NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS
CONTENTS 1 THE BASIC IDENTITY INTRODUCTION THE GENERALIZED RAMANUJAN IDENTITY THE WEIERSTRASS ELLIPTIC FUNCTION NOTES 2 THE DI?ERENTIAL EQUATIONS OF P, Q AND R RAMANUJAN’S DI?ERENTIAL EQUATIONS RAMANUJAN’S 1?1 SUMMATION FORMULA RAMANUJAN’S TRANSCENDENTALS U2N AND V2N THE IMAGINARY TRANSFORMATION AND DEDEKIND’S ETA-FUNCTION 3 THE JORDAN-KRONECKER FUNCTION THE JORDAN-KRONECKER FUNCTION THE FUNDAMENTAL MULTIPLICATIVE IDENTITY PARTITIONS THE HYPERGEOMETRIC FUNCTION 2F1(1/2, 1/2; 1; X): ?RST METHOD 4 THE WEIERSTRASSIAN INVARIANTS HALPHEN’S DI?ERENTIAL EQUATIONS JACOBI’S IDENTITIES AND SUMS OF TWO AND FOUR SQUARES QUADRATIC TRANSFORMATIONS THE HYPERGEOMETRIC FUNCTION 2F1(1/2, 1/2; 1; X): SECOND METHOD 5 THE WEIERSTRASSIAN INVARIANTS, II PARAMETERIZATIONS OF EISENSTEIN SERIES SUMS OF EIGHT SQUARES AND SUMS OF EIGHT TRIANGULAR NUMBERS THE HYPERGEOMETRIC FUNCTION 2F1(1/4, 3/4; 1; X) THE HYPERGEOMETRIC FUNCTION 2F1(1/6, 5/6; 1; X) THE HYPERGEOMETRIC FUNCTION 2F1(1/3, 2/3; 1; X) 6 DEVELOPMENT OF ELLIPTIC FUNCTIONS JACOBIAN ELLIPTIC FUNCTIONS RECIPROCALS AND QUOTIENTS DERIVATIVES ADDITION FORMULAS 7 THE MODULAR FUNCTION ? MODULAR EQUATIONS MODULAR EQUATION OF DEGREE 3 MODULAR EQUATION OF DEGREE 5 MODULAR EQUATION OF DEGREE 7 MODULAR EQUATION OF DEGREE CHAPTER 1 ?Q5 + WHERE Q = 1 + 1 ?Q3 + , AS WELL AS A NICE DEDUCTION OF A CONVOLUTION PROPERTY FOR DIVISOR SUMS HERE, AND THROUGHOUT THIS BOOK, Q IS A COMPLEX VARIABLE THAT SATIS?ES |Q| < 1 LET F(Q) = Q1/24(1 ?Q)(1 ?Q2)(1 ?Q3) RAMANUJAN DETERMINES ORDERS OF SEVERAL ARITHMETICAL FUNCTIONS USING THE ?(N) NS = Y P PRIME 1 ??(P)P?S + P11?2S AND MAKES THE FAMOUS CONJECTURE [91, (105)]: |?(N)| ?N11/2D(N), WHERE D(N) DENOTES THE NUMBER OF DIVISORS OF N THE CONJECTURE WAS PROVED HALF A CENTURY LATER BY DELIGNE [51], USING THE ADVANCED METHODS OF ALGEBRAIC GEOMETRY IN [91, (17)] RAMANUJAN HAS GIVEN THE FOLLOWING IDENTITY (1 5) = 4 COT ? QN (1 ?QN)2 COS N? + NQN 1 ?QN (1 ?COS N?) NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS THE BASIC IDENTITY HE GIVES A COMPLETELY ALGEBRAIC PROOF OF THIS IDENTITY HE ALSO GIVES THE IDENTITY [91, (18)] 8 COT2 ? 2 + (1 6) = (1 7) WHERE Q = 1 + J5QJ 1 ?QJ IN THIS CONNECTION, RAMANUJAN OBTAINS HIS IMPORTANT DI?ERENTIAL EQUATIONS Q DP DQ = P 2 ?Q , Q DQ DQ = PQ ?R JQJ 1 ?QJ RAMANUJAN’S SIMPLE PROOF OF THE PRODUCT FORMULA (1 7) HAS ESCAPED THE ATTENTION OF SCHOLARS WEIL [108, P 34] AND LANG [80, P 249] BASE THEIR PROOFS ON THE CLASSICAL IDENTITIES OF JACOBI-WEIERSTRASS THEORY SERRE, IN THE GENERALIZED RAMANUJAN IDENTITY THE IDENTITY (1 5) MAY BE GENERALIZED INTO A FORM WHOSE PROOF BECOMES VERY SIMPLE LET ?1(Z) = 12 + X N ? QNZN (1 ?QN)2 , (1 8) WHERE THE SYMBOL X N ? DENOTES THAT THE SUMMATIONS ARE OVER ALL NON-ZERO IN- TEGERS N BY THE RATIO TEST, THE SERIES DE?NING ?1 CONVERGES FOR |Q| < |Z| < 1, WHILE THE SERIES DE?NING ?2 CONVERGES FOR |Q| < |Z| < |Q|?1 THE GENERALIZED RAMANUJAN IDENTITY IS THE FOLLOWING: FOR ANY THREE COMPLEX NUMBERS ?, ? AND ? SATISFYING ??? = 1 AND |Q| < |?|, |?|, |??| < 1, WE HAVE ?1(?)?1(?) ??1(??)(?1(?) + ?1(?)) = ?2(?) + ?2(?) + ?2(?) (1 9) THIS MAY BE PROVED AS FOLLOWS LET THE LAURENT EXPANSION OF THE LEFT HAND SIDE OF (1 9) BE ? X M,N=?? CM,N?M?N SUBSTITUTING THE EXPANSIONS OF ?1(?), ?1(?) AND ?1(??) IN THE LEFT HAND SIDE OF (1 9) WE EVALUATE THE COE?CIENTS CM,N DISTINGUISHING FOUR SEPARATE CASES CASE (I): MN(M ?N) ?= 0 THEN USING (1 9) WE OBTAIN CM,N = (1 ?QM)(1 ?QN) ? (1 ?QN)(1 ?QM?N) ? (1 ?QM)(1 ?QN?M) = 0 CASE (II): IF M ?= 0 AND N = 0 THEN CM,0 = 2(1 ?QM) ? (1 ?QM)(1 ?Q?M) = QM (1 ?QM)2 NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS THE BASIC IDENTITY SIMILARLY, IF M = 0 AND N ?= 0, THEN C0,N = QN (1 ?QN)2 CASE (III): M = N ?= 0 THEN CM,M = (1 ?QM)2 ? (1 ?QM) = QM (1 ?QM)2 CASE (IV): M = N = 0 THEN C0,0 = 1/4 ?1/2 = ?1/4 THE RIGHT HAND SIDE OF (1 9) CONSISTS OF THREE TERMS BESIDES THE CONSTANT TERM, WHOSE COE?CIENTS ARE EASILY SEEN TO BE IDENTICAL WITH THE CORRESPOND- ING TERMS OF THE LEFT HAND SIDE THAT ARISE FROM CASES (II) AND (III) THE SIGNI?CANT THING WE OBSERVE IS THAT THE BULK OF THE TERMS ON THE LEFT HAND SIDE, NAMELY THOSE WITH MN(M ?N) ?= 0, CANCEL OUT THIS COMPLETES THE PROOF OF THE IDENTITY (1 9) WE NOW TRANSFORM THE FUNCTIONS ?1 AND ?2 INTO APPROPRIATE FORMS WHICH EXTEND THE RANGE OF THEIR DE?NITION FOR THE FUNCTION ?1, WE HAVE: ?1(Z) = Z?M 1 ?Q?M (1 10) = Z?MQM 1 ?QM = ZQN 1 ?ZQN ? Z?1QN 1 ?Z?1QN (1 11) THIS SHOWS THAT ?1 IS ANALYTIC IN 0 < |Z| < ?EXCEPT AT Z = QR, R = 0, ±1, ±2, , WHERE THERE ARE SIMPLE POLES THERE IS AN ESSENTIAL SINGULARITY AT Z = 0 FROM (1 11) WE MAY DEDUCE THE SYMMETRIC PROPERTY ?1(Z) + ?1(Z?1) = 0, AND THE QUASI-PERIODIC PROPERTY ?1(Z) ??1(QZ) = ?1 ZQN 1 ?ZQN ? Z?1QN 1 ?Z?1QN = 1(?) = 2?2(?) + ?2(1) (1 14) NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS THE BASIC IDENTITY LET ?1 AND ?2 BE THE FUNCTIONS DE?NED BY ?1(?) = 2I?1(EI?) AND ?2(?) = 2?2(EI?) (1 15) FROM (1 10) AND (1 12), WE IMMEDIATELY DEDUCE THE FOURIER EXPANSIONS ?1(?) = 1(A) + 2?? 1(A) = ?2(A) + 2?2(0) (1 22) THIS CAN BE SEEN TO BE EQUIVALENT TO RAMANUJAN’S IDENTITY (1 5), ON USING THE FOURIER EXPANSIONS FOR ?1 AND ?2 IN (1 16) AND (1 17) THE NEW FUNCTIONS ?1 AND ?2 AND THE GENERALIZED RAMANUJAN IDEN- TITY (1 21) FORM THE FOUNDATION FOR THE DEVELOPMENT OF ELLIPTIC FUNCTIONS ITS FORM INDICATES THAT (1 21) IS OF RELEVANCE IN OBTAINING SOME QUADRATIC TRANSFORMATION FORMULAS; IN FACT RAMANUJAN [91, (20)] OBTAINED THE SUM OF FOUR SQUARES FORMULA FROM THE SUM OF TWO SQUARES FORMULA USING (1 5) AND DEDUCED SEVERAL OTHER INTERESTING RESULTS 2?2(0) (1 23) NOW SUPPOSE A + B + C = 0 WE MULTIPLY (1 21) BY 2 AND ADD TO (1 23) TO GET (?1(A) + ?1(B) + ?1(C))2 + 2(?? 1(A) + ?? 1(B) + ?? 1(C)) = 2?2(0) (1 24) WE APPLY THE DI?ERENTIAL OPERATOR ?/?A ??/?B TO OBTAIN 2(?1(A) + ?1(B) + ?1(C))(?? 1(A) ??? 1(B)) + 2(??? 1(A) ???? 1(B)) = 0, (1 25) OR ?1(A) + ?1(B) + ?1(C) = ???? 1(A) ???? 1(B) 4(?? 1(A) ??? 1(B)) (1 26) COMBINING EQUATIONS (1 24) AND (1 26) GIVES ??? 1(A) ???? 1(B) ?? 1(A) ??? 1(B) 2(?? 1(A) + ?? 1(B) + ?? 1(C)) (1 27) WE DE?NE1 THE WEIERSTRASS ELLIPTIC FUNCTION ?BY ?(A) = 2(?2(0) ??? 1(A)) (1 28) THEN (1 27) BECOMES ?(A + B) = ??(A) ???(B) ?(A) ??(B) ??(A) ??(B) (1 29) THIS IS THE ADDITION THEOREM FOR THE WEIERSTRASS ELLIPTIC FUNCTION THE ABELIAN FORM OF THE ADDITION THEOREM CAN BE DERIVED SIMILARLY APPLY ?2/?A?B TO (1 25) TO OBTAIN ?? 1(A) ?? 1(B) ?? 1(C) ??? 1(A) ??? 1(B) ??? 1(C) = 0, WHERE A + B + C = 0 1THE STANDARD DE?NITION OF THE WEIERSTRASS ELLIPTIC FUNCTION WILL BE GIVEN IN THE NOTES AT THE END OF CHAPTER 1, AND ITS EQUIVALENCE WITH THE DE?NITION IN (1 28) WILL BE ESTABLISHED THERE NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS THE BASIC IDENTITY THIS, EXPRESSED IN TERMS OF THE WEIERSTRASS FUNCTION, ASSUMES THE FORM BN N! ?N (1 30) THE ?RST FEW EVEN BERNOULLI NUMBERS ARE GIVEN BY B0 = 1, B2 = 30, B6 = 30, B10 = 2, B3 = B5 = B7 = = 0 THE SERIES EXPANSION OF THE COTANGENT FUNCTION IS GIVEN BY (?1)NB2N (2N)! ?2N?1 THE EXPANSION OF ?1(?) IN POWERS OF ? MAY NOW BE OBTAINED BY EXPANDING THE FOURIER EXPANSION (1 16) IN POWERS OF ? WE ?ND THAT ?1(?) = QJ 1 ?QJ SIN J? (1 32) = J2N?1QJ 1 ?QJ ? ? ??2N?1 THE INITIAL TERMS IN THE EXPANSION ARE GIVEN BY ?1(?) = JQJ 1 ?QJ ? ? ?? (1 33) ? ? ? ? 5! ? = 2? ? P ? ? Q 3! ? R 5! ? , JQJ 1 ?QJ , Q = 1 + 2! + (1 37) THE ?RST FEW TERMS IN THE EXPANSION OF THE WEIERSTRASS FUNCTION MAY BE DETERMINED USING (1 28), (1 33) AND (1 36), AND THEY ARE GIVEN BY: ?(?) = 2 (?2(0) ??? 1(?)) (1 38) = J2N+1QJ 1 ?QJ ? ? ??2N = ?2 + Q 240?2 + R 12 × 504?4 + NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS THE BASIC IDENTITY NOW WE WILL OBTAIN THE DI?ERENTIAL EQUATION SATIS?ED BY THE WEIERSTRASS ELLIPTIC FUNCTION WRITE THE GENERALIZED RAMANUJAN IDENTITY (1 21) IN THE FORM (?1(A) + ?1(B))?1(A + B) ??1(A)?1(B) = 2(?2(A) + ?2(B) + ?2(A + B)) EXPAND BOTH SIDES IN POWERS OF B AND COMPARE THE COE?CIENTS OF B2 TO OBTAIN 2?1(A)??? 1(A) ?P 6 ?? 1(A) + 3???? 1 (A) = ??? 2(A) ??1,2 NEXT, DI?ERENTIATE (1 22) TWICE TO GET 2?1(A)??? 1(A) + 2(?? 1(A))2 + 2???? 1 (A) = ??? 2(A) ELIMINATING ??? 2(A) FROM THE LAST TWO EQUATIONS GIVES 4(?? 1(A))2 + P 3 ?? 1(A) + 3???? 1 (A) = 2?1,2, WHICH WE WRITE AS 2?? 1(A) + P 3???? 1 (A) = 2?1,2 + P 6???(A) = 2?1,2 + P 144 (1 39) WE USE (1 38) TO EXPAND BOTH SIDES OF (1 39), AND THEN EQUATE THE CONSTANT TERMS, TO GET Q 144 = 2?1,2 + P 6???(A) = Q 144 (1 40) THIS IS EQUIVALENT TO (1 6) MULTIPLY BOTH SIDES BY 12??(A) AND INTEGRATE, TO OBTAIN (??(A))2 = 4?3(A) ?Q 12?(A) + K, (1 41) FOR SOME CONSTANT K ONCE AGAIN WE USE (1 38) TO EXPAND (1 41) IN POWERS OF A, AND EQUATE CONSTANT TERMS THIS GIVES K = ?R 216 THEREFORE (??(A))2 = 4?3(A) ?Q 12?(A) ?R 216 (1 42) THIS IS THE DI?ERENTIAL EQUATION FOR THE WEIERSTRASS ELLIPTIC FUNCTION PUTTING G2 = Q/12 AND G3 = R/216 WE OBTAIN THE DI?ERENTIAL EQUATION IN ITS STANDARD FORM: (??(A))2 = 4?3(A) ?G2?(A) ?G3 (1 43) NOTES IN SEVERAL INSTANCES, REFERENCE HAS BEEN MADE TO RESULTS IN RAMANUJAN’S PAPER [91] THE READER IS REFERRED TO THE COMMENTARY BY B C BERNDT [94, PP 365–368] FOR A SUMMARY OF THE IN?UENCE OF RAMANUJAN’S PAPER AND THE ACTIVITY IT HAS GENERATED RAMANUJAN’S IDENTITIES (1 3) AND (1 4) ARE ON P 266 OF HIS ?RST NOTEBOOK [92], IN CHAPTER 17, ENTRIES 8 (IX) AND (X) IN HIS SECOND NOTEBOOK [92], AND IN HIS PAPER [91, EQ (65)] RAMANUJAN’S IDENTITY (1 5) CAN BE EXPRESSED IN TERMS OF THETA FUNCTIONS, AS FOLLOWS IT IS EASY TO CHECK THAT THE THETA FUNCTION U(?, T) = ?I ? X J=?? (?1)JQ(2J+1)2/8EI(2J+1)?/2, WHERE Q = E?T, (1 44) IS A SOLUTION OF THE HEAT EQUATION 2?U ?T = ?2U ??2 THEREFORE, LOG U SATIS?ES THE NON-LINEAR PARTIAL DI?ERENTIAL EQUATION ?LOG U ?? (1 45) IN THE NEXT CHAPTER WE WILL PROVE JACOBI’S TRIPLE PRODUCT IDENTITY (2 36) THIS IDENTITY, WHICH WE ASSUME FOR NOW, IMPLIES THAT THE SERIES DE?NED BY (1 44) HAS THE FACTORIZATION GIVEN BY U(?, T) = 2Q1/8 SIN ? (1 ?EI?QJ)(1 ?E?I?QJ)(1 ?QJ) (1 46) IN 1951, B VAN DER POL [101] OBSERVED THAT IF (1 46) IS SUBSTITUTED INTO (1 45), THE RESULT IS RAMANUJAN’S IDENTITY (1 5) RAMANUJAN’S OWN PROOF OF (1 5) IN [91] HAS BEEN DESCRIBED BY G H HARDY [62, P 133] AS “A VERY CHARACTERISTIC SPECIMEN OF RAMANUJAN’S WORK” HARDY LIKED RAMANUJAN’S PROOF SO MUCH THAT HE REPRODUCED IT IN HIS LECTURES [62, PP 133–135] AS WELL AS IN HIS BOOK ON NUMBER THEORY WITH E M WRIGHT [63, PP 312–314] PROOFS OF RAMANUJAN’S IDENTITY (1 5) AND AN EXTENSION, BY CONTOUR IN- TEGRATION, HAVE BEEN GIVEN BY V RAMAMANI [87, PP 82–92] THE DE?NITION OF THE WEIERSTRASS ELLIPTIC FUNCTION GIVEN BY (1 28) IS NOT THE USUAL ONE THE STANDARD DE?NITION OF THE WEIERSTRASS ELLIPTIC FUNCTION WITH PERIODS 2? AND 2??, WHERE IM(?) > 0, IS GIVEN BY ?(?) = ?2 + X M,N ? (? ?2?N ?2??M)2 ? , (1 47) NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS THE BASIC IDENTITY WHERE THE SYMBOL X M,N ? INDICATES THAT THE SUMMATION IS OVER ALL INTEGERS M AND N WITH THE TERM CORRESPONDING TO (M, N) = (0, 0) OMITTED THE BASIC PROPERTIES OF THE SERIES DE?NED BY (1 47) ARE THE FOLLOWING: THE SERIES CONVERGES ABSOLUTELY AND UNIFORMLY ON COMPACT SETS THAT DO NOT CONTAIN ANY OF THE POINTS 2?N + 2?M?, M, N ?Z; THE FUNCTION ?(?) IS ANALYTIC EXCEPT AT THE POINTS ? = 2?N + 2?M?, M, N ?Z, WHERE THERE ARE POLES OF ORDER 2; AND ?(?) IS A DOUBLY PERIODIC FUNCTION OF ? WITH PERIODS 2? AND 2?? FOR PROOFS OF THESE STANDARD FACTS, SEE, FOR EXAMPLE, THE BOOK BY WHITTAKER AND WATSON [111, PP 434–435] WE WILL NOW SHOW THAT THE EQUATIONS (1 28) AND (1 47) ARE EQUIVALENT RECALL THE STANDARD RESULTS [3, PP 11, 12] OR [36, PP 218–222]: ? X N=?? (? ?2?N)2 = 4 SIN2 ? 6 THEN, CONSIDERING THE TERMS M = 0 AND M ?= 0 SEPARATELY, WE HAVE FROM (1 47) THAT ?(?) = (? ?2?N)2 ? ? X N=?? = ? X N=?? 4 SIN2( ? 2 ???M) ? SIN2 ??M + ? X M=?? SIN2( ? 2 + ??M) 12 + ? ? X M=?? = 2 (?2(0) ??? 1(?)) THUS, (1 28) FOLLOWS FROM (1 47) THE STEPS ARE REVERSIBLE, AND SO (1 28) AND (1 47) MAY BE REGARDED AS BEING EQUIVALENT NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS CHAPTER RAMANUJAN’S DI?ERENTIAL EQUATIONS RAMANUJAN PROVED THAT THE FUNCTIONS P, Q AND R IN (1 34) SATISFY THE SYSTEM OF DI?ERENTIAL EQUATIONS Q DP DQ = P 2 ?Q 2(R + 1) + ?0,R, (2 2) WHERE M, N AND R ARE NON-NEGATIVE INTEGERS, AND BN DENOTES THE NTH BERNOULLI NUMBER RAMANUJAN HAS PROVED THE FOLLOWING: THEOREM 2 1 (I) IF R ?1, THEN S2R+1 IS A POLYNOMIAL IN Q AND R OVER THE ?ELD OF RATIONAL NUMBERS THIS IS A CLASSICAL RESULT, E G , SEE THE LECTURES OF K WEIERSTRASS WRITTEN UP AND PUBLISHED BY H A SCHWARZ [96, PP 10, 11] (II) IF M + N IS ODD, THEN ?M,N IS A POLYNOMIAL IN P, Q AND R OVER THE ?ELD OF RATIONAL NUMBERS THIS IS A NEW RESULT, DISCOVERED BY RA- MANUJAN DEVELOPMENT OF ELLIPTIC FUNCTIONS ACCORDING TO RAMANUJAN WE WILL NOW GIVE RAMANUJAN’S PROOF OF THEOREM 2 1, AS WELL AS HIS PROOF OF THE DI?ERENTIAL EQUATIONS FOR P, Q AND R WITH THE NOTATION DE?NED ABOVE, THE SERIES EXPANSIONS FOR ?1, ?2 AND ?GIVEN BY (1 32), (1 35) AND (1 38) CAN BE WRITTEN AS ?1(?) = (?1)N (2N)! ?1,2N?2N, (2 4) AND ?(?) = ?2 + 24 + ?0,1 = ?P 24, S3 = 3 (2 8) BY INDUCTION USING (2 7), WE CAN EXPRESS S9, S11, S13, , AS POLYNOMIALS IN S3 AND S5 (EQUIVALENTLY, AS POLYNOMIALS IN Q AND R), WITH RATIONAL COE?CIENTS IN GENERAL WE HAVE S2N?1 = ?B2N 4N + ?0,2N?1 = X KI,JQIRJ (2 9) NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS THE DI?ERENTIAL EQUATIONS OF P , Q AND R 1(?) + 2 EXPANDING BOTH SIDES IN POWERS OF ? USING (2 3) AND (2 4) AND COMPARING COE?CIENTS OF ?2N, WE OBTAIN ?1,2N = 2N + S2J?1S2N+1?2J, N = 1, 2, (2 10) IT FOLLOWS THAT ?1,2N IS A POLYNOMIAL IN S1, S3, , S2N+1, WITH RATIONAL COE?CIENTS SINCE S7, S9, , ARE POLYNOMIALS IN S3 AND S5, IT FOLLOWS THAT ?1,N IS A POLYNOMIAL IN P, Q AND R, WITH RATIONAL COE?CIENTS TAKING N = 1, 2 AND 3 IN (2 10), WE OBTAIN ?1,2 = 1, ?1,4 = 10S5 ?8S1S3, ?1,6 = 3 ?12S1S5, WHERE (2 8) HAS BEEN USED IN THE LAST STEP BY (2 6), THESE ARE EQUIVALENT TO 288?1,2 = Q ?P 2, 720?1,4 = PQ ?R, 1008?1,6 = Q2 ?PR (2 11) BY INDUCTION IT MAY BE SHOWN THAT ?1,2N = X KI,J,?P IQJR?, WHERE KI,J,?IS A RATIONAL NUMBER, I, J AND ?ARE NON-NEGATIVE INTEGERS THAT SATISFY I + 2J + 3?= N + 1, AND N ?1 FURTHERMORE, IF N ?2, THEN I ?1 WE NOW OBSERVE FROM THE DE?NITION (2 1) THAT Q D DQ ?M,N = ?M+1,N+1 (2 12) THEREFORE, THE IDENTITIES IN (2 11) MAY BE REWRITTEN IN THE FORM 288 Q D DQ?0,1 = Q?P 2, 720 Q D DQ?0,3 = PQ?R, 1008 Q D DQ?0,5 = Q2?PR, DEVELOPMENT OF ELLIPTIC FUNCTIONS ACCORDING TO RAMANUJAN AND SO, ON APPLYING (2 6) WE DEDUCE THAT Q DP DQ = P 2 ?Q Y 1?I<J?N (RI ?RJ)2 THE DISCRIMINANT OF THE CUBIC EQUATION T3 ?PT ?Q = 0 IS KNOWN TO BE 4P3 ?27Q2; SEE [52, P 153] THEREFORE, THE DISCRIMINANT OF THE WEIER- STRASSIAN CUBIC (SEE (1 42)) 4T3 ?Q 12T ?R 216 = IS A CONSTANT MULTIPLE OF Q3 ?R2 AN IMMEDIATE CONSEQUENCE OF RAMANU- JAN’S DI?ERENTIAL EQUATIONS IS Q D DQ LOG(Q3 ?R2) = 3Q2 Q DQ DQ ?2R Q DR DQ (2 14) = NQN 1 ?QN INTEGRATING, WE OBTAIN LOG(Q3 ?R2) = C + LOG Q + (1 ?QN)24 (2 15) NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS THE DI?ERENTIAL EQUATIONS OF P , Q AND R J11QJK = C1Q3 + C2R2, (2 18) FOR SOME CONSTANTS C1 AND C2 THE VALUE OF B12 WAS GIVEN IN (1 31), AND THE VALUES OF C1 AND C2 CAN BE DETERMINED BY EQUATING THE CONSTANT TERM AND THE COE?CIENT OF Q ON BOTH SIDES OF (2 18) THE RESULTING FORMULA MAY BE EXPRESSED IN THE FORM 691 + ?11(N)QN = 441Q3 + 250R2 (2 19) NEXT, BY (2 15) AND (2 16), WE HAVE ?(N)QN = Q3 ?R2 (2 20) IF WE MULTIPLY (2 19) BY 12, MULTIPLY (2 20) BY 455, THEN SUBTRACT ONE FROM THE OTHER, WE OBTAIN 8292 + 26 × 33 × 5 × 7 × (?11(N) ??(N)) QN = 691(7Q3 + 5R2) (1 + ZQ2N?1)(1 + Z?1Q2N?1) (2 22) SINCE THE PRODUCT CONVERGES ABSOLUTELY AND UNIFORMLY ON COMPACT SUBSETS OF THE REGION 0 < |Z| < ?, IT HAS A LAURENT EXPANSION OF THE FORM F(Z) = ? X N=?? CNZN IN THE ANNULUS 0 < |Z| < ? ONE CAN EASILY SHOW THAT F(Z) = QZF(ZQ2) NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS THE DI?ERENTIAL EQUATIONS OF P , Q AND R (2 25) THAT SEPARATES THE TWO IN?NITE FAMILIES OF POLES BY LAURENT’S THEOREM, ? HAS AN EXPANSION OF THE FORM ?(Z, ?, ?) = ? X N=?? CN(?, ?)ZN (2 26) DEVELOPMENT OF ELLIPTIC FUNCTIONS ACCORDING TO RAMANUJAN IN THE ANNULUS (2 25) ONE EASILY NOTICES, USING (2 24), THAT ?(Q2Z, ?, ?) ?(Z, ?, ?) = (1 + ?QZ)(1 + Q?1Z?1) (1 + ?Q?1Z?1)(1 + QZ), OR (1 + ?QZ)?(Z, ?, ?) = (? + QZ)?(Q2Z, ?, ?) (2 27) IF WE PUT ?(Z) = (1 + ?QZ)?(Z, ?, ?), WE SEE THAT ?(Z) IS ANALYTIC IN THE LARGER ANNULUS |?Q| < |Z| < |?Q3|?1, (2 28) BECAUSE THE POLE OF ?(Z, ?, ?) AT Z = ?(?Q)?1 HAS BEEN ELIMINATED NOW ?(Z) = (1 + ?QZ) ? X N=?? CN(?, ?)ZN = ? X N=?? (CN(?, ?) + ?QCN?1(?, ?))ZN, AND BY (2 27), ?(Z) = (? + QZ) ? X N=?? CN(?, ?)Q2NZN = ? X N=?? (?CN(?, ?)Q2N + CN?1(?, ?)Q2N?1)ZN BOTH OF THESE EXPANSIONS FOR ? ARE VALID IN THE LARGER ANNULUS GIVEN BY (2 28) COMPARING COE?CIENTS OF ZN GIVES CN(?, ?) + ?QCN?1(?, ?) = ?CN(?, ?)Q2N + CN?1(?, ?)Q2N?1, HENCE, WE OBTAIN THE RECURRENCE RELATION CN(?, ?) = Q2N?2 ?? 1 ??Q2N QCN?1(?, ?) ITERATING, WE OBTAIN CN(?, ?) = (1 ??)(Q2 ??) (Q2N?2 ??) (1 ??Q2)(1 ??Q4) (1 ??Q2N) QNC0(?, ?), (2 29) FOR ANY POSITIVE INTEGER N FROM THE DE?NITION (2 24), IT IS CLEAR THAT ?(Z, ?, ?) = ?(Z?1, ?, ?) AND COMPARING COE?CIENTS OF Z?N ON EACH SIDE GIVES C?N(?, ?) = CN(?, ?) (2 30) NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS THE DI?ERENTIAL EQUATIONS OF P , Q AND R (1 ???1Q2N?2) (1 ??Q2N) (1+AN?1BN)(1+ANBN?1)(1?ANBN) (2 37) AND POINTS OUT THAT IT IS A SPECIAL CASE OF THE 1?1 SUMMATION FORMULA (2 35) THE FORMULAS (2 36) AND (2 37) MAY BE SEEN TO BE EQUIVALENT BY SETTING Q = (AB)1/2 AND Z = (A/B)1/2 IN (2 36) NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS THE DI?ERENTIAL EQUATIONS OF P , Q AND R (1 ?(?Q)N)2 = ? X N=?? (3N + 1)Q(3N+1)2/3 DEVELOPMENT OF ELLIPTIC FUNCTIONS ACCORDING TO RAMANUJAN PROOFS OF THESE IDENTITIES WILL BE GIVEN IN APPENDIX B NEXT, LET Z = ?1 IN (2 38) TO GET (1 + QN)2(1 ?QN) = ? X N=?? QN(N?1)/2 = 1 ??Q2 C0(?, ?) NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS THE DI?ERENTIAL EQUATIONS OF P , Q AND R JMKNQJK UN ?P NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS THE DI?ERENTIAL EQUATIONS OF P , Q AND R ?J,K,?P JQKR?, THE IMAGINARY TRANSFORMATION AND DEDEKIND’S ETA- FUNCTION IN THIS SECTION, SUPPOSE IM(?) > 0 AND LET Q AND P BE DE?NED BY Q = EXP(2?I?) AND P = EXP(?2?I/?) THE FUNCTIONS ?1(?) AND ?(?) DE?NED IN CHAPTER 1 ALSO DEPEND ON Q, AND HENCE ON ?, AND THEY WILL NOW BE DENOTED BY ?1(?, ?) AND ?(?, ?), RESPEC- TIVELY FROM (1 28) AND (1 36), WE HAVE ?(?, ?) = ?2?? 1(?, ?) ?P(Q) 12 , WHERE THE PRIME DENOTES DI?ERENTIATION WITH RESPECT TO ? BY THE PERIODICITY AND QUASI-PERIODICITY PROPERTIES OF ?1 GIVEN IN (1 19) AND (1 20), IT FOLLOWS THAT ?(? + 2?, ?) = ?(?, ?) = ?(? + 2??, ?), THAT IS, ?(?, ?) IS DOUBLY PERIODIC WITH PERIODS 2? AND 2?? MOREOVER, BY CONSIDERING THE POLES OF ?1 (SEE THE LINE AFTER (1 20)), IT FOLLOWS THAT ?(?, ?) HAS POLES OF ORDER 2 AT EACH POINT ? = 2?M + 2?N? FOR ALL INTEGERS M AND N, AND NO OTHER SINGULARITIES THE FUNCTION ?(?/?, ?1/?) HAS THE SAME PERIODS AND POLES AS ?(?, ?), AND IT IS NATURAL TO EXPECT THAT THESE FUNCTIONS MUST NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS THE DI?ERENTIAL EQUATIONS OF P , Q AND R ? (2 55) = ? 4 Q(P) ?2 + 12 × ? 6 R(P) ?4 + IS ANALYTIC AT ? = 0, BECAUSE THE POLE OF ?(?, ?) AT ? = 0 HAS BEEN ELIMINATED BY DOUBLE PERIODICITY, THE LEFT HAND SIDE OF (2 55) IS ANALYTIC FOR ALL COMPLEX VALUES OF ?, AND BY LIOUVILLE’S THEOREM IT MUST BE CONSTANT BY LETTING ? = 0 THE CONSTANT IS SEEN TO BE 0 THEREFORE, FROM (2 55) WE HAVE THE TRANSFORMATION FORMULA ?(?, ?) = ? AS WELL AS THE TRANSFORMATION FORMULAS FOR THE COE?CIENTS: Q(Q) = ? 4 Q(P), R(Q) = ? IS ENTIRE BY THE PERIODIC AND QUASI-PERIODIC PROPERTIES (1 19) AND (1 20) WE SEE THAT G(? + 2?) = G(?) + 2I? AND G(? + 2??) = G(?) + ? ? ? 4?I? DEVELOPMENT OF ELLIPTIC FUNCTIONS ACCORDING TO RAMANUJAN IS ENTIRE AND DOUBLY PERIODIC BY LIOUVILLE’S THEOREM IT IS A CONSTANT FUNC- TION, AND SINCE ?1 IS AN ODD FUNCTION, THE CONSTANT IS ZERO THUS ?1(?, ?) = ? + ? 4?I? (2 56) EXPANDING IN POWERS OF ? USING (1 33) AND COMPARING COE?CIENTS, WE AGAIN OBTAIN THE RECIPROCITY FORMULAS Q(P) = ? 4Q(Q) AND R(P) = ? 6R(Q) (2 57) AS WELL AS THE MORE SIGNI?CANT RESULT P(P) = ? 2P(Q) + 6? ?I , (2 58) WHERE Q = E2?I? AND P = E?2?I/? TAKING ? = I IN (2 58) WE GET P(E?2?) = ? , WHICH, WHEN WRITTEN EXPLICITLY, GIVES A FORMULA POSED AS A PROBLEM BY RA- MANUJAN E2? ?1 + E4? ?1 + E6? ?1 + = 8? (2 59) WE TAKE THIS OCCASION TO PROVE THE FUNDAMENTAL RESULT (2 56) BY A SIMPLE METHOD WHICH DOES NOT MAKE ANY USE OF COMPLEX FUNCTION THEORY IT MAKES USE OF A NATURAL DUPLICATION FORMULA BY (1 11) AND (1 15) WE HAVE ?1(A, ?) = 4I 1 + ? 1 ?? + ?QN 1 ??QN ? ??1QN 1 ???1QN ! (2 60) WHERE ? = EIA, Q = E2?I? REPLACE A BY A + ? (I E , ? BY ??) AND ADD THE RESULT IS ?1(A, ?) + ?1(A + ?, ?) = 1 ??2 + ?2Q2N 1 ??2Q2N ? ??2Q2N 1 ???2Q2N ! IN THIS, REPLACE A BY A + ?? (I E , ?2 BY Q?2) AND ADD THE RESULT SIMPLI?ES TO ?1(A, ?)+?1(A+?, ?)+?1(A+??, ?)+?1(A+? +??, ?) = ?I 2 +2?1(2A, ?) (2 61) BY THE QUASI-PERIODIC PROPERTY (1 20) WE ALSO HAVE ?1(A, ?) + ?1(A + ?, ?) + ?1(A ???, ?) + ?1(A + ? ???, ?) = I 2 + 2?1(2A, ?) (2 62) NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS THE DI?ERENTIAL EQUATIONS OF P , Q AND R 2 + ? + ?? ???(0) ?4M HENCE THE MAXIMUM OF |??(A)???(0)| ALSO OCCURS AT A0/2, AND BY THE SAME ARGUMENT AT A0/2N, N = 1, 2, 3, BY CONTINUITY OF ??, THE MAXIMUM OCCURS AT A = 0 BUT |??(0) ???(0)| = 0, AND SO ??(A) ?0 THUS ? IS CONSTANT, AND SINCE ? IS AN ODD FUNCTION, ?(A) ?0 THIS AGAIN PROVES (2 56), THIS TIME WITHOUT USING COMPLEX FUNCTION THEORY WE USE (2 60) TO REWRITE THE TRANSFORMATION FORMULA (2 56) IN THE EXPLICIT FORM 4 COT A 2 + EIAQN 1 ?EIAQN ? E?IAQN 1 ?E?IAQN = 4? COT A 2? + EIA/?PN 1 ?EIA/?PN ? E?IA/?PN 1 ?E?IA/?PN + A 4?I? DEVELOPMENT OF ELLIPTIC FUNCTIONS ACCORDING TO RAMANUJAN INTEGRATING BOTH SIDES WITH REPECT TO A, WE OBTAIN 2 LOG SIN A 2 + LOG(1 ?EIAQN) + LOG(1 ?E?IAQN) = 2 LOG SIN A 2? + 4?I? SIN A = ? I EXP ??I ? + (1 ?PN)2 (2 69) NUMBER THEORY, TRACE FORMULAS AND DISCRETE GROUPS THE DI?ERENTIAL EQUATIONS OF P , Q AND R = I EXP ??I (1 + QN) = (1 ?PN?1/2), (2 71) (1 + PN), (2 72) (1 + QN?1/2) = (1 + ?ZQ2N?1)(1 + ??1Z?1Q2N?1)(1 ??Q2N?2)(1 ???1Q2N) = (?1)N 1 ??Q2N Q ?Z N = ? X N=?? (1 ?TQN?1)(1 ?TQN)(1 ??QN?1)(1 ???1QN) = ? X N=?? TN 1 ??QN , (2 75) DEVELOPMENT OF ELLIPTIC FUNCTIONS ACCORDING TO RAMANUJAN PROVIDED |Q| < |T| < 1 THE PRODUCT IN (2 75) IS SYMMETRIC IN ? AND T, BUT THE SERIES IS NOT SYMMETRY CAN BE RESTORED TO THE SERIES AS FOLLOWS: ? X N=?? TN 1 ??QN = T?N 1 ??Q?N = ??1T?NQN 1 ???1QN = 1 ?? + (1 ?TQN?1)(1 ?TQN)(1 ??QN?1)(1 ???1QN) (2 76) =