Number Theory in the Spirit of Ramanujan

•

•

Number Theory in the Spirit of Ramanujan

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

01 Ramanujan. 2006

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

"'"

30 Serge Tabaclmikov, Geomecry and billiardA, 200~ 29 K r istopher Tapp, Matrix ~JIII 1'0. un
§2.2.

Elementary Congruences for .. (n)

28

§2.3.

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

31

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

39

§2.5. The Parity of p{n)

43

§2.6. NoteI

49

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

§3. 1.

Lambert Series

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

" " " 56

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

Squ~

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

Note!!

Challter -I. ~4.J.

Ei!!enstein Series

Bernou!li Numbers and Eisenstein Seriftl

§U . Trigonometric Series §4.3.

71

,.n 85 85 87

A Clas- of Series from Ramanujan's Lost Notebook

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

67

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

§3.8.

63

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

§4.5. N _ Chapter 5.

97

'IOS"

The Connection Between Hypergeomeuk Fuoctioll!l and Theta Fullctioll8 109

S5.1.

Definitions of Hypergeomelric Serie. and Ellipt ic Integrals

1119

§$.2.

T he ""ain T heorem

11.

§5.3.

Principle:e of Duplication and Dimidiation

120

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

Eiselllltein Series

I?!

Notftl

128

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

Introduction

133

§6.2.

Sums of Squarl:'!l and Triangular Numbers

134

§6.3. Modular Ecluatiollll §6.4.

Notes

Chapter 7. §7.1.

TIle R.ogen-Ramanujan Continued I'rllCtion

Definition and HisjpficaL BacQ:round COpyngrued Matenal

140

150 1$3 153

SPIRIT OF RAMAN UJAN

§7.2.

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

§7.3.

Thll Rogers-Ramanujan F\mctiOIlll

vii 155 I~

§7.4. Identities for R(q)

161

§7.5.

166

Modular Equations for R {q)

§7.6. Notes

167

Bibliography

m

Index

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Copyrighted Material

Preface

Generally acknowledged ILl! India's greatest mathcm3tician, Srill;""'5&. Ramanujan is IIlOIlt often thought of 88 a number theorist, although he made substantial cont ributions to analysis and several other areas of mathemMi(;8. For ffiO$t number 1hoor;$ts, when Ramanujan 's name is mentioned, the paTlitioll and tau functions immediately comc to mind. His interest in these arithmetic fUllctiollll was inextricably intertwined witb his primary interests of theta [undioM and other qseries. In fact, most of Ramanujan's researeh in number theory 81"O@e out of q-seriesllnd theta functions. Theta fun The purpose of this book is to provide an introduction to this large e:rpanse of Ramanujan's work in number theory. Needless to say, we shall be able to cover only a very small fraction of Ramanujail's work on theta. functions and q-serie5 and tbeir oolurections with number theory. Ilowever, after de,-eloping only a few facts about

Copyrighted Material

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n. C.

BERN DT

q-sene.

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

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

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

~

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

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

SPIRIT 01-' RAMANUJAN

mUes IIOUth-.outhwes~ of MOOr38. where her h\l!lband WNI a clerk in t he office of a cloth merchant. At t he age of twel~, Ramanujan borroo»'ed • copy of the -'Ond part of Loney'. PI4~ 1hgonomdry [149] from an oIdeT student and worked all the problems in it. Thi$longtime popullLl tJ:'xtbook in India ha!! much more in it IhM iU titll:' suggests. For exampll:', infinite !!eriel!l and elJ:'mentary funcl.ioDII of a oomplex ..ariable are two of iUl topics. At the llI!le of about fi fteen, he borro¥.'ed from t he Kumbakonam College library a copy of G. S. Carr'. A $ynOp~1-I of Blernenta'1l Re~td~ in Pu.re Ma themlltic. [64], which sen'ed &'I his primary IIOUJ"(:e for learning mathematiC!. CarT ....aII a tutor in London and compiled this! compendium of 44 17 rClult.l ( .... it h very few proo&) to facilitate hia tutoring. At the age of sixteen, Ramanujan entered the Government College in Kumbakonam. By t hat lime, Ramanujan ....... com pletely devoted to mathematies and CODlIequently failed hill aamillAtions at the end of hia first ~1Ll , be
and

IIUTl1illl

homes.

Ramanujan'. health turned slightly upward when ill 1918 he became the !!«ODd Indian to be elect.ed all a f ellow of the Royal Society and the 61'!1t Indian to be cboeen .as a fellow of Trillity College. Af· ter World War I ended, in 1919, Ramanujan returned home. but his health continued to deteriorate. IlIld on Apri l 26, 1920 Rruuanll,jll./l died at the age of 32. Doctol1l 111 both England and India had difficulty diagnosing Ramanuj"n'. i1llless. lIe was treated for tuberculosis, but a se,'CTe vi_ tamin de6ciency. liver calleer, le&-ersily of Madrllll. It traIlSpired that Ramanujan'! Collected Paper. (192) were published in 1927, bu t hit! notebooks "nd other m"lluscripts were not publ ished. Sometime in the late 1920s, G. N. Watson and B. M. WiLson began the ta.'5k of editing Ramanujan 's notebooks. The second notebook, being a revised, enlarged edition of the 6n;t, was thei r primary focUll. Wilson WII5 assigned Chapten; 2-14, and Watson waB to examine Chapters 15-21. Wilson devoted his efforts to this task until 1935. when he died from an infection at the early age of 38. Watoon wrote over 30 paper!! inspired by t~ notebooks before his interest evidently waned in the late 1930s. Thus, the project w.u never oompleted.

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

D . C. DERN DT

xvi

Wat.iOn probably the m(lllt popular and frequently used text 00 anal· ysis in the 20th century [221[. Rankin hl'd succeeded Wal90n as /I.IllSQn Professor of Mathematics at the Universi ty of Birmingham, where Wa\.80n served for most of hill careoer, but was now Professor of Mathematics at the Uni>-ersity o f GIMgow. Both Whittaker and Rankin went to Watson'~ attic office to examine the papenl left by him, and Whittaker found the aforementioned manUllCript by Ra.-manujan. Rankin suggested to Mrs. Wal90n that he might sort her late husband's papeni and send those worth preserving to Trinity Col. lege Li brary, Cambridge. During the next three years, Rankin sorted through Wst.son's papers :;ending them in batches to Trinity College Li brary, with Ramanujan's manuscript being sent on Deeolmbcr 26, 1968. Not realizing the im portance of R.a.manujan's papeni, neither Rankin nor Whitta.ker mentioned them in their obituaries of Wat-son [195J, [222J. T he next question is: How did Watson come into pQIlSeS/:Iion of this sheaf of 138 pages of Ramanujan's work? We mentioned above that in 1923 the University of Madras had sent a package of Ramanujan's p&peni to Hardy. Most likely, this ~hipment contained the uJost notebook." Of the over 30 papenl that Watson wrote on Ramanujan's ",ur k, two of hill last papers were devoted to Ramanujan's mock theta functions. which Ramanujan dillcovered in the last year of hill li fe, which he des.

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

(1.1.4)

[~]

(q )"

,- {

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

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

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

=

I,

$IIow thllt

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

I.

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

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

~

I.

[m"] ~["- ']+q·["-'l 1 m m~

Exercise 1.1 .6 . Stalnd, employing UmIl1G 1.1.6 and Inducllon on n, ~ that I:' ) u a ~~e(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

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

ilUltructi~

e.xereil!e.

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

(a~ )oo ( blz) ""

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

Proc«dmg like _

~

~

~::oo eR: .

did in the proof of Jhml
(2.L I)

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

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

(2.\.2)

p(5" + 4)

(2.1.3)

p(7n

(2.1.4)

== 0 (mod :» ,

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

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

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

8 .C. BERNDT

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

Let"

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

~. ~ 11 C and (2.1.~)

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

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

:s

(2.1.6)

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

In 1938, C . N. WII.L90n {2IS] publjghed 11. proof of (2.1.6) for a ". t '" 0 rind gll.Vl!! 11. more detll.iled \'~rsion of Ramanujan 's proof of (2. 1.6) in the 7 and that indeed the proof of ~m 2.2.3 de~rar.es this.

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

T(23n - r)

E!!:

0 (mod 23).

Pn)Of. By t he bitlOmial theorem , ~

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

po'I"erl

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

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

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

(q; q)~

..f:,

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

,.,

..f:,

(lmq"" .

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

(2.3.9)

(2j

+ 1)2 + 3m 2 ::

°

(mod S).

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

WO!

see that

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

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

- ) - (q;q)""

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

(23 .. 1 I)

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

..,

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

-

1

= _ 1_

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

= (q;:)1

( 1+2

f (_l)mqm')

eop~tfJl:rMaterial

we

find that

(q;q)""

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

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

f:. pt5" +

(2.3. 12)

R-o

4)q" _

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

P roof. We begin by writing

(2.3.13)

(

I/~·

I /~)

q(.r ; ~ ) .. 00

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

(q)q2 / ! .

To

Re this, UIlC the pentagonal number theorem (1.3.18) in the numerator on the left side of (2.3.13) ./Uld. if n is the index of summation. divide the ltrms Into rtllidue classe!! modulo~. If ,, ;;; O, 2 (mod5). then the p
(J: +

+ Jzq~/'

3J~q)

J~

1[3 / '(3J I

+

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

J?

l1q + q2J1

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

.

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

B. C. BERNDT

36

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

(2.3.19)

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

1 /~)

1/ &.

..,

=

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

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

(2.3.20) (1 _

q~/~)(l

_

However, i f n

wqn/~)( l _

= Srn, then the correspooding terms are

instead of I - qft that

(2.3.21)

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

IT..

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

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

(q5 ;qS).,.,

(q:q)!,

= (rr;qS)~ '

We now examine the product on the right side of (2.3. 19). Sin;z + 1J;J)o4..,~ - ~ (log N )e,

(2.5. 3)

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

#- { n

(2.5.4)

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

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

.. nd. for each (2.5.5)

(We pa.~wexplain the notation :>. We write F(N) :> C(N) , ifand only if there exists a nositi~ ..cplW1.ant (,such that F(N) 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 2.6. Notes Theorem 2.2. 1 hBB been slightly refilled by M. R.. Murty, V. K. Murty , and T. N. Shorey [165]' using a more sophisticated argumcnt They also ohtain lower hounds for t he \l8.Iuetl of "T {n ) .... hen T(n) is odd. Anothcr proof o( Theorem 2.3.1, rivalling Rarnanujan's first proof in simplicity, has been giycn by J. Drost [84). See M. O. Hirschhom's paper [120) for still anothcr clementary proof. Many referellce'! \.0 further proofs of both T~ffl;m~J M.l~(f.worem 2.3.4 can be found

8 .C. BERNDT

50

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

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

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

=

0, 1,2 amounting to

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

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

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

JlO"'~1'lI

of & waB obviously

We have not glven a proof of Ramanujau's congruerw:e P{ l lfI+6):E 0 (mod 11). The 010111 elementary proof is d ue to L. Win-quisL [221] and uses Winquist'g Identity. Further proofs of W inqui!;t's identity h&\~ beo:-n found by HirsIl1I1mary of previous results provided in these papers. The best lo_r bounds for t he number of even and odd valu"," of p(n) in arithmetic progressions are (2.5.6) ]170) and (2.5.7) (61, respectively, while the most gelleral theorems of this _ t are foulld in [55] and (56).

Copyrigl!/ecj Material

B. C. BERNDT In hiB unpublished manuscript on p(n) and r(n), Ramanujlln asserted and, in some CIISeS, proved furt her congruences for T("), mOl!\ involving divisor functions. References to most of the many papen written on this subject can be found in Berndt and Ono's ao:colll1t of Ramllllujan'! manuscri pt [50). We now have a complete understand_ ing of congruences for r(n) through the theory of t- adic representa. tions, IInd an lI
Theorem 2.6.1 . For

an~

po8itive integer n,

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

r(n)

iC

(fJ)=-1 and 2 l ordpn, 0 (nlod 23), if then! eri..u 0 prime p ,,,ch that p = 2xl + xV + 310'2 and or 0 if and only if tWill pnm~ p congruent to 3 mOOulo 4 ;n the canomcal fadon:anon ofn appttlr~ flIith an etlen t:lpOnenl [172. p. [,[,. Theorem 2.1:;1. First P roof of Theore m 3.2. 1. Using the Jacohi triple product identity, Theorem 1.3.3, ",-e 6rsl doouce that 2 (a - l /a)(a q; q)""(a - 1q;q)",,,(q; q)"" = a(a 2q; q)""(a- 2;q),,,,(q; q)".

-

- L

n __ OQ

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

Copyrigl!/ecj Material

S PIRIT OF RAMAN UJ AN

f. + f: )(_lr

= (

n w_ OO

.. w_oo

" ."'''

.. odd

-

L

=

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

-

L

a· ,,+l q,,(l ,,+I) -

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

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

,

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

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

.. ~ of

({2r

~ -2"l( Lq" • __ ...

..

2. - I)' '' ..•... )

+ I)' - (ZI)2 )q,· H>.. r

(

1 + 4q -

d) L~ q>'+'

dq

' .-coo

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

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

: q2 )"" ) ,

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

"'''e

~

L

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

--~

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

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

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

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

(-q' q' ):!.,

copyr@l/edMGffJfldl

SP IRJT OF RAMANUJAN

"

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

t SPIRIT OF RAMANUJAN

63

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

.r( 2~-1 ) _

77

= q and b = q~ t.o find

that

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

q6{1n _l)

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

(3.713) .

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

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

f

n

q2 _l -

... 1

1

ofI 2n - 1)

!I(1" - I)

f: f:

=

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

", .. 0 ... 1

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

lished (3.8.3) using the theory of modulll1 forms. An arithmetic proof of (3.8.3) can be found in [128]. We provide a proof of (3.8. 3) in Chapter 6; see Theorem 6. 2.12. The connection between formula.s for 8ums of squares and sums of triangular numbeTS w88 further solidified by P. Barrucand, Cooper, and Hirschhorn 132J, ",ho proved that

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

where

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

l:S;k::;7.

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

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

82

O. C. BERNDT

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

In·

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

s:

Formulas for r. (n) , when It is odd, have an entirely different Havor. GIIWIS found a formula for r3(n), which Wall put in a mOll! concrete form by G. Ei$elllltein [88J. [90, p . SO~I, who aJao gave $!I analogoUII formula for .)(n). Chapter 4 in E. Croeswald's book [101] is devoted to the study of r3(" )' while p8I!:e!I 128 and 129 in the same text provide information about rll (n) . For recent work on formuJas for r_ (n ), when k Is odd, !lee papelll by ~ ['7 1], [18J. Theornn 3.6.2 is due to Jaoobi . Tb.! lint proof of ~ 3.6.3 was found by Legend", [139, p. 133J. Further proofs ~ gh_ by Cauchy [65, p. 572[, [66, p. 64] and Plana [179, p. 147). For aD elenulntlUy proof ~ (ID an extension of !ton Identity of Liouville, SOle a paper by J . G. lIuard, Z. M. OU, 8 . K. Speannan, and WilIiams [128]. Jarobi [131] claimed tha.~ V. Y. Bol.lniakowsky first proved Theorem 3.6.3, bu~ he did not gi,-e a reference. LIIl [1 43] derh'ed a formula for the number of repre:lentation8 of a poIIitive integer by 8 triangular numbers. T he analogue of Lq;range·. theorem is the theorem of Causa [98, p. 497] stating that every positi\.., integer et.n be repr8ented III a sum of three triangular numbers. An elegant proof via q-aene. hM been gh~n by Andre .... [16]. Williams [2.0.2 + C.o.3, .... here Ill, .0. 2, and 113 denote triangular numbers, represenu all positive inwgers. They are

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

SPIRIT OF RAMANUJAN

83

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

:s

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

SPIRIT OF RAM A N UJA N

89

Using the elLllily Vl!rified elementary iden~ities

q'Hk

(

l _ q.. H

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

(I in (4.2.4), we find t bat

, _

e" -

q" 2(1 -

q")

q" 2(1

(402.5)

.,

,,_I (

" qnl ''--

2(1q" qn)

...

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

+ l - q~

+

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

(I ~"qn) 2

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

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

=I _ qn

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

WO!

find t bat

,

'"'

'"'

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

(

4

(4.2.6)

1

.,

kqk

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

This is the first of tbe two primary trigonometric series identities that WO! need. Using (4.2.6), we e'ltablisb a recurrence formula for the functiol\!l Sr. Appearing in the recurrence relation are tbe fun"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
Q3 _ R2 =- Q(I +5J)2 _ {p+5J)2 _ Q _ pl +5J (4.4.5) But, by (4.2.23) and the binomial ~heonlm, (4.4.6) Q' _ R2 = 1T28q(q' q):14 = 3q (q; q)~ + 5J = 3q (q~; 1-, by Lemma 5.1.10, ~ (611 extend the definition of F (z ) t.o % .. O .... d;l: _ lbytetting

F(O) .. 0

F ( I ) _ I.

and

Lem ma 5.2. t. For 0 < % < I, (5.2.2)

F (%') - F '

Cl

:;l:Z}l) .

..

P roof. In (5.1.7) replace %by (I - %) / (1

, %

+ %).

Obeen:e that

- l -(I +%p'

u

,

( I+%), -> I - Z , 1 +;1:_ - '-.

1+.

Hence,

(5.2.3)

~

arrive .t

1 ~%2FI G , ~;I;I - (I :Z%)l) -

,F1

G'~ ; I ; I - zl).

Now divide (5.2.3) by (5.1.7) t.o deduce that

(5.2.4)

2

2F'(i , l ; I:I-~) (

,F1 1 , 1 ; 1 :(I !~j f)

-

,F1 ( , , 1; 1 _ %'

,F'I

, : 1;%2

Multiplyif18 both . ides of (5.'2.4 ) by - If , exponentiatif18 both tides, a.nd invoking the definition of F (%) horn (5.2.1), ~ complete the proof of (5.2.2). 0

Lemma 5.2.2. 1/%, 0 < Z < I, ", defined by (5.'2.5)

(5.'2 .6)

1- %

r:;:-; "

",2( _ q)

'P"(q) -: A - A(q),

116

8. C . DERNDT

Proof. By (1.3.32). (3.6.7), and the definition of A in (5.2.5). l' .,4(_,') _ { A (q ) - ..,.(q1) -

=~ =

11+» '

2,..{q)op( - q) } ' q)

~(,) + ~(

4~

(I+JT.)

,_ I _ z 3.

o

This concludes tbe proof. Lemm a 5 .2.3. 1/"

(5.2. 7)

F

= 2"',

when! m

U alll!llallllly(ltive illllyu,

(_'..,.(q) (-0») _F" (_'""{q") 1-,"»).

P roof. ~laeing z by (I - z)/( 1 + z) in Lemm.. 5.2. \, find that

1Io"l!

readily

(5.2.8) Applying Lemma 5.2.2 to (5. 2.8), _ find thllt (S. 2.9) Iterate (5.2.9) to deduce that

F(,\'(q)) _ F Z (AZ(qZl),. f'2' (,,2(l'») _ .. _ F2~ (,,3 (q'~»), that i$ to say,

o Lemma 5 .2.4.

1/ n

_ 2"' .

..men! m U anI! nOf\llt9(1111~ IIlllye,., IAen

(5.2. 10)

Proof. From the definition of F (. 1.7). and (:>.3.3),

: = , F L G' ~; I;Z)

- , F O, ~; I: ( I :~),) L

_( I

+ J%ihFL G , ~;I;l':')

_ (1 + H)z'.

(~.3.6)

Th ..... solving (5.3.6) for : ' and Wling (5.3. 1). we find th&t (5.3. 7)

,: ::=~=

1 + v:s:'

:: ~

1+ L-

L+

I( ~I __ ,l+V I _ :r:.

_ I

-I

Them 'em 5.3 . 1 ( Princi ple of Duplication ). Supp!»e that tlllO .ret8 of parumdl!l"l, :r, r . and:: ami T , V and:'. are retatM /lr the cqualtOfl.f (5 .2.27), (5. 2.29), and (5.2.30) WltlL:r, 1/, and: rep/aud b1/%'. V, and T. re.opectlllO!l,. SuppoJe thelllatuh an equallon of /he form n {:r',

v,

t /) _

O.

aml:r L.!I retatM to:r' It, (:>.3.2). Then. by (5.3,1), (5.3.5), and (5.3.7), we obtain an equation of tilt foroL (5.3.8)

n (( 1-11=» ' I

+ .,JI

:r

I I +v1-.o:): ~ ) _ 0. , 21/,-,(

T heorem 5.3.2 (Principle of Dimidi&tion ). A I In the premotll theorem, , uppose that two ,et8 of poromettrl, :r. r, and: and:r' ) ami T , are ntalM 1>11 the ~1l!r}M&klbJl29), and (5. 2.30) ...~th:r,

v,

122

D. C . BERNDT

V' and :r replaced bv z.', an etj'U4to.cm o/liIe lorm

11.

and %', rup«t'w/y. SUPpole they " tuf)

U(%', y' , i): O. and

Ill\!

re!lerU lIIe rotu 01 z , 11, and

%

""Ill /hole 01 z' ,

11,

and

%', re-'J'«hw/y. Then, by (5.3.2), (5.3.5), and (5.3.6), we abtom an equotlon olllle larm (S.3. 9)

' I" 1 ) o ( (I + J%)1' 21/, (1 + v'i"):r

-

o.

5.4 . A Catalogue of Formulas for The ta FunctiollS a nd Eisens tein Series l/sin& (5.2.:19), (5.2.27), the princi ples of duplica~ion and dimidia~ion from tbe previoua KCtion, and element,.,.y theta function identities, such as thale liven in T heorem 1.3.10, we can derive a plethora of evaluation. of the functiolUl~, ,p, J, and X lit d ifferent ~n of the argument q in lerTlU! of %. z, and q. l/!ing the tool! mentioned above, rea.den should be able 10 easily derive each of t he formulll!l below. Proof, of all t he reIIults in Theorems 5. 4.1- 5.4.4 can be found in [33, IIp· 122- 125, Entriell 10-12]. After we state the formulll!l, ...·e offer a few proofs in iIlWltration.

Theorem 5 .4 . 1. Ilz. q, and (5.2.30), tlIen (il (u)

(Hi) (iv)

(v) (vi)

(vi i)

%

ore nUltttl

bv (5. 2.27), (5.2.29),

tp(q) - .;;.,

tp( - q ) _ ';;'(1 _

be,,;"-! to find that one of /he two linmrly independent 6o/utiQn.I 1.'1 2 FI(~'~; 1; x). Those readers unfamiliar with solving ordinary differ(!ntial equa· t ions with regular singular points may use the definition of : = 2F, (i , i;l;r) t(l e3IIiJy check that: = 2FI(i,i;1; x) is indeed 8 solution of {S A. I ). Theo rem 5.4.8. If 11 i, defined by (5.2.30) and ~ if defined b~ (5.2.29), then (5. 4.2) Proof. Uaing (5. 2.27), (5. 2.29), (1.2.4), and Theorem 5A.3(H), (iv), we can resdily esta.blish the identity (5.4.3) Take the losarithm of both aides of (5. 4.3 ), differentiate with resp«t to q, and multiply both side! by q to deduce that q

Ih

00

4nqh

00

nqn

- + C COII2q; + k c. In the oourse of these investigations, ... series for the rectification of the ellipsis occurred to me, remll1kable for its simplicity, as well lIS its rapid convergency. As [ believe it to be new, I send it to )":Ill . Landen's transformation was introduced by J. Land(m in a paper written in 1771 [137] but developed more completely in hi:! pa_ per [1381 publ ished in 1775. This trlUlBformation was crucial in our proof of the fundamental Theorem 5.2.8. The importance of Lan· den's traruformation is conveyed by G. Mittag-LefHer, who, in his survey sUicle [l60J on elliptic functions written in 1923. emphasizes, uEuler's addition theorem and the transformation theorem of Landen and Lagrange ""ere the two fundamental ideM of which the thoory of elliptic functions was in po5IIe!i5ion when this thoory WM brought up for renewed consideration by Legendre in 1786. ~ Born in 1719, LandeR was appoiRted as the llUld-D.geRt to the Earl Fitzwilliam, 11 pOI>t he held until his retirement t"u years hefore his death in 1790. According to an edition of Encyclopedia Britannica published in 1882, ~He [Landen] lived a very retired life, and saw little or nothing of society; when he did mingle in it, his dogmatism and pugnacity caused him to he genera.lly shunnM." Landen made se,~r&l contribut ions to the Ladies Diary , which was published in England from 1704 to 1816 and "designed principally for the amusement and instruction of the fair sex." As was common with other contributoTli, Landen freqlllmtly used pseudonyms, such as Sir Stately Stiff, Peter Walton, WaltonieTL'lis, C. Bumpkin, and Peter Puzdem, for problems he proposed and solved. The largest portion of each issue was de,uted to the presentation of mathematical problellll5 and their 10Jutiollll. Despite i19 name , of the 913 contributors of mathematical problems and IIOlutions O"I~r the years of ill< publication. only 32 were women. For additional information about Landen and the Ladic6 Diary, see a paper by G. Almkvist and the author [11]. Reader!! are also recommended to read G. N. Watson's article. The marqui6 and the Iand-lIgent; a tale ollhe eightunth century [21 7]. (You know the

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
~ ~= (£)B1

Exerel.se 5.5 .2.110

11

< o ,fj

11

~how

~

in the theory


T he next exen::ise is a form of the addition of tlliptk funCl.iona.

..,[

..

r sin'41

_ ~'F,(". b) 2 2 ' 2 '"

The follOlVing exercise gives the elasllical duplkalion /ormula for elliptic integrala, Exercise 5.5.3.I/O < 0, {J< t". ondc:ototan{{J/2) ..

Vl

r!in'a,

proll!! that

J.

2 11

0

dO

Vi :uin'l~ =

EJeerclse 5.5,4 , I/ O < a ,fj < (I + 2O).i no , ,.-ow Mat

J.' Vi .. OIl

!.-

2Otinitjl '

and ( I

+

2Osin 2 o )3inP

The t ransformation abow; la called GaU$ll's trlllUlfo rmation alld is similar in foml to Landen', t ransformAtion given in Exercise :>. 1.9. Historically, the t heory of elliptic functions 1lf0lM! from t he prolJ.. Itm of in''erting i nro~~~, such as the incomplete

B. C. BE RN DT elliptic integral of th'" first kind given in Definition ~.1.8. These inversion problems "''ere motivated by the well_known inver!lion of the trigonometric integral

. J.'

# O< x V L qn... = L: L ~.I

-

... 1

tPq".

"I"

" I d odd

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

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

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

\0

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

proor. It will be simpler from

~1nI

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

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

Copyriglted Matena!

""l!

complete

0

B. C. BERNDT

140

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

r:=

K

L

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

~d

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

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

11

(6.3.2)

~F,(!,!; I ; 1

n

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

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

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

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

(6.3.4)

m

=

'f','(('))

VI q"

= .:!.. z..

We note t hst the method of establishing modular equations briefly dt)8(:ribe tJ

P roof. If """ mske the indicated robfltitutions in the definition (6.3.4) of the mult iplier, we find that [6,3.5)

n

2FI( . ; 1; 1 -(3)

;;; =~ FI (! ' ; 1;1 - 0-)"

Rearranging (6.3.5) and using (6.3 .4 ), we ded uce that

lFI( , 1; 1; 1 _ 8) ~FI( !'

:1 ;11) ,

i.e. , """ obtain the defining relation (6,3.2) for a modular equat ion.

0

We now discuss modular equations of degree 3. We need several identities for Lambert series.

Theorem 6.3.3. If {j} denote" Me ~re ! ymbol, th( n q""H I

)

ql2n+IO '

(ii ) (iii )

(iv) Part (i) ill identical to Theorem 3.7.7, and part (iil Can be found in [3.7.8).

Copyrighted Material

B.C. BERNDT proor o r ( Ill). RePJac;1II q2 by q in (1.3.60) and employilll the J. cob; triple product identity, equation (1.3. 11). we find that (6.3.6) ~

".,'(-q)/(-q) _

= d:d

L

(6n + I )q(h'+oo)n

{z/(q',8,q/ ,8J}1'. 1 d

- / (q, q2 ) d, (IOS {tl(q1:8 .q/:8)})1'.1 - / (q,q1)

(I + : : (loc( _ q/ z8; q' loo( _q2t';~)00{q3; t¥')00)1._1 ) '

Ullilll the Maclauri n teries for log(l + ,). inverting the order of SUDr m.tion, and then summing the resultin& geomeuic &eries. we find that, for 101 < t , ~

(6.3.7)

IOS(-o; q3)"" _

L log(1 + aq3.. ) .~

00 "" (- I)"'(atf")'" ~ - LL m .. _ 0 ... _ 1

00

(_ Cl )'"

-- _ L.,m (1

"-r

U8;1II (6.3.7l in (6.3.6), we find that

'P'{-q)/(-q) _ I (q,q' )

(1 - f: ~

(- q:-I)"

d: .. _ I n(l

q3" )

(6.3.8) where we eJi:pllnded the summand.!! in goometric ..rie- and then iD''ert«!. the order of tummation. Now , by Theorem 1.3.9, (1.3. 15), and COpyr;ghted Malenal

SPIRIT OF RAMANUJ AN

143

(3.2.7), ~(q)/(q)

I(

-r(q)' -q; -q)"" (q; q3)",,( q' ; q-l )",,(-q3; q3)-

q,q~)

_~'(q )(_q,_ q)"" (q, q)"" I (q; q)"" (q; q3)",,{ q'; q3)"" ( q3; q3)""

' I ) {q~; q3)"" -~ q ( q3; q')"" _

(6.3.9) ~lacing


Taking the product of the cube of (6.3.18) and (6.3. 1~),

(6.3.1 9)

.-

(m- l )(3+m)3

"=

(m- l )'(3+ m)

1ft!

find that

16m 3 ' while taking the product of the cube of (6.3.1:;) and (6.3. 18), "''e find

,ha,

(6.3.20)

16m

p

'

Next multiply the cube of (6.3.17) by (6.3. 16) to deduce! that (6.J.21)

1 - 0= (m+l ) (3-m)' 16m 3

'

aDd lutly multiply the cube of (6.3.16) by (6.3. 17) to fiDd that (6.3.22)

1_P

=

(m

+

1.'::: - m ).

pl'OQr of ( Iv), By (6.3. 19), (6.3.20), and (6.3. 18),

o

&.:!m which (Iv) is immediate.

P l'OQf of (v ). Substituti~ (6.3.11) into (6.3. 19), (6.3.20), {6.3.21}, and (6.3.22), _ readily deduce t he four formulas In (v). 0 Pl'OQf or (vi). Uli", fint (6.3.19 ) and (6.3.20) and 8O!(X)ndly (6.3.21) and (6.3.22), 1ft! find, re!lpecth'ely, that

(6.3.23)

"'" (6.3.24 )

~)1 /3

(o

_m{m - I)

I_P) I12 _ ( 1 -0

3+m m(m+ 1). 3 m

If we lubstitute (6.3.23 ) and (6.3.24 ) in the right lide of the first equality of (vi), 1ft! easily Vl'Tify itS troth. The $ find elt&ctly t he value of R(e--';;;)." . GOpyrighJed Matenal

SPIRIT OF RAMANU J AN

157

The mea.ning of this IlI.'jt statement W!L'! not clarified until 1996 when Berodt, H. H. Chlln, and L. - C. Zhllng [46] demonstrllte-')/ :

• _ _ 00

L ~

~ _1_ (_ It "'"'a r .H ' Q(r' - r +.,- .)/2 ('1,'1)"" ' ,'__ 00 ~

(7.4.3)

L

""

where, for

- 00

"e

h.

The previous exercise relating p(Sn

+ 4)

to Fibonacci numbers

was suggested by M. O. Hil'llChhom (125J. In [124J, he used the fact 51/~"+t and a variation of the argument we gave aoo..~ in our proof of Theorem 7. 4.5, t.o give a similar proof of Theorem 7.4.[>. See also (2.3.25). It is remarkable that /f>rIH and f'V> ..+ 'U obey the same CQngruencell as p(Sn + 4) and p(2:'m + 24), respectively. Ramanujan's

original conjocture (2.1 .5) in the case that.s = 53 is p(125n + 99) = o (mod 12[,). Unfortunately, in general, it is not true that fn~ ..+flIl is divisible by 125, and so the analogy f&iLs for 53,

7.5. Modular Equation s for R(q) Recall that a modular equation of degree n can be thought of Ill! a relation among theta functiol\$ with argument q and theta functions with argument q". We dose this chapter by offering some of Ramanujan's modular equatiol\$ for R(q). First, on page 326 in his seoond notebook [1931, [38, p. 12J, Ramanujan recorded a relation between R(q) and R (q~ ) in different notation , which Will! also proved hy Rage", [199, eq. (~.4 )]. Thoo rem 7 .5 .1.

ut .. := R(q)

andv: ""

R(q~ ).

Th en

Also on page 326, Ramanujan adroitly defines the parameter

(7.5.1) and states the rollowing two elegant and symmetric relations [38, p . 13] . Copyrighted Material

SP IRlT OF RAMANUJAN

167

Theorem 7 .5.2. With k d~fined. by (7.5. 1),

Jl&(Q) _ k(I - ,)2 I +k

anG

W(Q2 )_ k7(1+k) I- k

.

In h~ 10et notebook 1194), Ramanujan recorded te'-eTal exqu~ite identities for theta fuDC;tions in the arsument k 01 (7.S.I ); _ 119,

Sects. \.8, 1.9}. The next beautiful modular equation of degree 3 is found on page 321 in R&manujan 's socond IlOtebook 1193}, [38, p. 17), and 'NIIS a1so efltablillhed by Rogera [38. p. 392). T heorem 7 .5.3.

ut u :_ R(q) and 11 :_ R(q3 ). ell _ u 3)(1

+ 10113) _

TMn

3u~1I7 .

Lastly, we conclude with 11 modulu equation of degree 5 for R(q) thllt R&manujan oommunical.ed in his finlt letter to Hudy (107. p. xxvii), [5 1, p. 29] , and that is found on page 289 of Ram&.nujlUl 's teCOlld notebook [19 3], [3 8, pp. 19-20). Again . this modulu eqllll-

lion WM first esta.blisbed by Rose'" [199, p. 392J. FOI' reference! to fw1her proofs, !lee [38. p. 20] or [51 . p. 43]. T heor e m 7 .5.4 . ut u: " R(q) "'nd v:. R ("). Th~n " 1 - 211+4v3 _ 3v3+v·

u - Il l +311 + 4v2+2v3 + ... .

7.6 . No tes The Roge.-.-R&m!l!lujan continued fraction _ one 01 R&manujan'. favorite functions. We have relatO!
in the form of a paper "On the continued fraction

giving a full proof of the principal and mO'lt remarkable tbeorem, vi~. that the fraction can be expre>sed in finite t.emlll when 1: = e-· ..... , when.!! is rational.

However, Ramanujan never

folk",~

HllTdy 's a.hice.

The iLisUJry of the famoWl Rogers- Ramanujan identities (7.3. 10)

i!I now weLl known. They were originally di.scov/1Ud Ch...., equ.ali
[7] S. Ah!gren , TM part1hon ",ncfion modulo Mmpas'/. ",'¥r. M, MMh . Ann , 3 18 (2000), 795-800.

[8) S. Ah!gren an
1'IIl:orJ o} P<J.rl,I, OfU, Addi8On- Welley, Reading. MA , 1976; re~ : Cambridre-n, TIle

/1:;1 G. £. Andre ..... C~ fh>lIoen"., Put"..".., Mem. Amer. Math. Soe" No. 301 , 49 (1984), Amcrio:aD Mathoe""'tial Society, PIO'Vidmoo, RI , 1984.

[16J G. E. Aoo,"", Eo.nb! nurn (1986), 2&-293.

= 6 + .il. + 6 ,

J . NumbH Thy. 23

[17) C . £. Andrews, On tM proof. 0' the Ro,en- R"",o"lIJ'In ,oI F\"'~h" ..... Univer. ity PIUII . Cambridge. 1999. (19J G. E. A"d ",WI .IId B. C. Berodt. R"manll)ltmhmu, S. Kanemitsu and K. Gyijry, eds., Kluwer. Dordr«:ht . 1999, pp. M-49.

R"m""...,...." '""""'nw:t.I /o-r !he part,hQn /tmt:twn. ""d 11 , . ubmitted for public.tlon.

[401 8 . C. Berndt, ......nJo.!>. 7,

[4 1) B. C. Berndt . S. Bhargava, and F. C . G...-van , R....... nlOan '. 111(01"1"" 1>/ dl'phc /*
pOwer ,~"u

",rta... ",aIlCn" 0/ E .. e...lcm .er. .... TT .....

A~r.

(~ ), 4379-44 12.

[43) B. C. Bemd l . P. BiaJelr., and A . J . YeI!, Form tdo.l 0/ Ra ..... n..]
e.......

(~]

a. C. Berndt and A. J . Yet , A J>Il9C 0 .. t ..1I .enu ,.. &m.m ... ]duct •.w.!dy, J. In_ d;IIn Mllth. Soc. 6 1 ( 1995),226- 228.

[58) S. Bhllrga,,. Md C. Ad;gll. S,mple proof' of J4QObi', !\IIO ~nd fou .. lqure Ih~, Inter. J . Mlltn. &\. Sei. Tech. 19 (1988), 77!J-782 .

(59) S. Bhllrga..... C. Adigll, ""d D. D. Somuhekara , R..",..,.enJa!,on of an intre ..... •n Pure A-f.,lhem..,/lCI.

(65J A. c..ucby, Mrnw,;re . ur I'.,pplication .I.. calcul de. ,.tWl..., .,,, ~Iopptme"t .I" produ,t,o cmnpoJ6 d 'un nombre ;"fi"; de f".:I~rs, C. R. AClId. SeL (Paris) 17 (\ 843), 572- 581 . [66J A. Caucby, O~....e" Ser. I, Vot. 8, C aulhier_Vill ...., Pari., 1893. [67] A. Cayley, An Ekmen/a'1l 1h:
[81] P. De1i&11e. La con,..,t ..... de !Veil. I, Insl .

lIaU\~ Elud.,. Sci.

Pub!. Math . (1 974 ), no. 43, Z73--307. (82] P. C . L.. Diridlll!l , Rtt~. 'ur dnlef"OU apphcotlQnl ft /',,1IIa.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,

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