS)-

B. C . BERNDT

144

Replacing q by - q in (6.3.10) lind then using the calculation atxr.-e, we finish the proof. 0

to

In the theory of modular equatiom of degree 3, it is advantageous introduce the parameter p defined by

(6.3_11 )

m "" I

+ 2p.

Theorem 6 .3.4 (Modular Equa tions of Degree 3). Let {J have deqru 3 over Ct, and let m denote the multiplier 0/ degree 3. Then

(il (ii) (iii)

(iv)

14~

SPlRlT OF RAMANUJAN

Proof of ( i ) . Using (iii) of Theorem 6.3.3 twice, once with q repiau
(6.3.12)

by Theorem 6.3.3{iv). Employing Thoorems ~.4 .I (i), (iii) aIld ~.4.2{i), we trll!l$Cribe (6.3.12) into the form / ~3/2 %3 2(1_ a )3/8 _,_+\

z~/2

''S/~'(~ - 2.~::-';:/"''' I .~/~q~)':;;/'

~~ '2( 1 -11)1/8 -

which upon oirnplification yi
2 -1n z~n(tllq )l'8 '

tru- first. equAlity !If (i).

The seoond equality of (i) ;5 the rociprocal of the first . Proof of (n ). From Theorem 6.3.3{ii),

,

,

-<,- (0) ( 1+'" ( q)~


(6.3.13)

1-0)")

- 1+q ~

0

B. C. BERNDT

'<6

by T heorem 6.l.l(i). Coowrting (6.3.l3) into a modular equation by applications of Theorems 5.4.I( i), (ii) ;md 5.4.2{iii), we deduce that .jZ1 ZJ - .,hjzJ{(l - o){l - PW l4 '" .jz] Z3 q(o j q}I/ 4{(J/ q3)1/ \

from whicll (ii) follows.

0

Proof of ( iii). Using Theorem 6.3.3(ili), ~(q)

"''l

find that

"'" ( (_I )"q3n+l (_ I)"q3nH) op(tf) +2'f( ;f) =3+6 L 1+( q)3"+! +1+( q)JnH
"-,

+ J2

~

L "=1

(q6n+2

q-Gn+4)

- 1+;fn+2+1+;fnH

~ 3('+2t.(j) ,+;"')") = 3rp(q)
(6.3.1 4)

by Theorem 6.3.3(ii). Using Theorem SA _I (i), (iii), (6.3.14 ) in the form 3/2 1/2 ZJ

Zj

"'~

may re"'rite

l /2( , - 0 )'" + 2 =31/2 ( 1 - (J)llS = 3.fi"iZl. Zj

Simplifying, we Qbtain the second equality of (iii). The first equality of (111) Is the reciprocal of t he second. 0 Before embarking on the proofs of the remaining modular eqU!It ions, we provide some useful parameterizatioIL'l that greatly facilitate the derivation of modula.r equations of degree 3. from Theorem 6.3.4(iii),

(6.3.15) Thus, from Thoorem 6.3.4(i) and (6 .3.15), (6.3.16 )

(\ _~3 )'/8

=1+ (~)1/8 = m;1

Next, from Theorem 6_3.4 (iii), (6.3.17)

( {l_oi\l/$

3-m

"&ob'lfiHhtJd Mater1Bf1

•

SPIRJT OF RAMANU JAN

147

Hence, from Thoorem 6.3.4(i) and (6.3. 17),

(~) '/' = 1+( {\~~3)'/.

(6.3.IS)

3+m 2m •

Taking the product of the cube of (6.3. IS) and (6.3.15), we find that ,( m:::.:~'~}(~3i+~m,,-)' 16m' ' while taking the product of the cube of (6.3.15) and (6.3.IS), that p = (m - I)3(3 +m). (6.3.20)

Q=

(6.3.19)

"'"e

find

16m

N""e multiply (6.3.21)

th~

cube of (6.3.17) by (6.3.16) to deduoe that I ~Q = ( m + I )(3~m)3 16m3

and lastly multiply the cube of (6.3.16) by (6.3.1 7) to find tha.t (6322) ..

1 _"' = (m +l)3(3_m). ~ 16m

P roof of ( iv). By (6.3.19), (6.3.20), and (6.3.18), m 2 a_{J=(,,?_I) 3+m 2m =(m1 _ 1)

(0')'1 7i ' ' o

from which (iv) is immediate.

P roof o f (v ). Substituting (6.3.11) L,to (6.3.19), (6.3.20), (6.3.21 ), and (6.3.22 ), we readily deduce the four formulas in (v ). 0 Proof of (vi). Using first (6.3.19) and (6.3.20) and seoondly (6.3.21 ) and (6.3.22), we find, respecth-eiy, that

P ) (

L/2

(6.3.23) ~d

(6.3.24)

Q

= m(m _ I ) 3+ m

'_ P)L12 = 1 ( Q

m(m I I ).

3

m

If we substitute (6.3.23) and (6. 3.24) in the right side of the first equa.lity of (vi) , ""e easily verify its truth. The second formula follows from the first by the ~lJrt,Ip.!Ml\l:iPn. 0

B. C. BERNDT

148 Prool of (v ii). From (6.3.19) and (6..3.20),

(0 .13")" .

= (m -

1~~3 + m ),

and from (6.3.21 ) and (6.3.22),

{(1 - 0 )( 1 _

.B)~p/. =

(m

+ 1~~3 -

m).

Hence,

(6.3.2b) by (6.3 . •9)-(S.3.?!). This

pl'O\'eS

the fint equality in (vii ).

By the principle of reciprocation, the second equali ty follows from

the first. Lastly, from (6.3.19)-(6.3.22),

(6.3.26)

1 + (0 /1)' / 2 + {(I - 0 )( \ _ ,ll)} L/2

= 1+

(m - 1)' (3 + rn )2 16m2

(3 + (m .j. 1)2 16m 2

m)2

=

"m,'~+;;-3L J' 8m 2

we multiply both sides of (6.3.26) by !, lake the squan root of both sidell, and appeal to t he !'init eq\l.llJity in (6.3.25). "''f! complete

I(

the proof of the third equality of (vii).

gives t ....,o further modulM tqW!.tioIl$ of degJft prototypes of modulu equations of further orders.

The next

3 that

Me

t~

Theorem 6.3.5. If P :",, (lf.n!1( I _ n )( l _,Bj} ' / B
(6.3.27)

0

SPIRIT OF RAMANUJAN

149

If then

(6.3.28)

~ = 2(P - ~).

Q-

Proof. From (6.3.19)- (6.3.22), we easily find that p2

= (m 2

1){9 _ m 2 )

-

8m' Thus, PQ -

m2 _ 1

- 2-12

Eliminating m from thili last pair of equatiollS yields

P Q

-

Rearranging thili last. equality, we readily deduce (6.3.27). T he proof of (6.3.28) is similar. From (6.3.19) and (6.3.20),

pl =(m

1)(3+m) 4m

and Q2 = m (m - l). 3+m

It follow! that

PQ = m-l 2 Eliminating

m

P Q

3+m 2m

from thili last pair of equatiollS, we find that p Q

2+PQ 2PQ+I '

which lIpon rearrangement yields (6.3.28).

o

With the use of (6.3.11 ) in conjunction with (6.3.15) and (6.3.17), it folloW!! that p > 0 and p < I, respectively, Or equivalently that 1 < m < 3. From the formul8.'l for () iUld fj in Theorem 6.3.4 (v}, we readily find that , for 0

da = 2(1 _

p}2{2 + p)1 >0

~OPyrighlJdMf1);ria/ -

,so

B. C. BERNDT

~d

dO = 6p1(l + p)2 > dp (1+2p)2-

o.

There is consequently a. one-to-one oorresp(lDdence between" and p and al90 between (J and p when 0
Theorem 6.3.6. kt p (6.3.29) 2FI

O, ~; l ; p

~ d~finM

Cz::pf)

by (6.3.11 ). Then

= (I +2pJ,F,

G,~; 1;p3

C2:~)).

Proof. Recall the definition (6.3.4) of the multiplier m and use The.Orem 5.2.8 and (6.3.3) to write thill definition in the form

2Fl ( ~ ' ! ; 1;0') = m2F'1( ~ ' ~; 1;p). Now use the repr(lS('ntatiOIlS for m , 0, and fJ 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 tbe second proof of T heorem 6.2.4, the proof of Theorem 6.2.9, and Exercise 6.2.7.

The theory of modular equations begins with Legendre's (139, vol. 1, p. 229j modular equation of degree 3 in 1825. namely, (6.4.1 )

(ap) I/4

+ {(l- o l( I -

(J )}1 / 4 = I.

In the century that followed , several mathematicians , including A. Berry, A , Cayley, A. Enneper, E. Fiedler, R. Ft-icke, C. Guet'l'laff, M. Hanna, C. G. J. Jacobi, F. Klei n. R. Russell, L. Schliifli, H. Schwter, L. A. Sohru:ke, G. N. Watson , and H. Weber, contributed to the growing list of modular equatiQn5. Ihw:e.-er, the mathematician whQ diso;wered far mQre modular equations than any Qf thew mathematicians was Ramanujan . whQ constructed over 200 modular equations. As indicated immediately above, the fonn of the modular eqll&tion given in T heorem 6.3. 4(ii) is due to Legend re [139, vol. I, p. 229).

and Can also be foundd3pWfg'hlia9.t¥J~~l text on ellipt.ic functiollS

SPI RIT OF RAMANUJ AN

151

(67, p. 100J and in Jacobi's epic wOfk 1131 , p. 68]. This type of modular equa tion has been established for &C"eral nth", cl"!';T""" In particular, formulas due to Schroter in hi.'l dissertation 1206] are useful in establishing such formulas: see also 134. pp. 66-72J. Thew modular equations are also called ~of Russell_type," after the Engli.'lh mathematician R. Russell , who derived seve ral modular equations of thi.'l sort. 1202], 12031. For example, tht modular equations of degrees 5 and 7 of this type are given by, respectively 134, pp. 280, 314], (a p) l /2

+ {(1 -

a )( 1 -

13»1/2 + 2{16aP( I- 0')(1 _ pnt /ti = I

(0:,8)1/8 + {( I _ 0: )( 1 _

i3W /8 '"

I.

The parameterizations for a and fJ given in (6.3.19) and (6.3.20), respecth"ely, ""ere first discovered by Legendre (139, \-01. 1, p. 223J and rediscovered by Jacobi (13 1, p. 25J. A. we have seen, the parameterizations of a and fJ in terms of the multiplier m are extremely useful in deriving modular equations of degree 3. Similar parameterizations exi.'lt for a and ,8 in the theory of modular equations of degree 5 (34, pp. 280-2881. but for higher degrees we do not know of any further parameterizations, which reHcct.s t he fact that witb increasing n , the shapes of modular equations become iucreasingly more oomplex, and finding them becomes increasingly mOTe difficult . The paper [105J by M. Hanna provides a summary of much of the "-ork w;wmplished on modular equations up until 1928. Ramanujan summarized much of his work on modular equations in fra~ent8 tba t were publi.'lhoo with his lost notelx>ok [194J On pages 55 and 350- 352. For accounts of these fra~eDts. see Chapter 17 in the book by AndreW>! and Berndt (19 J. In particular, the formulas for m in T heorem 6.JA("i) are examples of a large class of Similar formulas for m. AI~h<Juf,1I "." II ....., pm,,;deU .. bi,,,!,l,, vllJOf based on parameteriz.ation8 for a and p, in general, we do not know Ram anujan's methods for deriving them . In our Ix>ok (34), we used the theory of modular forms to verify sc\-eral of tbem ; Ramanujan most likely did DOt ~§ill'e'tf Material

B.C.BERNDT

'"

The modular equation in (6.3.27) is of "Schliilli-type~ [204]. and Ramanujan recorded many modular equations of this sort in his notebooks and lost notebook {HJ3], [1114]. See [111, Chap. j ', ], (M , Chaps. 19-201. lind !37, Chap. 25J.

In this short monograph prov;din& an introduction to Ramanu. jan's thoory of theta functions, we do not have the space to give applications of modular equations. The proofs of Ramanujan 's ""ngruences for p(n) modulo arbitrary powers of 5, 7, and II depend upon modular equatioru; (194], (501. [218), [28J. Many of the re5ults in the following Chapter 7 crucially utilize modular equations of degree 5.

Copyrighted Material

Chapter 7

The Rogers- Ramanujan Continued Fraction

7.1. Definition a nd His torical Background A continued fraction is an expiession of tbe IIOI"t (7. 1.l)

.. +

_______0·"_______

'"

_____'"e'_____

. 'hich is commonly written in the more compact rorm (7.1.2) The continued fraction (7.1.1) or (7.1.2) may I.('rmi nate, i.e., t he frae(iOTUi do nol continue indefinitely. f or example,

t!-

is a terminating continued lTaction. Mote illl.ert':$l.ing are Lhoseoontin\ted fractions that do not terminal.e, i.e., infinite (':Ontioued fractions. Copyrighted Material

153

B. C . BERNDT

154

Suppooo that we define the sequences p .. and Q n , " > -1 , by

+ Il"P.. _2

Pn = b"P.. _ l

Qn = b"Q .. _l + o"Q .. _. ,

P_ 1 = I, Exe r cise 7. 1.1.

Q_I = 0 ,

Pro~

n

> 0,

"
Po = bo,

Qn = I.

tluIt

(7.1.3)

The" if · -p" I ,m

n_"" Qn exists, we say that the continued fracticm (7. 1.2) oonvergcs; otherwise it di\~rges. Clearly, the first task in de,doping a theory of continued fractioJl.'i is to derive criteria for OOIl\'eTgen~ and divergence. In particular, when the numerators an and denominators b.. are functions of a complex variable, an eJ!tensive theory has been developed, and it continues to evolve. It is not the purpose of this monograph to develop such a theory, and 80 we refer lcaders to the excellent text by L. Lorent zen and H. Waadeland [151 . Chapter IJ for many criteria for convergence and divergence. In particular, see [15 1, p. 35, Theorem

3J. /l.1ost mathematics students first enCOunter continued rractions in a course in elementary number thoory. T he first infinite continued fractions that students may be asked to e\"8.1uate are those in the following exercise. Exe rcise 7.1.2. Prove thai (7.1 A )

"'" (7.1.~)

1

1

1

vv'5;';'+"1

-

-

1

./5 - 1

1+ , +, +, + . .. = - 2 1

1- -

1 + 1 - 1 +' "

,

T hese are, in fact , special cases of perhaps the mOOSt interesting continued fraction in mathematics, the Rogcrs-Ramanujan oontinued fraction. which first appeared in a paper b¥ L. J . Rogers [1981in 189-1. Copyrighted Matenal

SPIRlT OF RAMANUJAN

155

Definit ion 7.1.3. Tile Roger$- Ramanujan continued /ruction R(q) i3 defined by ql /~

(7.1.6)

provided tJw.t it

q

ql

rt

R(q):=-I- + l+l+ 1 + ... ' con~ergu. Furthermore, set

(7.1.7) .rnl

(7.1.8)

T(q): = _I_ = I +!! q2 qJ F(q) 1+1+1 + ·"

Readers will immediately ask, "W hy does q l/~ appear in the defi. nition of R(q)!" T he reason is that R (q) belon.g:s to the wOlrld of theta functiOl08, and R(q)'s modular properties are lOore symmetric and el· egant with q l/~ appearing in its definition than if ql/$ were absent. However, there are oe<:asi008 when the factor ql / 5 is nOlt helpful, and SO we then use ooe of the representations (7.1 .7) Olr (7. 1.8).

1.2. The Converge nce, Divergence, and Values of R(q) In the definitiOln above, we ha''e nOlt indicated where R(q) com'erges or where it diverges. By standard thoorems (151 , pp. 35, 94J, R(q) comoerges in the complex q--plane fOlr jq( < I , and it di''erges for (q( > 1. What happens if 1111 = I? Taking the redprocals of {7. 1.4} and (7.1.fl), "oe find, respe<:thoely, that (7.2. 1)

_ R(-I) '"

.J5 + 1,

2 where we have taken the root (_ I j1/~ '" - 1 in the definitiOln of R(q). FUrthermOlre, I. Schur [207, pp. 319-321), (208 , pp. 117- 136) and Ramanujan in his notebooks (1931 established the fOlliowi ng theorem (38, p_ 35). Theorem 7.2.1. Recall that T (q) i3 deji'led by (7.1.8). Let q be a primitive mth root of unity. If m is !I multiple of fl . T (q) divergu. Otherwise, T (q) converges and

(7.2.2 )

cWr~TJ8~~e;:;;)'$,

B.

l~

whe"" 0 denote! the Legend"" symbol residt>e 0/ m modulo a.

('W-)

c . BERNDT

and p i& /he leMt p
Thus, for a root of unity, we know when R(q) com'erges and when it diverges. What about other points on Iql '" I? We do not know the answer in general, but D. Bowman and J. McLaughlin (61) have found an uncountable set of measure 0 on the unit circle (not includ· ing anth roots of unity) where R(q) di''erges. It is conjectured that R(q) diverges on the unit circle except for those points of convergence described in Theorem 7.2.l.

All mentioned above, the RogeIll- Ranumujan conti nued fraction fiIllt defined by Rogers, who proved a few of itll properties. How· e''er, most of the results that ""'e know about this continued fraction are due to RAmanujan. $everal theorems about R(q) appear in his notebooks [193), but his lost notebook [194) contains considerably more material On R(q); see 119, Chapters 1- 5). In his first t,,"<) letten to G. H. Hardy [192, PI'. xxvii, xxvii( , [51 , pp. 29, 57), RAmanujan communicated !;
R(e- 2-) = J5 +2.;g _ ..;52+ 1

(7.2.3)

.w (7.2.4) where we ha''e set

S(q) = - R(-q).

(7.2.5)

In his second letter to Hardy [192, p. :o:viii), [5 1. p. a7]. Rrunanujan further asserted that

R (~-2_.;5)

=

..;5

I+V53/4(~)~/2 _ 1

2

In both letters, RAmanujan claimed that [192, p. xxvii), [51 . pp. 29, 57], "It is always possjble to find el<..u;tly the value of R(e-·.;n).~ c;opyrighfed Material

."

SPIRIT OF RAMAN UJ AN

The meaning of this last statement "'as not clarified until 1996 ",hen Berndt, H. H. Chan, and L.--C. Zhang [46\ demonstrated that if n is a positive integer and if the requisite c!ass i nvariants could be determined, then R(e - ~"r.;:) could be explicitly evaluated. (Class in,-aciants ~ certain mul tiple!! of Ramanujan's functioo X( ±~- · ';;;:) . Historically, they wer~ first brought into prom inence by H. Weber [ZZOj, who explicitly calculated many class invariants and who used them to generate Hilbert class fields. Ramanujan explicitly determined over 100 class invariants and used them to calculate ,-alues of certain quotients oflheta funct-lons and t he Rogers-Ramanujan continued fraction. Ramanujan's work on cla.ss in'"3riants is described in [38, Chapter 34]. For ao introductioo to class invariants and Ramanujan's applications oftbem , see [47].) Both in his notebooks [193] and his lost notebook [194], Ramanujan recorded many values for Lhe Rogers- Rama.nujan oontinued fraction. For example, On pages 204 and 210 of hill lost notebook [194], Ramanujan offered the \'lI.lue

where S(q ) ;5 defined by (7.2.5). For proofs of the5e clainu;, see 144J, [46J, and [19 , Chapter 21. Those readers with some background in aI(.~l"aic nwnber th~ry and who h"w: been exccptio""lI y oOOerV3nt will have observed t hat each of the \'3lu~ for R (q) that we have quoted is a unit in some algebraic number field. T his is t rue for all \'lI.lues of R(e-·.,{ii), when n is a positive inleger; see [46] for 1\ proof. We record one further result on R(q) communicated from India by Ramanujan to Hardy. Let 0" and {J be pooitive numbers such that u{J = ",1. In his second letter to Hardy, Ramanujan 119Z, p. xxviii], 151, p. 57] claimed that (7.2.6)

(I

+2.;s +

R(~-2"»)

(I

+2.;s + R(e- 2t1)) = 5 +z.;s,

which was first proved in print by Watson IZI6). T his result is also recorded as Entry 39(i) in Chapter 16 of Ramanujan's second notebook; see [34 , pp. 84-851 for another proof and furt·her reference!:l, 10 and also see !19 , p. 9'\!:ofjp~hMa WlriID0WS the value of R(e - )

s .C. BERNDT

>5'

for II certain 0, then (7.2.6) enables one to immediately calculate lin· other value R(e - 2t1 ). Thus, besides being II beautiful formula, (7.2.6) cnabl"" o ne to obtain two

valUC3

from one.

1.3. The Rogers- Ramanujan Functions The Rogers- Ramanujan continued fraction is intimately connected with the famous Rogers-Ramanujan functions G(q) and H(q), which are defined by ~d

(7.3.1 )

Both Rogers [1915J and RamanuJan 1193, Vol. II , Chapler 16, Sect. 15), [34 , p. 301 proved that

(7.3.2)

R (q) = q l/5

~:::.

which we nOw prove. In fact , we first prove II finilt form of (7.3.2), which can be found lIS Entry 16 in Chapter 16 of Ramanujan 's second

notebook [193J, [34, p. 311. and from which (7.3.2) follows as an immediate corollary.

Theorem 7.3. 1. For mch nonnegatil'e integer n, let (7.3.3)

jJ

,= p,,(a,q):=

(("+1)/2)

L

k_O

t

(q).. _ H ID q , (q)k{q)n_2Hl

(n/2) ( )

(7.3.4 )

k'

• k (HI )

v;= v .. (a , q):=L qt ) kt )q b .O

q ~ q n - l~

,

where {'I'] denote. the great""t integer less t/w.n or equol to

forn::,,:l, (7.3.5)

, aq

J.' = l+aq

1 +1 +·

1/

",,"

+ 1

Proof. For each nonnegative integer t, define l(n _ '+I)1~1 ( )

k

k (.+ k )

qn_r ~+Ia q b O {q) k(q) n- r _2Hl Copyrighted Material

F. := F. (a, q):=

L

'1'.

Then,

SPIRlT OF RAMANUJAN Observe

'"

tha~

(7.3.6)

Fo =

jl

Also note that and

(7.3.7 )

We nOW develop a recurrence relation for F•. When ""e combine the two sums in the first step below. we use the fact that 1/( q) _ , = 0 by ( \.3.35). To that end,

(7.3.8)

= aq

.+' r.+ ''' l ·

Using (7.3.6), (7.3.8) rcpeat.cd.ly, and lastly (7.3.7), we conclude that

B.C.BERNDT

160 = l + aq I aq

= 1+I

,

+

all

I

aqn _ 1

+ .. + F,,_ J/Fn aqn - '

aq~

+

-I

+

+

I

",.

+

I

o

(7.3.9)

Proof. Let n

+ ...

---> 00

o

in (7.3.5).

T he continued fraction in (7.3 _9) is called the Genernliud R amarlUjan Continued Huction.

lWg~r3-

T he renowned Rogers-Ramlltlujall identities, which we do not prove in this book, are given by (7.3.10) I 'od C(q) .. (q;q'J""(q";q ' J",,

Using (7.3.1 0),

"'~

I

H (q) = (q';q"",,{
immedia tely deduce the elegant representation for

R(q) in the next tboorcm. Theorem 7.3.3. We have

(7 _3_1l ) Proof. Set a" I in Corollary 7.3.2, take the reciprocal of both sides, use the definitiOflll (7.3. 1), and lastly use (7.3. 10). 0

T he results in this se(:lion can be greatly generalized in that maoy more general oontinued fractions can :,e rep!'('5ented as quotients of two q-series . See, in particular, i19 , Cbapter 6J and 134, pp. 30- 31]. Copyrighfed Maten'al

SPIRIT OF RA MANU J AN

161

7.4. Identities for R(q) Ramanujan discovered tW(l beautiful identities connecting R{q) and l / R(q), which we prove in this lIe<:t ion The fir8t is given below. Theore m 7.4.1. Recall that T(q) .. defined in (7.1.8). Then (7.4.1) Com(>3ring (7.4.1) with (2. 3.13) in tbe proof of Theorem 2.3.4,

we see that the unidentified aeries J, (d and J1 {q) are equal to T (q) and - I / T (q ), respectively. We now prove a more general theorem rrom wh ich Theorem 7.4. 1 follows by specialization. It will be oot:>-enient to introduce tbe nota·

,..,

(01 , , 012 , ... , a", ; q) ... = (a, : q ),.,{a2; q)_ ..• {a ... ; q)"" .

Theorem 7.4.2. For anll complex n~kr a, (7.4.2) (a, a2,q/ a, q/ a 2, q; q)""

"

(a~q;() ... (a -~ q~ ;qS)""

(lI~q2 ; q&) ... {a-~q';q5 ) ""

-(<{, v )"" {q;,f) ... (q( ;of )"" II (q' ;of )""{q';of )",, _a' (a5.f;.f)...,(1I - 5q' ; ,fl"" + 013 (a5q( ; .f)_{o-Sq;

I"' ''I~I'';''I~

r)",,) .

I""I~I"' ''I~

Bef~

proving Theorem 7.4.2, ,,~shoo.> thM Theorem 7.4. 1 fol· low! immediately from Theorem 7.4.2. Proof of T heo re m 7 .4 . 1. If,,~ replaoe II by
forthwith.

0

The decompositions in Theorems 7. 4. 1 and 1.4.2 ~ called ~ dwtelioM , because in the former thecrem, the .eries tenru! of (q;q )"" are separated out in powen of q accordi", to their residue classes modulo 5, and in the latter theorem , the terms are separated out in powenl or a acrordin~r",~rQf.11es modulo 5.

162

B . C . BERNDT

proor or Theore m 7.4.2. Using the Jaoobi triple prod uct idellti ty (1.3.1I) twice, we 6nd that

, I I ' .) _ (a,a,qa,qa,q,q",, _

(a,q/a,q;q)""{a~,q/al,q;q)",, ()

q; q ""

~

=

~

L

L

1 (_I)"a'q(" - 'l/l (_ I )'a1'q(" -')1l (q;q)"" ' _ _ 00 • _ _ 00 ~

L (q;q)"" ,.,__

(_Ij" t 'a· tl 'q(.,- .t" - '1/ 2

= "."'-

00

~

(7.4.3)

=

L

n __ oo

a"c,,(q),

where, for - 00 < n < 00, I

c,,(q):= (q;q)"" '. '~-"" .tl.~

..

We now determine c,,(q) a.crording 5

to

the residue class of II modulo

First, oonsider the residue class 0 (mod 5). Replace II by 5... and make the change of variables r = II - 2t and s = 2n + t. Note that r + 2s .. 5.... Then, simplifying and applying the J aoobi triple product identity (1.3.11 ), we find that

L ~

I (_I )nH q((n-2' )' (q;q)"" ' _ _00

(_l)~q(.n'

(7.4.4)

-

(n-lt)+(2 .. H)'_(2"t,j)1l

3nl/2

(q; q~ )",, ( q';.r)"" .

5«:o1ld, oonsider the residue class I (mod 5). Set r = II - 21 + I and 3 = 2n + t, so thtJtfpyri(;flmJWlIiJnafhen. u[lQn simplification

SPIRIT OF RAMANUJAN

163

and the use of the Jaoobi triple product identity (1. 3.11), we see that C~n+1(q) =

1

~

L (_1)-+'+1

(q;q)oo , _ _ <»

X q((n - Z'+1l' -(n -2' +'1+(2n+' l' _(2n+1))12 l )n +1 (~ .. '-n )/2 <» = (q (_ I }'q{~" - 3'l/2 {q;q)"" ''' - 00 (- 1In+ 1q(~n' - .. )/2 = . f(_q,_q4) (q, q )"" (- 1)n+ 1q(~'" - n )12 (q2; qS }<XI«(rl;~) "" .

L

(7.4.5 )

It should now be clear how to calculate the three remaining cases. Exercise 7.4 .3. ProlJ(! that (7.4.6)

( - 1)".j- lq(~n'+nl/2

CSn+2(q) =

(q2;~)
( _1)"q(m'+3,,1/2

(q;tf)oo(
(7.4.7)

CS"H(q) =

(7.4.8)

CSnH(q) = O.

Substitute (7.4.4)- (7.4.8) in (7.4.3) !LIId use the Jacobi triple product identity ( 1.3.11 ) foUl" times to conclude that (a, a2 , q/a, q/a 2 , q; q)."

B. C. BERNDT

164

Prom Theorem 7.4.2, we derive the second major identity involv. ing R (q) or T (q). Theorem 7 .4.4. lVe have 5

(7.4.9)

~

ql O

_

(qS ; ~ )~

r {q ) - 11q - ~ (q) - (Q2~; q15 )~ '

Proof. The proof is almost identical to II portion of the proof of Theorem 2.3.1. We therefore forego most of the details. Write (7.4.1 ) in the form (7.4.10)

( q, q23 ,9 , q' , 'I~'~1 , 'I "" "'T(

(qZ5 ;q25)""

51 __ q

q

q' T(~ )

Let W be any fifth root of unity, lind replace q by wq in (7.4.10) to find that l--1 . 4 5 . ...5) " ( "'q,w 22 q , W q-,w q ,'I . '1 "" - T ( ' I "q (7.4.1 1) ('12$ , '125)"" q -wq - T (qS) Now multiply aU five "'lualiti.." (7 .4.L l ) tog~th=. 0" ~he left oide

of Our product, IlOO precisely the same argument that was used in (2.3.18)- (2.3.21 ) to arrive at the right-hand side of (7.4.9) . For the right-hand side of ou r product of the expressions in (7.4.11 ), employ the same argument that was used in dtducing (2_3_23 ) from (2.3.22). We then obtain the left-hand side of (7.4.9) to complete the proof 0

Although the proo£S are essentially the same. the use of Theorem 7.4 .4 leads to a somewhat cleaner and more satisfying proof of The<)rem 2.3.1 than the proof em ploying the pentagonal number theorem. Jacobi 's identity. and the division of power series in Chapter 2. We ~"m[ll .. te OUT work on the Rog....- Ramanujan continued f.""tion in t his section by giving this alwrnath-e l>roof of T heorem 2.3.1. Theorem 7.4 .5. For each nonnegative inleger n.

(7.4.12)

p(tm+41=- O{m04 5j.

GOPyrigh ed Material

lOS

SPffilT OF RAMA NU JAN Proof. Write (7.4.9) in the form

(q>li.qn ). ( ,. ) ... rs(qi) _ llq5 _ -'i;'", (q5 ;q5)t, 1'1{q5) .

(7.4.13)

1=·

Divide both s ides of (7.4. 13) by (7.4.1 ) in the form

(q;q) ... -

(qU:q2~)oo (T(¢)- q - T~» )'

Then, U!lins: IonS divillion and tbe abbreviated oot.ation '" = T(~ ). _ 6nd that f:P{n)q~ _ (q15 : q15)~ r& (qS) ~_

(qS;q6)t,

11<1 -

T{qS)

_ (q15: q:z:5 )~ (",.

(If : q~ )t,

q

qlo/r&(~ )

q3fT(q5)

+ qz3 + 2q2",2 + 3q32; + 5q~

_~ +2q8 _q7 +rt) '" ",2 r' ",4'

(7.4.14)

We !IO?' extract those terms from both sidell of (7.4.14) that involve only the po-t.-e1"S t nH , n > 0, to ded uce that ...

(7.4.1$)

( 36.

25)6

"' p($n+ 4)q--S"H_5q4 q , q ~

"".

(q6; q~)t,

DIVid mg both sId es of ( 7.4.1.5) by q' and ",.,I""j,,1S 'I" loy q, we com-

plete the proof.

0

Theorenu 7.4 .1 and 7.4.4 ""ere found by G. N. Watson [215]. ]216] in Ramanujan's notebooks lind p~"Cd by him [215] in order kl eflubl.i$b (7.2.3) and (7.2.4). Observe thllt if _ can e\'alll.ll.t.e the riSht side of either (7.4. 1) or (7.4.9) rOT a certain \-alue of q, then we can find the \'8.lue of R(q) by limply «>I\'inS" quadratic equalion. Exercise 7.4 .6 . '17tlllhifttd FibonaCCI n.,mber& 1ft. n > 0 , are definc;l

b)'/o= b - 1and (H .IG)

£Stab/WI th e lolllJI
2: l!tq" -

I

'.

...0 (A)P)'I'/{}J llell 'Milenal

lql <

I.

8. C. BERNDT

166 Next

pro~

that

-=

Conclude thai / Sn H O(modS). In p
O(mod25). ( To prom: !hu k>st oongrumoe, you will probably nl'd to I/.!e furth er pro~rt;es of Fibonacci numb..rs. ) The previous exercise relating p(5n + 4) to Fibonacci numben; was suggested by M. O. Hirscllhorn [125J. to [124J, he used the fact 5115"+4 and a variation of the argument we gil"'" aoo.-e in our 1'1Oof uf Tin,,,]',,,,, 7.4.5, ..... I'.ive" silUil~r "rour or The<>rem 7.4.5. See al50 (2.3.25). It is remarkable that '~H and h~n +24 obey the same congruences as p(Sn + 4) and p(25n + 24 ), respedively. Ramanujan 's original conjecture (2.1..5) in the case that,s = 53 is p(12Sn + 99) ~ o {mod 125). Unfortunately, in general, it is not hue that fn~ .. +99 is divisible by 12[', and so the analogy fails for 53,

7.5. Modular Equations for R(q) Recall that a mooular equation of degrN! n can be tllQught of as a relation allloLlg theta f" ...:tiuu:s wit h "'-11"'''~''l" and the"" functioIL'l '. rith argument '1ft. We dose this chapter by offering some of Ramanujan's modular equations for R (q).

First, on page 326 in his second notebook [193J, [38, p. 12], Ramanujan reoorded a relation between R(q) and R (q2 ) in different notation, which was also proved by RQgers (199, eq. (5.4)). Theor em 7.5.1.

ut u ' ''''

R (q) and v := R (q2 ). Then 2 .~

--

v_,,1

V+ ,,2 '

Aloo on P<'ge 326, narnanuj an adroi t ly deli".,,; the pan.."et.er

and states t he following two elegant and symmetric relations (38, p . l3J.

Copyrighted Material

SPIRIT OF RAMANUJAN

167

Theore m 7.5.2. With k defined by (7.5.1),

J?5{q) -k(:::r

and

J?5{~) =k2C+:) .

In his 1000t notebook (194), Rrunanuj/Ul recorded!le\-eral exquisite identities for theta functions in the argument k of (7.5.1); see !l9, Sect.s. 1.8, 1.9). The nut beautiful modulM equation of degree 3 is found on page 321 in Ram&lluj&ll's sooond notebook [193]. [38. p . 17). and was.u.o established by Rogers [38, p. 392]. ThAOrElll 7.5.3.

Let It :=- R(I,I ) and tI: .. R{q-l). TIIen (tI-

,,3){1

+ w') _

3,,~~.

Lastly. we ooncill(k .... ith a modular equation of degl ~ 5 for R(q) that Ramanujan communicated in his first letter to Hardy [107, p. xxvii), (51 , p. 29]. and that is found on page 289 of Rrunanujan 'l second notebook [193J, [38, pp. 19-20J. Again, this modular equation was first. established by Rogei . 1199, p. 392). For referenoes to further proofs, !Iee [38, p. 20) or [51, p. 43). T heoNl ill 7.5. 4. Let u:. R(q) and ~:_ R(rt). TIIen ~

1 - 2t1+ 4 t12_3v"+~~

" = tll+3t1+4t1~+2"'+v4'

7.6. Notes The Rogef$-Rrun/Uluj&ll oontinued fraction was one of RamanujlUl'l fa>-orit.e functions. We have related to readers IIOme of it.s IJlOfit fa&cinating properties. but Ramanujan recorded many further re!!ult.ll, esp«ially in his lost IlOtebook [194]. We hope readCi'll will be stimu· lated to read about these discoveries in Chapters i - 6 of (19), which is the most com plete souroe of theorems and referencea on the Rage ...... Ramanujan continued fraction. An eqM)OIltory aca>l.u,t vf "",,,,,,a1I~ orems on the Rogers-Ramanujan CQntinued fraction ean be found in [45J. A !ubset of the TeIlulta proved in [19) are established in [49]. Hardy WlIS intrigued by Ramanujan 's claims about thc RogersRamanuj&ll oontinuecC~I!~"NIU6nly wrote to RBmanujan

B. C. BERNDT

'"

urging him to write a pa~r about it. In a letter probably written On 24 December 1913. Hardy exhorted {51, p. 871

)fyou will send me your proof written out carefully (so that it is easy to follow), I will (assuming that I agree ,,~th it-of which [ have very liule doubt) try to get it published for YOll in England. Write it in the form of a paper "On lac continued fraction

1 + 1 + 1 + "

"

giving a full proof of the principal and most rem"rkAhl~ th...,r"m. viz. that the fraction can be expr med in finite terms ",he., :J: = e - w ';;;, when.!! is rationaL

However, Ramanujan ncver followed Hardy 's advice. The history of the famolls Rogers- Ramanujan identities (7.3.10) is now well known. They werc originally disoo'1'red by Rogers [198J and rediscovered by Ramanlljan, who a t first did nOt have proofs of them. One day while at Cambridge, Rarnanujan was perusing back issues of the Pr0cee4ing$ of the Lo"da" Mathematical Society and found Rogers 's paper [198) giving pr()Ofs of (i.3.10). Ramanujan soon fOWid his own pr()Ofs and published them in [189), [192, pp. 214- 215J. Far further historical accounts, see Hardy's book [107, pp. 00-99). Andre ....s·s text [14. Chapter 7). or Berndt's book [34 . pp. 77- 79). Many proofs of the identiti~ nOw exist; a description and classificat-ion of all known proofs up to 1989 can be found in Andrews's paper 117). The identities ( 7.3.10) ha\l~ beautiful combinatorial interp~ta­ tions. In the definition of G(q) given in (7.3.1), write ,,1 = 1 + 3 + ... + (2n - l ). The 6rst identity in (7.3.10) is equh-alent to the assert ion that the number of partitions of a positive integer N jnto distinct partS with differences at least 2 equals the number of partitions of N into pacUi congruent to either 1 or 4 modulo 5. For example, there ar~ 6"e partitions of 9 into distin<;t pacl..>! with differences bet ....een parI..>! at least 2. namely, 9, 8+ I , 7 +2.6 +3, and &+3+ I. The 6,... partitiom; of 9 into parts congruent to either 1 or 4 modulo & are 9 , 6+ I + I + I .

HH I, 4 + 1 + I + I + tthyil3~'~~ Mater;~" For the second identit)',

SPIRlT OF RAMANUJAN

'"

in the definition of H (q), write n(n + 1) = 2+4 + .. ·+2n. The second Rogers- Ramallujan identity in (7.3.10) i~ an analytic statement of the f1lCl. that t he number of partitions of N into distinct parts with differences at least 2 and with no I's is equal to the number of partitions of N into parts congruent to either 2 or 3 modulo 5. For example, the three partitions of 8 into distinct parts with parts differing by at least 2 and with no 1'8 are 8, 6 + 2, and 5 + 3, while the three partitioDll of 8 into parts congruent to either 2 Of 3 modulo 5 are 8, 3 + 3 + 2, 8nd2+2+2+2.

Exercise 7.6.1. Pro~ that I~ Roger3- Rumanujan idrnt itiu have the combinatorial interpretation.! de.IJcriW in !he p~in9 pamgraph. Theorem 7.4.2 is due to Hirschhorn [1231, and the proofs of Theorems 7.4. \ and 7.4.2 that we ha''e gh·en are also due to Hirschhorn [123]. The only other proof of Theorem 7.4. 1 known to us is by Wat.son [215J, who employed the quintuple product identity. Hirschhorn's proof is !lQmewhat simpler. The proof of Theorem 7.4.\ given by Berndt [34, p . 267J is similar to that of Watson. Qill deduction of Theorem 7.4.4 from Theorem 7. 4.1 is the same as that given in the aforementioned works of Watson, Bendt , and Hirschhorn. Exercise 7.4.6 arose from the oombined efforts of Hirschhorn, P. C rut<:her , O. Yest Chan, and thc ..·.lthor.

An approach to the Rogers-Ramanujan continued fraction via modular fomlS has beo:!n written by W. Duke [85].

Copyrighted Material

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[183J V. R· ma",...,,; , Some 1d~'llU C....jcctw-cd ~ S ...... _ Ra ....... ujcuo FatUla .n Hio wlAograplletl NDfu C""""",letl ....th Partili<m 'I'Iw:or}I and Elliptic. M~ " Flo...",..... - 17Ieu- Proof. - IruefWfl"':di.... ..rII v ............. Othu T&p1U '" rAe 'I'Ito:>r) of Numkr, and S ome G....eraI· uati<mr Tht:m:>n, Ph. D. Th""i3, Uni""",ity of M)'f(m!, 1970. [1841 S. Ramanuj"'" ModulDr _ Iiofu .nd DpprvrimDlioM ID II , Quart . J Mat h. 45 (1\lI ~), 350-372. [ISS) S. Ramanujan, H.ghly armponte number., Proc. LoDdon Math. Soc. 14 (l91!» , 347--409. (186) S. Ramanujan , On ""Din "rith~1lC:Id jomeliOn.!, 1'1-..... PhiloA. Soc. 22 ( 1916), 1!>9--184.

(187) S. Ramanuja.., On liOn.! ," thc ~ ( 1917), 259-276.

CambricJ&t

CCf14'" In9O'l<"'lctriml ""M 0/

,,~ mben,

and th",r "PP""'· Tr..... Cambri
(188) S, Ramanujan, Some prop1" ... idml.li ..... ," arm~~ .. nt:ol~ P"",. c....bnd,s. P bUOOO. Soc. 10 (1919). 2)<1- 216. [190} S. RAmanujan , C.mgnomco p.ope' li..... 0/ parllll ..... , P roc. Loodoo M.lh. Soc. 18 (1920), '«nIck for 13 Man:h 1919, :oWI. (191 [ S. Ramanujan , C01IgrIOmco p,....."li ... of pam/,On.!, Math. Z. 9

(1921) , 147- 1.53. Copyrighted Material

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183

[192] S .. Ramanujan, C~lIttted PQP'=, Cmlbridge Univ.m;ity PTe6s, Cam. bridge, 1927; rePrinted by Chelsea. Ne .... York. 1962; reprinl.ed by the American M&thcmatic,,1 Society, Providence, RI, 2000. (193) S. fuunam.jan. Noteboob (2 volumes), Tat .. Institute of Fundamental R.e!oo>,.,..,b , Bombay, 19:>7. . [194) S. RamanujlUl , The Lo.t Notebook ond OrM<- Unp"WUhed Papa., Narosa, New Delhi, 1988. [1 9:» R. A. Rankin , Ceqrgc Nellilic IV"",,,,,. J. London Math. Soc. 4 1

( 1966),551-!>6S.

[J(6) R. A. Rankin , Modular Forms and fbnclion.o, Ca.mbt;';ge Univeroily Pr_, Cambridge, 1977. (197) R. A. RAnkin, The Roman"iom T /-ncIKm, in R""",nujan RwU ..... " , h ..... , ""d R. A. Rankin , eds .• A~&demic Press, San Diego. 1988, PI>. 245-268.

)198) L. J . Rogers, Second memoir on Ih<: apan.rion 01 certain infinite prod. """, Proc. London Math. Soc. 25 (1894 ). 318-343. )199) L. J . &.g..rs, On a type 01 modular relation, Proc. London M"'b. Soc. 19 (1920). 387- 397. )200) R. Roy, Review of Quantum CakuI ...., by V. Kac ....d P. Cheung,

Amer. Math. Monthly 110 (2000), 552-657. )201 ) J. M. Rushforth, C....".,..,.,ce Propcrli .. ol/he Parolion fbnclion and A •• lI<"ialM' FUnction.<. Doctoral Thesis, University of Birmingham, Birmingham. UK, 19:>0.

)202) R. R""""U . 0" 1<.1. - 11.1.' ",<>'" Ilcrm;u..:Mn V""""ndl,,ng.talcln for die el_ lipt"chtn Modular",nclion"", J. Reine Ange ..... Math . 72 (1 87(1), 360-

"'.

(205] B. Schoeneber&, SII'ptw: Modular F.nclion.<, Springer. Verlag, Berlin, 1974. [m [ H. Schroter, lk Aoquah<>inilJ.u.o MotfuiarilJ.u.o, Dio::Itrtatio Inauguralis, Alberti"" L;~ter",um UnivenilAte, R.egiomomi , 18M.

[20'] I. Schur, S in Batmy :roT additi""" ZothkntMonc und :roT Th«>rie d.,.. Kettcnbriiche, S. B. PreUSS. Akad. Wis9 . I'lIys.- Math. 1( 1. ( J917). 302-321. (208) I. Schur , C ... ammcltc AbhandlungC1l. BBnd II, Springer· Verla&. Berlin , 1973. [2m] A. Selberg. CoJlecf!ttYH'§h1'e»,,w!t§fflif'&er-Verb& , Berlin, 1989.

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B. C. BERNDT

(210) B. K . Spearman and K . S. Willi""",. TM timple
[211J M. V. Subbar&o, Some ",,,,,,rb "" 1m partitiQn junClion, Amer. Math. Monthly 73 ( 1966), 851-854. [212) Z.- H. Sun and K. S. Williams, R
""

[215J G. N. Watson, Th..,,,,,,,,, .lolM by Ramanuja" (VII ): Th~ on contl"u(d jrru:.lion$, J. london Mallo. Soc. .lr:n. J . Reine An&ew . ).Ialh. 119 ( 1938). 97-128. [219) R. Weaver, New rong"",n"".. far 1M. parhllOn jrmClifm, Ramanujan J. S (2001), 53~3 . [22OJ H. Weber. uhrl>11.ch dn- AIg<'bm, dntter Band, Chelsea. Ne ..· York, 1961. [221J E. T. Whittaker and G . N. Watson, A Ca .....e of Mc>dern ArI4IysU, 4th 00. , Cambridge Uni''el"6i(y p,est. Cambridge. \966. [n2J J. M. Whittaker. Gwrg<' Nevillc 1¥,wan, Biographical Memoirs of Fellows of the Royal Socirty 12 ( 1966), 521-<>30. 1223) S. Wigert, S .. r I 'oron- de grnnde .. r~" narnb .... dM di ......eur. d'"n "". licr, Ark. Mat . "'stron. Fya. 3 (1906-1907), 1- 9. 1224] K. S. Williams, Same Lam/.Jcrl .en:... e%pQns'ans af prnd!J
Ra,,,,,,, ..

'qu"""

1227[ L. Winqu ist, An elementary prool of p(llm + 6 ) J . Comb. Thy. 6 ( 1969). 56-59. Copyrighted Material

=

O( mod 11 ),

Index

"l.el. N_ 11-. ' 3~ addition theorem. 131 Adlp. Co. 23. 1'11, 83 A111c"'D, S .. ~4, 110, ~I A1mk _Iot, G .. 130 And,~. G . E .. 22- 24. 32, N. 82. 106. L08. L~I. 168 Apootol, T. M .• I ~ Asl:ey. R. A_. 25. 83 Atkin, A . O . L .. 28. M

B.iloy. W. 1'1_. 81 ~d.P.. 81 _ hype."",m."ic ..,,; ... 24 , 32 Boe,,,,,,,1h "umboo.. , ~ [1 ... , . ,-., S . • U, "". 7», 83, ,29 8 ..... k , P•• 106 bilo.~.&1 ... m ....ion, 15

01

-,,>nd kl"". 1\) ooo\lDu<>",b .. , P., Hill

'ho

C<x>"",.

0.11,,,.,, P.,

»

Dir ichlet, P. C. L .. 83 d,-",iQD., l$, Dobb~ . J . M .• 81 0.00', J . , 49

Duk<. W .• 168 duplico';on formula, 131

Eichho
.,(

f",o.~I_.

In

, """1Il<"~". 105 Ei .. n .... in. G .• 82 . 105 P.k lu.d. S . B .. 79 P.,i_n. K .. 22 Eul<,', identity.'. SII. GIl Euler', ..... Ution 4 P.ule,·. "''''&IIon''' numbe' tboQr~m ,

Bouni&k
0-0>.""', D ., 1M Boylan, M " 110

'beo'.....

y. A., 23. 82

Cay..,., A . • no

"

.-,,,,,i... ""..ion. 13

comb;'•

Chon, H. II.. M. H>1, 108. lUI, 157 et..o, O.-V ., 169 S. H., 24. 1\0, <>3, 80, 108

a......

ChO<)'.

'''''''pk«o ~"'p"< 'n"'V"

J_. 106

Cbowla, S .• ZS <:'10 ..... ''- ~ • 0l.4 d,d ..... bod. ~m. 22 .1_ in......-i ...... 151 «>mple'" e!lipl'" Inu,," of ,I>< ~m xl...!, 110. 140

Eul .. . L .. 23 E........ . R. J .,

~

F.-bf)'l0. 8J {os .. '" _ , . . . ; ... 05 F>-.., kli n. P .. l3

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"fa ..,.., . , . . l l~J

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t ... . J~,t.;§! 1~~~~. ir"'IHhtH i~m;~1ji1ij

~~jL H dh,J

SPIRIT OF RAMANUJAN ..bi_iod , _...., II, '13. 24 .. _ ...., 6 ",...dratic bm, 56, 13 qu"''''pIe pro
'"

~I'-s, ... ., :n Shonk D_. 43 ohin..d ~ . 2. 109 ~, T . N_ , 49

~ .. , V., I32 Soraubebra, D. D., !!3 s,..-m..., B . K., 79. S2 s.. __.... M. V. , 2:i , 43. 51 ..... of 'Ior ...... ~ d ..... ben, 12 ....... of Oq ....... 61 ....... of __ Oq"*'oa. 59 ....... of olx oquu", 63

:n.

Rackmact..r. IL , 22

RolI_. M _. :Z:; ~,

V .• 14, 1(17 !tarnaDaj.., illDOM, xiii. !cl.

life.

187

"'ch'

I - D V

Iooc _ _ , >
_ofoq_~, 1)4 ....... of ~ _mtwrs,

Rarnaruoj ... ' ........... maliGn. IS, 23, sa. 6"2. 64, 61, l1l.I RarnaruoJ ..... •• ce-rod tMu. f""ct;,XI.,

•

R.-aDuja,D '.
cou"",' ur .., 52 0&1-.1......9 R.onki.., fl. .... , ~ , 105 Rio " M ut.a fw>c.loa, 105 Robi''' ' 8.. 110. II 1ID&cn, 1.- J., I~, 1.\.8. 166-161 II.oc .....R '''''ouja,D """"..-d fro.ctioa. ~ 1 69 _ , 15<1- 157

dcIi""lon, 11>1> d'_C
'M

....... of'.... Oq_56 Sua, Z .- H .• ~2

Swi
'rob, P. Co . 101 ~w

..' .....

".,6

W_land. H., 1M Wahl. P. T " 80, 81 Wat.ooa . C_ N .• 23, 2:i, 21, 150, 165, 169 W_..... . fl.. 51 W eber, H ., 150, I S1 W....., , _ p-fubrt;"a, !1

lldI.c>e"",ber c. B ..

I~

Sc::hrOte<, H_. 150. 151

s.t....., I ••

r.o,

IlD.

W~ , S ~ 44,41

W illiams, K . S o. 52, 1'9. 1 1-33. lr.G WiDq,,""', idomi.y, r.G w",qu"". 1.-. so Wric .... &. ~ ., 1'9. 14

y_, y _. at Vee, .... J .• 40, 4., llXi. 1(17 V"';!yun . H., 50. 108 Vi. J _. 40

SUkOsy...... U

Sd.litli. 1.-, 1.100. 152

5S, 71 .

Zahor-eotrp • • D_ . 1'9

ZI>o.nc:, L.-e... 15T

155

Copyrighted Malerial

Titles

This Series

III

3<1 Bruce C . Berndt . Nwnber theory in anuj"", 2006

13 Rekha R. Too", ... , Le<:rur<:lO in ~mrtri<: cornbinMorKs. 2006 32 S heldon K . t z , Enumerat i.... pomeuy and otrint; thoo
31 Jobn McC.... ry, A first

CO\1rM ' "

topology: Contillui'y and din>msioo,

"""

JO Serp Tabacbnikov. Gellmetry and billiard.S, '2005 29 Kriotopoor Tapp, Matrix poUI» for uod.,....,nduat'"'" ~ 28 Emm.n ....1 Lesicn e. He&l ill ~ilily,

2005

'21 Reinhard mner , C . Sean Bobun, Samantba M c Collum, .nd T hea "", n Rood e. MMbo.", ... icaJ modem"" A cue ot udiea aPl>f'Ol'th. 2()(6 '26 Robert H udt , Edit .... , Six tbo.me5 OIl otari,..ion. 2QO.I '25 S . V . D..,.bin and B. D. Cbebotarevsky, Tt~iongoul»"" I>qinn~,

U

2QO.I

BruOl! M . Landm .. n and Aaron I\ob,."""n.

R~

tb.ory

01>

tho

int.egen.. 2004

23 S . K . Lando. Lectures

QO

C~ttMi",

functions, 2003

'22 Andreas Arvanitoyo:o'll"" An in' rod"",ioa '0 Lie gool» and tbe ~ry of homogeneous _ _ 2000 '21

W . J . K ti:SOI' and M . T . N.-..k, J>robl.. .... in 111: iDUgTatioo>. 2003

mat~,..icaI

""a1ysi11

20 Kia"" Huld< , Ekroo:m.o.ry aJe;braic ~l)", 2003 19 A . Shen and N . K . Veresbchagin, Computablo, functions, 200J 18 V. V . Yaschenlro. Editor , Cryptn, 2006

CUNoeI -

mri_ - manifolds .

loS G e rd F iocher, p~ ~hraic cun"", 2001 14 V . A . Vassiliev , Introduction to .opokcy . 2001 13 ~ri<:k J. A lmgren , Jr., Plilleau's proM.m: An in.itMion to vuifold tp:J<J>et ry.

200 1

1"2 \V . J . K . czor and M . T. N ........... PrabJo,,,,, in ma.heo-natical &DaI)":m R Conti" .. and dif£_ntiatioo, 2001

i.,.

For a rompJete li>il of l ilies in t his series, viM! t he AMS Boob~ at www ... nlS.org/ bookstore/ .

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