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Number Theory and Related Topics Number Theory And Related Topics NUMBER THEORY AND RELATED TOPICS Papers presented at the Ramanujan Colloquium, Bombay 1988, by ASKEY BALASUBRAMANIAN BERNDT BRESSOUD HEATH-BROWN IWANIEC KUZNETSOV RAGHAVAN RAMACHANDRA RAMANATHAN RANGACHARI RANKIN SATAKE SCHMIDT SELBERG SHOREY ZAGIER Published for the TATA INSTITUTE OF FUNDAMENTAL RESEARCH, BOMBAY OXFORD UNIVERSITY PRESS 1989 Oxford University Press, Walton Street, Oxford OXZ 6DP NEW YORK TORONTO DELHI BOMBAY CALCUTTA MADRAS KARACHI PETALING JAYA SINGAPORE HONGKONG TOKYO NAIROBI DAR ES SALAAM MELBOURNE AUCKLAND and associates in BERLIN IBADAN c Tata Institute of Fundamental Research, 1989 ISBN 0 19 562367 3 Typeset and Printed in India by B. A. Gala, Anamika Trading Co., Dadar, Bombay 400 028 and published by S. K. Mookerjee, Oxford University Press, Oxford House, Apollo Bunder, Bombay 400 039. Ramanujan Birth Centenary International Colloquium on Number Theory and Related Topics Bombay, 4-11 January 1988 REPORT An International Colloquium on Number Theory and related topics 1 was held at the Tata Institute of Fundamental Research, Bombay during 4-11 January, 1988, to mark the birth centenary of Srinivasa Ramanujan. The purpose of the Colloquium was to highlight recent developments in Number Theory and related topics, especially those related to the work of Ramanujan “such as the Circle method, Sieve methods and Combi- natorial techniques in Number theory, Partition congruences, Rogers - Ramanujan identities, Lacunarity of power series, Hypergeometric se- ries and Special functions, Complex multiplication, Hecke theory etc.” The Colloquium was organized by the Tata Institute of Fundamen- tal Research with co-sponsorship from the International Mathematical Union. Financial support was received from the International Mathe- matical Union and the Sir Dorabji Tata Trust, as in former years. The organizing committee of the Colloquium consisted of Professors M.S. Narasimhan, S. Raghavan, M.S. Raghunathan, K. Ramachandra and C.S. Seshadri and Dr. S.S. Rangachari. The International Mathematical Union was represented on the committee by Professors M.S. Narasimhan and C.S. Seshadri. The following mathematicians delivered one-hour addresses at the Colloquium: REPORT G.E. Andrews, R. Askey, B. C. Berndt, D. M. Bressoud, D. R. Heath-Brown, N. V. Kuznetsov, K. Ramachandra, K. G. Ramanathan, S. S. Rangachari, R. A. Rankin, I. Satake, W. M. Schmidt, A. Selberg, J. P. Serre, T. N. Shorey and D. Zagier. Professor H. Iwaniec could not attend the Colloquim but sent a paper for inclusion in the Proceedings. Besides members of the School of Mathematics of the Tata Institute of Fundamental Research, mathematicians from universities and educa- tional institutions in India, France, Canada, Japan and the United States of America were also invited to attend the Colloquium. The social programme for the Colloquium included a tea-party on 4 January, a classical Indian dance performance (Bharatanatyam) on 6 January, a film show and a dinner-party at the Institute on 7 January, a violin recital (Hindustani music) on 8 January, an excursion to the Elephanta Caves on 9 January and a farewell dinner-party on 10 January 1988. Contents 1. R.Askey : Variants of Clausen’s formula for the 1–14 square of a special 2F1 2. R.Balasubramanian : andK.Ramachandra : 15–26 Titchmarsh’s phenomenon for ζ(s) 3. B.C.Berndt : Ramanujan’s formulas for 27–34 Eisenstein series 4. D.M.Bressoud : On the proof of Andrews’ 35–44 q-Dyson conjecture 5. D.R.Heath-Brown : Weyl’s inequality, Waring’s 45–51 problem and Diophantine approximation 6. H. Iwaniec : The circle method and the Fourier 52–62 coefficients of modular forms 7. N.V.Kuznetsov : Sums of Kloosterman sums and 63–137 the eighth power moment of the Riemann zeta function 8. S.Raghavan : andS.S.Rangachari : On 138-175 Ramanujan’s elliptic integrals and modular identities 9. K.G.Ramanathan : On some theorems stated by 176–188 Ramanujan 10. R.A.Rankin : The adjoint Hecke operator II 189–210 11. I. Satake : On zeta functions associated with 211-231 self-dual homogeneous cones 12. W. M. Schmidt : The number of rational 232–240 approximations to algebraic numbers and the number of solutions of norm form equations 13. A. Selberg : Linear operators and automorphic 241–257 forms 14 T.N.Shorey : Some exponential Diophantine 258–273 equations II 15. D. Zagier : The dilogarithm function in geometry 274–295 and number theory VARIANTS OF CLAUSEN’S FORMULA FOR THE SQUARE OF A SPECIAL2F1 By RICHARD ASKEY* 1 Introduction One of the most striking series Ramanujan [10] found is 1 9801 ∞ (4n)! = [1103 + 26390n] . (1.1) [n!]4 (4.99)4n 2π √2 Xn=0 The first proofs of 1.1 have been given recently by Jonathan and Peter Borwein [3] and by David and Gregory Chudnovsky [5]. They have also found other identities of a similar nature, [4], [5]. As they remark, Clausen’s identity [6] 2 a, b 2a, 2b, a + b = 2F1 1; x 3F2 1 ; x (1.2) a + b + a + b + , 2a + 2b 2 2 plays a central role in the derivation of (1.1). Here n a1,..., ap ∞ (a1)n ... (ap)n x pFp ; x = (1.3) b1,..., bq = (b1)n ... (bq)nn! Xn 0 with (a)n =Γ(n + a)/Γ(a). (1.4) *Supported in part by an NSF grant, in part by a sabbatical leave from the University of Wisconsin, and in part by funds the Graduate School of the University of Wisconsin. 1 2 1 INTRODUCTION Ramanujan [11] stated an extension of Clausen’s formula a, b 1 √1 x a, b 1 √1 x 2F1 ; − − 2F1 ; − − (1.5) c 2 d 2 a, b, (a +b)/2,(c + d)/2 = 4F3 ; x c, d, a + b when c + d = a + b + 1. When c = d and the quadratic transformation a, b 1 √1 x a/2, b/2 2F1 ; − − = 2F1 ; x (a + b + 1)/2 2 (a + b + 1)/2 is used, the result is (1.2). The first published proof of (1.5) is due to Bailey [1]. 2 David and Gregory Chudnovsky have been asking me if there are other results like Clausen’s formula, where the square of a 2F1 is repre- sented as a generalized hypergeometric series. There are other instances, ans one will be given explicitly. The method of deriving it is probably similar to Ramanujan’s method of deriving Clausen’s formula. As a warm up, here is how I think Ramanujan derived (1.2). There are two chapters in Ramanujan’s Second Notebook devoted to hypergeometric series. The first formula in this first of these two chapters is the sum of the 2-balanced very well posited 7F6. This is a fundamental formula, as Ramanujan knew, since he started with it. This sum is a, 1 + (a/2), b, c, d, e, n 7F6 − ; 1 a/2, a + 1 b, a + 1 c, a + 1 d, a + 1 e, a + 1 n − − − − − (1.6) (a + 1)n(a + 1 b c)n(a + 1 b d)n(a + 1 c d)n = − − − − − − (a + 1 b)n(a + 1 c)n(a + 1 d)n(a + 1 b c d)n − − − − − − and e = 2a + 1 + n b c d, (1.7) − − − 3 The phrases very well poised and 2-balanced are defined as follows. A series a0, a1,..., ap p+1Fp ; x (1.8) b1,..., bp is said to be k-balanced if x =1, if one of the numerator parameters is a negative integer, and if p p k + a j = b j. Xj=0 Xj=1 The series 1.8 is said to be well poised if a0 +1 = a1 +b1 = ... = ap +bp. It is very well poised if it is well poised and if a1 = b1 + 1. Observe that the condition (1.7) comes from the series being 2-balanced. Dougall [7] published the first derivation of (1.6). Ramanujan’s dis- covery was probably later, but not much later. To derive Clausen’s formula, first consider 2 n a, b ∞ n (a)k(b)k(a)n k(b)n k 2F1 ; x = x − − (1.9) c = = (c)kk!(c)n k(n k)! Xn 0 Xk 0 − − ∞ (a)n(b)n n, a, b, 1 n c n = 4F3 − − − ; 1 x . (c)nn! 1 n a 1 n b, c Xn 0 − − − − − n The 4F3 series that multiplies x in the expression in (1.9) is well 3 poised. While a well poised 3F2 at x = 1 can be summed, and a very well poised 5F4 can be summed when x = 1, a general well poised 4F3 at x = 1 cannot be summed. However when the series is 2-balanced it can be summed. To see this, first reduce the very well poised 7F6 to a well poised 4F3. This is done by setting d = a/2, c = (a + 1)/2. Then (1.6) becomes a, b, e, n 4F3 − ; 1 (1.10) a + 1 b, a + 1 e, a + 1 + n − − (a + 1)n((a + 1 2b)/2)n((a + 2 2b)/2)n(1/2)n = − − (a + 1 b)n((a + 1)/2)n((a + 2)/2)n((1 2b)/2)n − − 4 2 THEFOURBALANCEDVERYWELLPOISED 7F6 (a + 1)n(a + 1 2b) n(1/2)n = − 2 (a + 1 b)n(a + 1) n((1 2b)/2)n − 2 − (a + 1 2b) n(1/2)n = − 2 (a + 1 b)n(a + n + 1)n((1 2b)/2)n Γ(a +−1 2b + 2n)Γ(n + 1/−2)Γ(a + 1 b)Γ(a + n + 1)Γ(1/2 b) = − − − Γ(a + 1 2b)Γ(1/2)Γ(a + 1 b + n)Γ(a + 2n + 1)Γ((1/2) b + n) − − − This last expression can be used when a = k.
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