Bruce C. Berndt - Number Theory in the Spirit of Ramanujan - AMS - 201p

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Number Theory in the Spirit of Ramanujan

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

01 Ramanujan. 2006

33 Rekba R. Tbornaa, Uctura in !!«>rnelric combi""loriol>f'Ol'Ch. 2005

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

26

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

a.naI,....

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

Copyrighted Materiar

Contents

Preface

Cbapter I.

Introduction

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

6

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

Functio~

§1.4. Nole8 Chapter '2. Congnaenautiful pl"O\M'niel ... tisfitd by tlm continued frllCtiol'l will moti'1lte readers to turn to original ~ to learn mote about it.

=

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

8Il well IIl3 their in6nite versions

Iql <

I,

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

SP IRIT OF RAMANUJAN

xix

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

Copyrighted Material

Chapter 1

Introduction

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

(I.LI )

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

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

.-,

n(\n al l,

(Ul ),

n 2: I ,

••• (a)""

:=

(a ;q)"" :=

(1 -

Iqj < I.

.~

We roll q the btm, and if the identifioou 0, and usilll l. 'Haspi!al's ru~ while letting q tend to I, we find that

. (q-;q)~ . _ 1- _ t(" _ 1 -_ If+! c"c--' lom - - _ hm _ · .. _"C-'C-er, it is one of the mOISt import&.nt &.rit hmetic&.l functions in Ilum~r theory &.nd Ari8es ill ma"y oontexU, in particular. in the theory of modular forms. We study this e2,~~~RR/in Chapter 2.

B. C. BEItNDT

6

DeHnltlon 1.1.19. For pnitive 'nlfSOeY"' n and t , kt r~ (n ) "mote /he numbtr of ~pre8m tatKml 0/ n III 11 81irn 0/ k 'qlUlnl. ~ rep1UentatiDru ""Ill d.ffemtl urden ond dIfferent ngn.r lire aot(q) :- / (q, q)

(1.2.3)

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

labl < I.

~

E qn ("+ I)/~,

Cop~ed'JflJterial

S PIRIT OF RAM A NU JAN

7 ~

I( - q):·f(-q, - i )-

(1.2.4 )

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

L

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

equal.,~

01 (1.2.3).

St:t: IIl80

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

.;,t (q) _

(1.2.5)

L It(n)q".

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

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

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

enormoti,!l~ ti,!~JW

Lc:oJnnta 1.2.4 . We hQw.

(1.2.6)

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

(1.2.7)

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

(1.2.8) and,

.f n

(1.2.9)

.., Qn~ IDlegff,

f (a, bo) _

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

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

B. C. BERNDT

8

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

Iql. 1:1 <

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

(1.3.2)

(az),.,

-I') '"" ""

~

n

1:1<

~ An: ,

I.

n_ O

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

( 1 - z)F(z)

( 1.3_3)

= (I -

az l F (qz) .

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

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

"'

1 _ a.qn _1

(1.3.4)

An -

I

qn An_ I.

n 2: l.

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

An

( L '5 . )

Using (1.3.5) in (1.3.2),

_(a)n (q) .. '

,,~complete

n

2: O.

the proof of (1.3.1 ).

0

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

= a(a +

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

=

/(q) ( ) = /(_q2)

.z2)

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

"

_ >.2%9)

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

00,

~

(1.3.:>5) From the definition of

1(:)

I,

=

L

a~.:".

we find that

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

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

ThLl8, from (1.3.55),

B.C. 8ERNDT

20

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

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

(l3~

q6ft -~Q.3" _3

'"'

_

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

.•• _

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

( 1.3.57)

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

q~'" (1\,

( 1.3.58 )

" 3" +1

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

By the Jaoobi triple product identity, T heorem 1.3.3,

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

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

~

L

=

(1.3.59)

( _ I YHq'Jl.i-I)"· 'z'JH,

j"'--",

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

(q4: q4).. ao

L

"

(_ I y" qSi'-2j

(q4 ;,').."

_

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

and

~ince

Q, \ . ,

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

a, _

- I. Recalling aJao that

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

P uuilli ,belie value\! in ( 1.3.SS), ,..~ complete the proof of ( 1.3.52), after rep1adna: n by -n- I in tbeseoond sum arising from ( 1.3.:.5). 0 Corollary 1.3 .:U . &coli fh O.~ Although not stated as a conjecture, since that time the nonvanishing of 1"(n ) ha.'i been known as Lehmer's con· jecture. Theorem 1.3.1, the Q-binomiaJ theorem, is due to A. Canchy (66, p. 4:>J, ""'hile Corollary 1.3.2 was first proved by L. Enler [91 , Chap. 16). T he Jaoobi triple product identity was proved by C. G. J. Jaoobi in his Fimdamenta Nova [131], the paper in which the theory of elliptic functious was founded and one of the mOll! important papers in the history of mathematiC!!. Hov."tlver, the Jaoobi triple product identity was first proved by C. F . Gau~ (91, p. 464]. The proof that we have gh-en here is independently due to AndreWll (12) and P K Menon (155J. F. F'l"anklin devised a beautiful oombinatorial proof of Corollary 1.3.:> (o~ Corollary 1.3.8). See Hardy and Wright'S book [112, pp. 286-287] or Andrews's text [14, pp. IQ111 for F'l"anklin's proof. Jaoobi'" identity, Theorem 1.3.9, has an elegant generalization found by S. Bhargava, C. Adiga , and O. O. Somashekara ]60]. Ramanujan's I>PI sum mation theorem was fil"5t stated by Ramallujan in his ootebooks [193, Chap. 16, Entry 17]. In fact, he stated it in preci$ely the form giwlfi in Corollary 1.3.14. It was found by Hardy, who called it, "a remarkable formula with many parameters" and intimated that it could be established by employing the qbinomial theorem [101, pp. 222- 223] . Howe"er, it was not until 1949 and 1900 that the first published proofs were given by W. Hann ]103) and M. Jacbon (130), respectively. The proof that "re have given here is due to K. Venkatachaliengar snd was presented for the first time in the monograph by Adiga, Berndt, Bhargava, and G. N. Watson [51; see also Berndt's book [3 4 , pp. 32-34J. As with the other thoorems;n this chapter, there are now many proofs of the Ilh th"", rem. Almost all proofs employ the q-binomial theorem at some stage. W. P. JohllSOn (132) has written an interesting semi-expository paper showing how Cauchy could have diso;xr.-ered Ramanujan's 1>P1 summ ation formula (but did not). Rams.nujan left uo clues about his proof Copyrighted Material

8. C. BERNDT

"

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

ilUltructi~

e.xereil!e.

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

(a~ )oo ( blz) ""

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

Proc«dmg like _

~

~

~::oo eR: .

did in the proof of Jhmlenu on the right aide ~

multipll'S of 5.

O~n~

that

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

Thus, I

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

1ft

that (2.3.6) ill true

+ 1)2 •

0 (mod 5).

III particular, 2k + I • O{mod5 ), whkb , by ( 2.3. ~), Implies t W I ~ coefficient of ~+5, 11 ~ 0, in q(q; q)!c, is a multiple of~. The coefficient of r/"+$ on the right side of (2.3.4) is therefore &00 • mult iple of ~, i.e., p(~1l + 4) is a multiple of S. 0 We now Ki~ a .geC;z and '-'3, we have counted each contribution t.o C 2 t",·ice. i.e., in fact, C, = -5. Using our values Cl _ - 5 = C2 in (2.3.12), "''eo deduce that (2.3.23)

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

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

(2.3.24)

F

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

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

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

(2.3.25)

8 .C. BERNDT

38

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

(2.3.26)

(q; q)t,

1

Jf(q)

=

11q + J~ (q)q~'

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

0

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

Theorem 2.3.6. For

e~ry

nonntgal1t>e inttger n,

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

(2.3.27)

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

we find that (2.3.28) ~

LP(5n

('~)'

~

(q;q)oo

....0

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

.. _ 0

p(n)q" (mod 25).

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

+ 24) == 0 (mod 25).

o

This completes the proof.

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

r(5n)

E

O{mod5).

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

L,., T(n)qn :

". 1

q(q; q )~

(.)2~'"

= q (:: q)

00

== q(q~; ~)!,

copyrigh1e'3 Ma/enal

L p{n)q" (modS).

n_ g

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

(2.3. 7)

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

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

for some posith'e number K. Ablgren proved (2.3.7) with K = I) An elegant, elementary proof of (2.5.7) when K = 0 was established by O. Eichhorn [86]. The lower bound (2.5.7) is currently the best known result for odd \"Blues of p(n). Our goal in thill section is to prove the resul ts (2.3.3) and (2.3A) of Nioolas, Ruzsa. /lIld SkkCizy 1170] by relati\'ely simple means. Our proof is a special instance of an argument devised by 6erndt, Yet! , and A. Zahareso;u 155[, woo prm-ed oonsiderably more general theorems that are applicable to a wide VlU"iety of partition functions. Except for one step, when we Ullll!t appeal to a theorem of S. Wiger! 12231 and RIImanujan 1185), our proof is elementary and self·ooutained. Th eorem 2.5. 1. For ooch /ix(d c u.';!h c > 2 log 2 and N I.21 )

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

~

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

(q;q )"".

~_!

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

SPJRlT OF RAMANU JA N in A, then

(2.;'.22)

1 = F(X)

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

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

F (X) = I +xn,

+ X'" + ···+X", + ... ,

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

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

(2.;.24) ..d

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

(fxn,) + f (x" ( I

, .. ,

(2.5.26)

=

13.. - I)/1 +

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

.... I

f:. --,

(xmI3m -!)/2+

x m (3 m +l)12) .

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

(2.5. 27)

X" ,

(I + ~ (x ..

{3n -' J/2

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

•

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

nj

+ !,,(3n -

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

which we put in the form

(2.5. 29)

2nj ~

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

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

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

8. C. BERNDT

"

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

n,3m + 3'1 -

1)

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

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

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

\.0

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

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

Oc (Nr;;Cv:)

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

at least.

0< (N! -~ )

numbers n, ~

N that are needed

to match

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

x

Theorem 2.5.1.

0

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

G(X} '''' X ""

+ X "" + ... E

A.

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

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

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

Hlmj:S: NI =

N - #{nj

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

#Imj

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

we find that

1+ G(X)

= =

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

I~ (I+ ..t , X

AJ' +

(X ft (3ft - l){l

+ Xft IJn+l)f2) )

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

- •-r,.;opyrigl!/ecj Matenal

SPIRIT OF RAMA NUJ AN

"

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

(2.5.33)

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

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

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

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

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

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

+.

x

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

(

1 + 4q -

d) L~ q>'+'

dq

' .-coo

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

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

: q2 )"" ) ,

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

"'''e

~

L

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

--~

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

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

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

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

(-q' q' ):!.,

copyr@l/edMGffJfldl

SP IRJT OF RAMANUJAN

"

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

tame parit y, we deduce t hM ~

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

".,

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

r

(q ; q )~(

2R(q)

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

(3.4.9)

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

( q2; ql )!,

=

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

(ql ; ql)~

=(

q'; ql)g" ,

wheTe we have appealed to Euler's identity (1. 1.9) in the last step. 5e 4 are more complicated than thOllC for k S 4. For t hose who have some familiarity wit h modular forms, we remark that t he generating function {f'2i(q) for r1. (n ) is a modular form of weight k. For k S 4, t he dimension of the spa« of modular forms in which Copyrigl!/ecj Material

SPIRIT OF RAM A N UJA N

89

Using the elLllily Vl!rified elementary iden~ities

q'Hk

(

l _ q.. H

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

(I in (4.2.4), we find t bat

, _

e" -

q" 2(1 -

q")

q" 2(1

(402.5)

.,

,,_I (

" qnl ''--

2(1q" qn)

...

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

+ l - q~

+

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

(I ~"qn) 2

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

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

=I _ qn

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

WO!

find t bat

,

'"'

'"'

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

(

4

(4.2.6)

1

.,

kqk

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

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

T,

(4.2.40)

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

8 . C. BERNDT

98

these formulaa form

Me

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

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

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

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

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

..,

= To(q},

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

We

II(IW

state Ramanujan's six formulu fOT Tn followed by •

ooroJlary and his gener&! formula.

* .,

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

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

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

(q, q).,.,

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

q;q

1'1 IT IO(lq)

q;q ""

(vi)

+ 44 8P R _

132Q2,

00

ITI1 {lq)

q;q "'"

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

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

+ 80256PQR -

2712Q3 - 9728Rl . Copyrigl!/ed Material

SPIRIT OF RAMANUJAN

'03

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

Theorem 4.4.1. For eodi IWflnegaliw inleg",(q), namely, ~

L

>ir, .• (q):=

(_ I) .. - Im'n'q""·

~

F, .• (q):=

L

(2m _I )'n' q(~m - l)nn,

...... __ OCI

where (q( < I. Ramamani employed the classical theory of elliptic functions in her work. In extending Ramamani '. study of (q), in her thl)Oji.9 [102], 11. Hahn chiefly employed ide ..... from Ramanujan's viewpoint in the theory of elliptic funct ions.

w, .•

H. H. Chan (68] deriwd an analogue of (4.2.2) .... ith cot replaced by C9C and used it to give new proofll of Ramanujan's famo\lll formulu

wd

.... here (i) denotes the Legendte symbol. These formulas lead, respectively, to proofs of Rarnanujan 's congruences p(Sn + 4) := 0 (mod 5) and p(7n + S) := 0 (mod 7). Z.--G. Liu [148] used the theory of ell iptic functions and associated complex analysis to deri>-e a trigonometric series identity invulviog thetB. functions that is analogoWl to Ramanujan's t rigonometric series identitiell. W hereall Ramanuj&n used his results to study sums of Squares, Liu used his identity to obtain repreJlentations for h.(n). The content of Section 4..3 i.9 taken from a paper by Derndt and Vee [53]. K. Venkat~3.8;~)ftJa~aflr;a~I -32] has gh-en

&.

similar

8 .C. BERNDT

112

Theorem 5. 1.6 (Landen'. Transronnalion). ForO

< :z:

t he held until his retirement t"u years hefore his death in 1790. According to an edition of Encyclopedia Britannica published in 1882, ~He [Landen] lived a very retired life, and saw little or nothing of society; when he did mingle in it, his dogmatism and pugnacity caused him to he genera.lly shunnM." Landen made se,~r&l contribut ions to the Ladies Diary , which was published in England from 1704 to 1816 and "designed principally for the amusement and instruction of the fair sex." As was common with other contributoTli, Landen freqlllmtly used pseudonyms, such as Sir Stately Stiff, Peter Walton, WaltonieTL'lis, C. Bumpkin, and Peter Puzdem, for problems he proposed and solved. The largest portion of each issue was de,uted to the presentation of mathematical problellll5 and their 10Jutiollll. Despite i19 name , of the 913 contributors of mathematical problems and IIOlutions O"I~r the years of ill< publication. only 32 were women. For additional information about Landen and the Ladic6 Diary, see a paper by G. Almkvist and the author [11]. Reader!! are also recommended to read G. N. Watson's article. The marqui6 and the Iand-lIgent; a tale ollhe eightunth century [21 7]. (You know the

Copyrighted Material

SPIRlT OF RAMANUJAN

131

identity of the 1and-agent; to augment your curiosity, '"' relTain from ~n, you the identity of the marqui$.) Ext!ciaoe 5.1.9 is just one of many beautiful transformation for_ !!Iul.. for elliptic intep-a1s, many o f which are due to Jaoobi and/ or Ramanujan [34, pp. 104-113J. We offer a few of thete transformation formulas as exen::i8etI. Exercise 5.5.1. I/O

.,'

< o,p e mtqrer n ,

- - -

Proof. By Corollary 6.2.11,

~'(q) -

2>V L qn... = L: L ~.I

-

... 1

tPq".

"I"

" I d odd

Equatill8 coefficients of qn+l on both sides above, ...,., complete the proof. 0

Recall that in Theorem 4.2.4 we establ ished the following fundamenu.l theorem.

T heorem 6 .2. 13. lYe have (6.2.9)

\0

Our representations from Theor=15 5.4.11 and 5. 4. 12 enable us give a very simple proof of Theorem 6.2. 13.

proor. It will be simpler from

~1nI

to UIe the argument ql insttad of q. Thus, 5.4.11 and 5.4.12. ,,"l! find that

Ql(q1) _ n2 (q2) .. ZI2(1_ %+ %1 )J _ Z12(1 + %)2( 1 _ !%)2(1 _ 2%)' _ -13 ' : 12%2( 1 _ %)2, (6.2.10) after a doNge of elementary algebra. On the other hand. by Theorem 5.4.3(iii), lns.? (q2 ;q2)!! .. 1728q2{"(_q') '" Z& . 3J q 2 z I22- 'z2(1 _ z )llt? (6.2.11 ) '" *3' : 11:.: 2 (1 _ %)'. Combining (6.2.10) ltDd (6.2.11 ) and replacing q2 by q. tbe proof.

Copyriglted Matena!

""l!

complete

0

B. C. BERNDT

140

6.3. Modular Equations Definitio n 6.3.1. Ld K , K' , L, and L' denQte oomplde elliptic integrat.. (If the jiNt kmd auodaled with the moduli k , k' := v'f="P, I, and ~, respectively, where 0 < k,t < L Suppou that K' l/ (6.3.1 ) n- = -

r:=

K

L

for lome poritive mteger n. A ",,'alion l>etw«n k and I induced by (6.3. 1) if called a modular equanon 0/ degru n. Following Ramllllujan , set 01 _ /,;'

~d

We o!'ten !lay that {J ha.s degree n o_-er a. for

UBing Lemma [>.1.3, we may replace the defining relation (6.3.1) modular eqUlltion by the equ ivalent relation

11

(6.3.2)

~F,(!,!; I ; 1

n

aj , F,(!,!;l;Q) ,..

,F,(!,!; 1; 1 -

P) ,Fl(!.!;I;,Bl .

Using (52.29) and the formulas from Section 5.4, we see !.hat 11 modular equation can be considered as IUl identity amongst theta functions with arguments q and theta functions with arguments qn. In fact, most often One e/jt ablishes a modular equation of degree n by 6r8t proving the requisite theta function identity. Then we use the formulas from Section ;'.4 to express theta funct.ioll.!i with argument q in terms of a , % = ~1, and ()XlSI:Iibly) q, and the theta functions wit h argument q~ in term. of (J, z.. , and (possibly) q", where (6.3.3) The multiplier m of degree n is defined by

(6.3.4)

m

=

'f','(('))

VI q"

= .:!.. z..

We note t hst the method of establishing modular equations briefly dt)8(:ribe1, [34 , p. 301 proved that (7.3.2) which we n ow prO\-"e . In fact, we fi rst prove a fin ite form of (7.3.2), which can be found 88 Entry 16 in Chapter 16 of Ramanujan 'a seo:lnd notebook [193], [34, p . 31). and from which (7.3.2) follow$ as an immediate corollary.

Theorem 7.3.1. Far each fWflllegative integer n , let (7.3.3)

~ := p. .. (o , q):=

1(.. + 1)/2)

L

i_O

(7.3.4)

v:= v" Ia , q) :;

~ I.'

IO q (q)t (q).. _2Hl

(q) .. _ H

,

I~I (q).. ~ ~a~q- (-+l)

f;o

(q)k(q)..

lk

'

whtrt r::t:] denolu the greatest integer less /f"", or equal to::t:. Then, / orn ;:O: l , (7.3.5)

!:: = 1+ ~ 1

v

+

a.q2 1

aq"

+ . .. + - ,- .

Proof. For each nonnegalive integer r , define l(n ~ .+I )/21

( ) k k(.+k) q n ~ . t+ l a q . b O (q)k {q) n - ' ~ 2HI Copyrighted Material

F. := F.{a, q):=

L

SPlRlT OF RAMANUJAN

Obeerve

'"

tha~

(7.3.6)

H - I'.

Fo- Jl.

F~

(7.3.7)

_ I

F" _ I_ I +aq" .

We now develop !\ recurrence relation for Fr. When we combine the tWO sums in the first step below, we use the fact th!t.t 1/(q)_1 _ 0 by (1.3.35). To that end, U ~ -'+I)I11 ( )

L

Fr - Fr +l .,.

~ t (rH )

q .. _r_ t +I O q (q).(q) .. - r - 2HI

t -o

1( .. - r)/l] (q),,_._kClq-"(HIH)

-E

(qJA(q),,_r _U

.

(q)t (q) ..

L,

.

(q) .(Q) ...

~

{q)1_1(q) ,,_r

1(.. - r+I )/2I (q ).. _r _kakq-"(rH ) ( I _ q.. _r _l+ 1

- L,

_._u

I

(1 - qk)

1( .. - r)l 0/ IIId a·jllnchona, in Theta .....""_ homo: I'hmI flit CIa""cuI to th., Modem, M. Ram Murty, ed .. CRM P"",. and Lect ure Notell. Vol. I. American Mat~m.tical Society, Providence, RI. 1993, pp. 1-63. [37] B. C . Btorndt. R""",,,ujan', NolelloooQ. PIIS1 IV , Spri/l&er-Verlag. N., .. York , 1994. [3S] 8 . C . ilernd t , York, I998.

Ro....."lI)un J . eo..... put. Appl. Math . 105 (1999), 9-24.

(46] B. C. Berndl , H. 11. Ch ....., and l..--c. Zhan" EJ'PliclI etIGI'IW'-"'" 11/ Ih~ R.lgtT"I- Romnllli)ll& Dnd mod\ll4lr fuM.t lll&, Bull. Amer. Mat h. Soc. 4 2 (2005). 137- 162. O\e", I",...,r oou"d on I~ nu",ber 01 odd VIII .... 01 the Of"dmo'll pol11110n luncllO"n. to a ppear .

[86[ D. Elchllo rn , A

[87[ G . Eille"'1.I!in, Ctnoue UnI..,.,uchung der unendllclltn Doppe/proillICIt, ,11" wkhco d.e elhpluciw:n Fu""' lOne.. ..t& Quol,mle.. ."'a..... n\etlilli'ngenden Doppe/mMn {a4 "'........IIa.th. (Oxford) (2) 12 (]96]), 285-290.

[100J B. Cordon ,

[101) E. G """,,,,,iliI , Rep"",enlahQnf of Inleg ..... Springer_Verlag , New York, H185.

duiUntll..icklung. I, ~iBlh. Ann. 114 (1937), 1~28; 11, Math . Ann . 114 (1 937),316-351. (11 4] M. O. Hirschhorn, A .. mple prIXlf of Jac.o/li', fOW"4'l"aI'C ~m. J. Austral. Mal.b. Soc. Ser. A 32 (1982), 61-(17. (1 ]5) M. D. Hirschhorn , A .,m"", proof of J4oob,', t...., .• 'l"4I'C theonom,

Amer. Math. Momhly 92 (1 985),

57~.

(1 16) M. D. Hirschhorn, A .rimpl~ p-roof of }acoin. " four'lHI"a ... theonom. Froe. Amer. Mat h. Soc. 101 (1987),436-438. [11 7] M. D. Hirschborn, A genen.lo:atlon of W.nquut·, id,",,1II1I and 4 con· JeCt.."" of Ramomlljan, J. Indian Math . Soc. 51 {1 987}, 4!!-M. [11 8] M. O. Hiracbhorn, R"monll)dtoa Ulmlohu,

Ralul~

Matb. 44 (2003), 312-339.

(137) J. LandeD. A dil"..",hon ""1I.C('m'''9 oet1a.n fl~£1W. ";'Ilcll "'"' .... ,."..Qb/e ~ !ht .."" oJ !he oonic . eet;""'; ..... erein "'"' mvul'gated ,om~ ne", Bn4 I14tfod !heoreml Jor co",put.ng ...eh fluent.., Phi"". Ttano. Royal Soc. London 61 (1771 ), 298~309, [138) J . Ll.IKIell , A>I m .... hgtion oJ .. 9""'nol th~ IIff find'ng IlK l/:>lgtiI oJ any IOI"t oJ 4nW come h~, ~ mel$oc. 5 (1962), 197-200 (1 64) M R. Murty, TM Rnmc"u)"n 1" /unchon. in Rnmanujon lkon.,t6l, G. E . And",,,",,. R. A. Allloey, B . C. Bernd!, K. G. R.o.rnan.o.th.o.n, Bnd R. A, Ranki n. !!dB. , Academic PreIIi!I. San Di~o, 1988, pp. 2(;9-288.

[lIiS) M. R. Murty, V. K. Mu'ty, "",d T . N. Sl>orey, Odd voJu~ 01 the Ro"",n,,)on ,·fund",n, Bull. !>lath. Soc. Fr""ce 11 5 (1987), 391-395. 1166) V. K. Murty, Rom",,"';"" ""d 1I"n3h-C"""dro, Math . Intell. 15 (1 993).33--39. 11 671 M. B. Nath&MQll , Elemen/"'1I M ethoo. ,n Nu""," 111""'1/. Spring~r_ Verlag, New York, ZOOO.

[168) M. NewmB.II , Penodialy mooulo m and di ..... ,blhty p"'p"r/,a "J p"rtili"" func/,,,n,

n-""". Amer . M.o.th . Soc. 97 (1 960) , 225- 236.

th~

llW) J .-L. NicolaOl, V4Ieu .. In,p"'I"U d~ la Joneli"" d~ pcol"l,/"", PIn), In_ tmoI' J>TOperf"iu of Ihe PtIJ"tot,,,,, f\",eh"" (I."d A .."",aled ""ncli""",, Doc1.oral "l"beoil , Unlvenity of Birmin,ham, Birmlngh&Jl] , UK, 1950. 12O'l] R. RUMen, On le), - Ie')'" ",od\olj}r tqlL(ltiofll, PrQC. London Matb. Soc. 19 ( 1887), 9O-1II . 1203] R. RIIIOIeIl, On modtJ4r Proc. Loodon 1>btb. Soc. 21

..

[

cvu