Number Theory and Related Topics [Paperback ed.] 0195623673, 9780195623673

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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 Combinatorial techniques in Number theory, Partition congruences, Rogers Ramanujan identities, Lacunarity of power series, Hypergeometric series and Special functions, Complex multiplication, Hecke theory etc.” The Colloquium was organized by the Tata Institute of Fundamental Research with co-sponsorship from the International Mathematical Union. Financial support was received from the International Mathematical 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 educational 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 square of a special 2 F1

2.

R. Balasubramanian : and K. Ramachandra : Titchmarsh’s phenomenon for ζ(s)

15–26

3.

B. C. Berndt : Ramanujan’s formulas for Eisenstein series

27–34

4.

D. M. Bressoud : On the proof of Andrews’ q-Dyson conjecture

35–44

5.

D. R. Heath-Brown : Weyl’s inequality, Waring’s problem and Diophantine approximation

45–51

6.

H. Iwaniec : The circle method and the Fourier coefficients of modular forms

52–62

7.

N. V. Kuznetsov : Sums of Kloosterman sums and the eighth power moment of the Riemann zeta function

8.

S. Raghavan : and S. S. Rangachari : On Ramanujan’s elliptic integrals and modular identities

138-175

9.

K. G. Ramanathan : On some theorems stated by Ramanujan

176–188

R. A. Rankin : The adjoint Hecke operator II

189–210

10.

1–14

63–137

11.

I. Satake : On zeta functions associated with self-dual homogeneous cones

211-231

12.

W. M. Schmidt : The number of rational approximations to algebraic numbers and the number of solutions of norm form equations

232–240

13.

A. Selberg : Linear operators and automorphic forms

241–257

14

T. N. Shorey : Some exponential Diophantine equations II

258–273

15.

D. Zagier : The dilogarithm function in geometry and number theory

274–295

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 ∞

X 9801 (4n)! . [1103 + 26390n] √ = 4 (4.99)4n [n!] 2π 2 n=0

1

(1.1)

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  2a, 2b, a + b  a, b         (1.2) ; x 2 F1  1 1 ; x = 3 F2  a + b + , 2a + 2b  a+b+ 2 2 plays a central role in the derivation of (1.1). Here   ∞  a1 , . . . , a p  X (a1 )n . . . (a p )n xn  ; x = pFp  (b1 )n . . . (bq )n n! b1 , . . . , bq n=0

(1.3)

with

(a)n = Γ(n + a)/Γ(a). * Supported

(1.4)

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   2 F1    ; ; 2 F1  2 2 d c    a, b, (a + b)/2, (c + d)/2   ; x = 4 F3  c, d, a + b

(1.5)

when c + d = a + b + 1. When c = d and the quadratic transformation    2 F1 

2

a, b (a + b + 1)/2

;

1−

√

     a/2, b/2 1 − x    = 2 F1  ; x 2 (a + b + 1)/2

is used, the result is (1.2). The first published proof of (1.5) is due to Bailey [1]. David and Gregory Chudnovsky have been asking me if there are other results like Clausen’s formula, where the square of a 2 F1 is represented 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 7 F6 . 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 ; 1 7 F6   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 , . . . , a p   (1.8) ; x p+1 F p  b1 , . . . , b p

is said to be k-balanced if x = 1, if one of the numerator parameters is a negative integer, and if k+

p X j=0

aj =

p X

b j.

j=1

The series 1.8 is said to be well poised if a0 +1 = a1 +b1 = . . . = a p +b p . 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 discovery was probably later, but not much later. To derive Clausen’s formula, first consider   2 X ∞ n X   a, b  (a)k (b)k (a)n−k (b)n−k 2 F1  ; x = xn (c)k k!(c)n−k (n − k)! c n=0 k=0   ∞ X  − n, a, b, 1 − n − c  (a)n (b)n ; 1 xn . =  4 F3  (c)n n! 1 − n − a 1 − n − b, c n−0

(1.9)

The 4 F3 series that multiplies xn in the expression in (1.9) is well 3 poised. While a well poised 3 F2 at x = 1 can be summed, and a very well poised 5 F4 can be summed when x = 1, a general well poised 4 F3 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 7 F6 to a well poised 4 F3 . This is done by setting d = a/2, c = (a + 1)/2. Then (1.6) becomes     a, b, e, −n  (1.10) ; 1 4 F3  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 THE FOUR BALANCED VERY WELL POISED 7 F6

(a + 1)n (a + 1 − 2b)2n (1/2)n (a + 1 − b)n (a + 1)2n ((1 − 2b)/2)n (a + 1 − 2b)2n (1/2)n = (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. Then    − k, b, e, −n  ; 1  4 F3  1 − b − k, 1 − e − k, 1 + n − k Γ(1 − k − 2b + 2n)Γ(1/2 + n)Γ((1/2) − b)Γ(1 − b − k)Γ(1 + n − k) = . Γ(1 − k − 2b)Γ(1 − k + 2n)Γ(1/2)Γ(1 − k − b + n)Γ((1/2) − b + n) holds for n = k, k + 1, . . ., and is a rational function of n, so it holds when n is replaced by continuous parameter −a. The result is    (2a)k (2b)k (a + b)k  − k, a, b, e  ; 1 = 4 F3  (a)k (b)k (2a + 2b)k 1 − a − k, 1 − b − k, 1 − e − k

4

(1.11)

after simplification. Recall that this series is 2-balanced, so e = −a − b − k + (1/2). One can take a = −k in (1.6) and then remove the restriction that one of the other parameters is a negative integer. However setting c = (1 − k)/2, d = −k/2 to obtain the 4 F3 leads to an indeterminate form, so it is better to reduce to a 4 F3 initially before letting a → −k. Both (1.10) and (1.11) are 2-balanced well poised series, but they are different in that different parameters are used to terminate the series. When (1.11) is used in (1.9), he result is Clausen’s formula (1.2).

2 The four balanced very well poised 7 F6 To find another formula like Clausen’s identity, we can look for another well poised series that can be summed. The obvious candidate is the 4-balanced very well poised 7 F6 . There are two natural ways to sum

5

this series. One is an easy consequence of (1.6), so it is a derivation Ramanujan could have easily given. We start with it. Set fk (b) =

(b)k (e)k (a + 1 − b)k (a + 1 − e)k

(2.1)

and use the 2-balanced condition e = 2a + 1 + n − b − c − d.

(2.2)

A routine calculation gives b(a − b) fk (b + 1) − (e − 1)(a + 1 − e) fk (b) (b)k (e − 1)k [b(a − b) − (e − 1)(a + 1 − e)]. = (a + 1 − b)k (a + 2 − e)k Observe that the last factor is b(a − b) − (2a + n − b − c − d)(b + c + d − n − a) 3a a 3a a a a a a = (n + − b − c − d + )(n + − b − c − d − ) − (b − − )(b − + ) 2 2 2 2 2 2 2 2 3a a 2 2 = (n + − b − c − d) − (b − ) = (n + 2a − 2b − c − d)(n + a − c − d); 2 2

so (n + 2a − 2b − c − d)(n + a − c − d)×   a   a, + 1, b, c, d, e − 1, −n   2 ; 1 × 7 F6  a  , a + 1 − b, a + 1 − c, a + 1 − d, a + 2 − e, a + 1 + n  2  a   a, + 1, b + 1, c, d, e − 1, −n   2  = b(a − b)7 F6  a ; 1  , a − b, a + 1 − c, a + 1 − d, a + 2 − e, a + 1 + n  2 − (e − 1)(a + 1 − e)×  a   a, + 1, b, c, d, e, −n   2  × 7 F6  a ; 1  , a + 1 − b, a + 1 − c, a + 1 − d, a + 1 − e, a + 1 + n  2

6

2 THE FOUR BALANCED VERY WELL POISED 7 F6

b(a − b + n)(a + 1)n (a − b − c)n (a − b − d)n (a + 1 − c − d)n (a + 1 − b)n (a + 1 − c)n (a + 1 − d)n (a − b − c − d)n − (2a + n − b − c − d)(b + c + d − a)× (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 − b − c − d)n =

5

or shifting e up by 1 and doing some algebra:  a   a, + 1, b, c, d, e, −n   2  F ; 1   (2.3) 7 6 a  , a + 1 − b, a + 1 − c, a + 1 − d, a + 1 − e, a + 1 + n  2 (a + 1)n (a − b − c)n (a − b − d)n (a − c − d)n = × (a + 1 − b)n (a + 1 − c)n (a + 1 − d)n (a − b − c − d)n " # n(n + 2a − b − c − d)(a − b − c − d) × 1+ (a − b − c)(a − b − d)(a − c − d) when the series is 4-balanced, or equivalently when e = 2a + n − b − c − d.

(2.4)

The second natural way to derive (2.3) uses a more complicated formula than (1.6), but the calculations from the starting formula are easier, and one can see how to extend the sum to the very well poised 2k-balanced series. The starting formula is Whipple’s transformation [14] between a very well poised 7 F6 and a balanced 4 F3 :   a   a, + 1, b, c, d, e, −n   2 ; 1 (2.5) 7 F6   a , a + 1 − b, a + 1 − c, a + 1 − d, a + 1 − e, a + 1 + n  2     − n, a + 1 − d − e, b, c (a + 1)n (a + 1 − b − c)n ; 1 . =  4 F3  (a + 1 − b)n (a + 1 − c)n b + c − n − a, a + 1 − d, a + 1 − e When e = 2a + n − b − c − d, the 4 F3 on the right is   −n, b + c + 1 − n − a, b, c    ; 1 4 F3  b + c − n − a, a + 1 − d, b + c + d + 1 − n − a

7

=

n X k=0

(−n)k (b)k (c)k (k + b + c − n − a) · (a + 1 − d)k (b + c + d + 1 − n − a)k k! (b + c − n − a)     − n, b, c = 3 F2  ; 1 + a + 1 − d, b + c + d + 1 − n − a (−n)bc × + (a + 1 − d)(b + c − n − a)(b + c + d + 1 − n − a)   1 − n, b + 1, c + 1   ; 1 ×3 F 2  a + 2 − d, b + c + d + 2 − n − a

The second 3 F2 is balanced, and so can be summed using the Pfaff- 6 Saalsch¨utz sum    − n, b, c  (d − b)n (d − c)n  ; 1 = .  3 F2  (d)n (d − b − c)n d, 1 + b + c − n − d

(2.6)

The first 3 F2 is two balanced, and so can be written as the sum of 2 terms by use of the transformation formula:    − n, a, b  (c − a)n (c − b)n  ; 1 = × 3 F2  (c)n (c − a − b)n c, d    − n, a, a + b + 1 − n − c − d  ; 1 . × 3 F2  a + 1 − n − c, a + 1 − n − d

(2.7)

For, when the series on the left of (2.7) is k-balanced, the third numerator parameter in the series on the right is 1 − k; so the series can be written as the sum of k terms when k = 1, 2, . . . For those unacquainted with (2.7), an argument giving a q-extension is in the last section. These series combine to give another derivation of (2.3) when (2.4) has been assumed. This method clearly extends to give the sum of the 2k-balanced very well poised 7 F6 , but the resulting identity is too messy to be worth stating until it is needed.

8

3 ANOTHER CLAUSEN TYPE IDENTITY.

3 Another Clausen type identity. To obtain the next Clausen type identity take the 4 F3 in (1.9) to be 4balanced, or take c = a + b + 3/2. As before, specialize (2.3) by taking c = a/2, d = (a + 1)/2 and make the series on the left 4-balanced. The resulting series is

=

7

   − n, a, b, e  ; 1  4 F3  a + 1 + n, a + 1 − b, a + 1 − e

(3.1)

(a + 1)n ((a − 2b)/2)n ((a − 2b − 1)/2)n (− 21 )n

× (a + 1 − b)n ((a + 2)/2)n ((a + 2)/2)n (− 21 − b)n     n(n + a − b − 21 )  × 1 + [(a − 2b)/2][(a − 2b − 1)/2](− 12 )    (a − 2b − 1)2n (− 21 )n 4n(n + a − b − 21 )(2b + 1)   . = 1 + (a + 2b)(a − 2b − 1)  (a + 1 − b)n (a + n + 1)n (− 12 − b)n

The replace a by −k and after the same argument given above, replace −n by a. The result is     − k, a, b, e (2a)k (2b)k (a + b)k ; 1 = ×A  4 F3  (a)k (b)k (2a + 2b + 2)k 1 − a − k, 1 − b − k, 1 − e − k (3.2) with A given by A =1+

(2 + 4a + 4b + 8ab)k + k2 − k 2(a + b)(2a + 1)(2b + 1)

(3.3)

or by A= and

k2 + (8ab + 4a + 4b + 1)k + 2(a + b)(2a + 1)(2b + 1) 2(a + b)(2a + 1)(2b + 1)

(3.4)

1 e = −k − a − b − . 2

(3.5)

9

Using (3.2) with A given by (3.3) in (1.9), we obtain     2  2a, 2b, a + b  a, b         = F F (3.6) ; x ; x 3 2 2 2   3 3   a + b + , 2a + 2b + 2  a+b+ 2 2    2a + 1, 2b + 1, a + b + 1  2ab x   + ; x  3 F2  5  (a + b + 1)(a + b + 3/2) a + b + , 2a + 2b + 3 2    2a + 2, 2b + 2, a + b + 2  abx2   ; x +  3 F2  7  2(a + b + 3/2)2 (a + b + 5/2) a + b + , 2a + 2b + 4 2 Using (3.2) with A given by (3.4) gives

  2       a, b  2a, 2b, a + b, c + 1, d + 1      ; x 2 F1  3 ; x = 5 F4  3 a+b+ a + b + , 2a + 2b + 2, c, d  2 2

(3.7)

where c and d are determined by

x2 + (8ab+ 4a+ 4b+ 1)x+ 2(a+ b)(2a + 1)(2b + 1) = (x+ c)(x+ d). (3.8)

4 Comments. After working out the above results, I went to a library to see if they 8 were new. The fact that  2     a, b     2 F1  1 ; x , n = 0, 1, . . . , a+b+n+ 2

(4.1)

is a generalized hypergeomatric series was proved by Goursat [8]. He also showed that   2   a, b   2 F1  ; x c

10

4

COMMENTS.

is a generalized hypergeometric series only when c = a + b + n + 12 , n = 0, 1, . . .. His proof that (4.1) is a generalized hypergeometric series uses Clausen’s formula (1.2), its derivative      a + 1, b + 1   a, b       2 F1  3 ; x 1 ; x 2 F1  a+b+ a+b+ 2 2    2a + 1, 2b + 1, a + b + 1    = 3 F2  ; x  a + b + 3 , 2a + 2b + 1  2

(4.2)

and the transformation

    a + 1 , b +  a, b    2 1/2   2 F1   a + b + 1 ; x = (1 − x) 2 F1   a+b+ 2

9

1   2 ; x . 1  2

Of course Ramanujan knew all of those facts. Goursat also used some recurrence relations. Ramanujan knew about some of the recurrence relations hypergeometric series satisfy, and almost surely derived some of his continued fractions from these recurrence relations. However Ramanujan did not use recurrence relations as much as he could have, or as often as he used other properties of hypergeometric series. While Ramanujan almost surely could have rediscovered Goursat’s result if he had needed it, it is more likely he would have used an argument like the one given above. Ramanujan does not seem to have found Whipple’s transformation formula (2.5). He did find a limiting case with one parameter missing, but we have not found (2.5) in any of the sheets of his. If there is another treasure like the sheets in Trinity College, I would not be surprised in (2.5) is there. Actually, I would be surprised if Ramanujan was very interested in Goursat’s result. What he really loved was not general results that could not be made very explicit, but beautiful formulas. I could imagine Ramanujan working out the details in section 3, but the resulting formulas are already starting to be messier than those he loved.

11

I sent an outline of the results in sections 2 and 3 to a couple of people, and George Andrews wrote back that the 4-balanced very well poised 7 F6 sum was found by Lakin [9]. The two proofs given in section 2 are easier than the two Lakin gave, so it is worth including them above. Lakin also found a basic hypergeometric extension of this sum. The derivation of his result from the q-extension of Whipple’s formula is the most natural one, so it will be given in the next section.

5 The 3-balanced very well poised 8ϕ7 . The analogue of Whipple’s transformation formula (2.5) was found by Watson [13]. It is √ √    a, q a, −q a, b, c, d, e, q−n 2 qn+1   a  √ 8 ϕ7   a, − √a, aq , aq , aq , aq , aqn+1 ; q, bcde  b c d e  −n aq   q , , b, c,  aq (aq; q)n ( bc ; q)n   de = aq  aq aq bcq−n ; q, q 4 ϕ3  aq  ( b ; q)n ( c ; q)n  , ,  d e a

(5.1)

where (a; q)n = (1 − a)(1 − aq) . . . (1 − aqn−1 )

(5.2)

and  X  ∞   a0 , . . . , a p (a0 ; q)k . . . (a p ; q)k xk   ϕ = ; q, x .  p+1 p  (b1 ; q)k . . . (b p ; q)k (q; q)k b1 , . . . , b p k=0

(5.3)

The series (5.3) is called k-balanced at q j if x = q j , one of the numerator parameters is q−n and a0 a1 . . . a p qk = b1 . . . b p . It is called balanced if k = 1 and j = 1. The series (5.3) is well poised if a0 q = a1 b1 = . . . = a p b p , and very well poised if it is well poised and if a1 = qb1 , a2 = −a1 .

12

5 THE 3-BALANCED VERY WELL POISED 8 ϕ7 .

The sum that corresponds to (1.6) occurs when the 4 ϕ3 in (5.1) becomes a 3 ϕ2 bey setting a2 qn+1 = bcde, and using  n   q , a, b  (c/a; q)n (c/b; q)n  = (5.4) ; q, q  3 ϕ2  (c; q)n (c/ab; q)n c, q1−n abc−1

to sum the resulting balanced 3 ϕ2 . Observe that the balancing condition is now 1-balanced in the q-case as opposed to 2-balanced in the hypergeometric case. The analogue of (2.3) requires a 3-balanced very well poised 8 ϕ7 at 2 q . To obtain this sum, use (5.1) with aq bcq1−n = de a

10

(5.5)

The 4 ϕ3 becomes n X (q−n ; a)k (b; q)k (c; q)k qk (1 − bcqk−n /a) k=0 −n

aq aq d ; q)k ( e ; q)k (q; q)k

1 − bcq−n /a

(5.6)

    q , b, c, bc(1 − q−n )(1 − b)(1 − c)q × = 3 ϕ2  ; q, q + (aqn − bc)(1 − aq/d)(1 − aq/e) aq/d, aq/e   1−n   q , bq, cq  ; q, q × 3 ϕ2  2 aq /d, aq2 /e

where

1 − bcqk−n /a = 1 − bcq−n /a + bcq−n (1 − qk )a−1 was used to break the series into two sums. The second sum on the right in (5.6) is balanced, and so can be summed by (5.4). The first is 2balanced at q, and a q-extension of (2.7) can be sued to sum this series. To obtain this transformation, recall a transformation of Sears [12]:  −n  !n  q , a, b, c  bc (aq1−n /e; q)(aq1−n / f ; q)n  × (5.7) ; q, q,  = 4 ϕ3  d (e; q)n ( f ; q)n d, e, f  −n   q , a, d/b, d/c   ×4 ϕ3  ; q, q d, aq1−n /e, aq1−n / f

13

when q1−n abc = de f . Let b, d → 0 in (??). The result is   −n !n   q , a, c e f qn−1 (aq1−n /e; q)n (aq1−n / f ; q)n  ; q, q = × 3 ϕ2  a (e; q)n ( f ; q)n e, f  −n   q , a, q1−n ac/e f   × 3 ϕ2  1−n ; q, q aq /e, aq1−n / f

(5.8)

When the left hand side is k-balanced, qn ac = e f q−k ; so that right hand side is the sum of k terms. Formula (5.8) with k = 2 reduces to formula (27) in [9]. The result obtained when the series on the right in (5.6) are summed is equivalent to (29) in [9] comes from the series on the left in (5.6) when 1 − bcqk−n a−1 is broken into the two parts 1 and −bcqk−n a−1 . Since these identities and the sum of (5.1) when it is 3-balaced at q2 and very well poised are given by Lakin [9], they will not be repeated here.

6 Acknowledgement This work was done while visiting Japan and Australia. I would like to 11 thank the many hosts who kept me busy enough so this work could be done inthe time between talks and visits.

References [1] W. N. Bailey : Some theorems concerning products of hypergeometric series, Proc. London Math. Soc. (2) 38 (1935). 377 - 384. [2] W. N. Bailey : Generalized Hypergeometric Series, Cambridge Univ. Press, 1935, Reprinted Stechert-Hafner, New York, 1964. [3] J. and P. Borwein : Pi and the AGM, Wiley, Canada, 1987. [4] J. and P. Borwein : Unpublished.

14

REFERENCES

[5] D.V. and G.V. Chudnovsky : Approximations and complex multiplication according to Ramanujan, Ramanujan Revisited, ed. G. Andrews et al, Academic Press, Cambridge, MA, 1987. [6] T. Clausen : Ueber die F¨alle, wenn die Reihe von der Form y = α′ β′ δ′ α β 1 + · x + . . . ein Quadrat von der Form z = 1 + · ′ · ′ x + . . . 1 γ 1 γ ǫ hat, J. Reine Angew. Math., 3(1828), 89-91. [7] J. Dougall : On Vandermonde’s theorem and some more general expansions, Proc. Edinburgh Math. Soc. 25(1907), 114-132. [8] E. Goursat : M´emoire sur les functions hyperg´eom´etriques d’ordre sup´erieur, Ann. Sci. Ecole Norm. Sup. ser. 2; 12(1883), 216-285, first part; 395-430, second part. [9] A. Lakin : A hypergeometric identity related to Dougall’s theorem, J. London Math. Soc. 27(1952) , 229-234. [10] S. Ramanujan : Collected Papers, Cambridge, 1927, 23-29. [11] S. Ramanujan : Notebooks, Vol. 2, Tata Inst. Fund. Research, Bombay 1957. 12

[12] D. B. Sears : On the transformation theory of basic hypergeometric functions, Proc. London Math. Soc. (2) 53(1951), 158-180. [13] G. N. Watson : A new proof of the Rogers-Ramanujan identities, J. London Math. Soc. 4(1929), 4-9. [14] F. J. W. Whipple : On well-poised series, generalized hypergeometric series having parameters in pairs, each pair with the same sum, Proc. London Math. Soc. (2) 24(1926), 247-263. Mathematics Department University of Wisconsin-Madison Van Vleck Hall 480, Lincoln Drive Madison, W 53706, USA.

TITCHMARSH’S PHENOMENON FOR ζ(s) By R. Balasubramanian∗ and K. Ramachandra∗∗

1 Introduction Under the title “On the frequency of Titchmarsh’s phenomenon for ζ(s)” 13 we have written seven papers [11, 12, 2, 1, 3, 4, 13] sometimes individually and sometimes jointly. the present article is a summary of these results. The function ζ(s)(s = σ + it) is defined in σ > 0 by   n+1 ∞  X  1 Z du   + 1 .  − ζ(s) =  ns s u  s − 1 n=1 n

The sum on the right can be easily shown to be an entire function by a repetition of the trick which we have employed to prove that this is an ∞ P analytic continuation of ζ(s) = n−s (σ > 1) in σ > 0. Thus the serious n−1

problem about ζ(s) is not the analytic continuation. But the conjecture ζ(s) , 0 in σ > 1/2 is really a very serious problem. [This is called Riemann hypothesis (R.H.)]. To serve as an introduction to our results we will first state some results (free from any hypothesis). We next recall some well-known consequences of Riemann hypothesis for comparison with these results. We will be concerned with the size of |ζ(σ + it)| in 1/2 6 σ 6 1, t > t0 where t0 is a large positive constant which may depend on parameters like σ and other constants like (arbitrarily small positive) ǫ when they appear. The letter A will denote an absolute positive constant and C will denote a positive constant independent of t 15

16

1 INTRODUCTION

but may depend on other parameters. There may not be the same at each occurrence. |ζ(1/2) + it| < tµ+ǫ (1) (µ = 1/2 is easy; µ = 1/4 is a little more difficult, the fundamental result µ = 1/6 is due to G. H. Hardy, J.E. Littlewood and H. Weyl [19]. There have been a number of important papers by various authors which reduce µ = 1/6, the latest being µ = 9/56 due to E. Bombieri and H. 1 Iwaniec [7] and a further reduction by 560 due to M. N. Huxley and N. Watt. A. result of N. V. Kuznetsov proved by him in a paper presented by him in this conference implies that we can take µ = 1/8). 3/2

|ζ(σ + it)| < (t(1−σ) 14

log t)A

(2)

(due to the ideas of I.M. Vinogradov, see A. Walfisz’s book [20]; see also H.-E. Richert [18]). |ζ(σ + it)| < tµ(σ)+ǫ

(2′ )

(various values of µ(σ) are obtained by various methods by various authors; see E. C. Titchmarsh’s book [19].) |ζ(1 + it)| < A(log t)2/3

(3)

(due to the ideas of I.M. Vinogradov, see A. Walfisz’s book [20].) We now state consequences of R.H. |ζ(1/2 + it)| < Exp(A log t/ log log t)

(4)

(due to J.E. Littlewood, see [19]) |ζ(σ + it)| Exp(C(log t)2(1−σ) / log log t), C = C(δ),

(5)

uniformly in 1/2 6 σ 6 1 − δ(δ > 0). (This is due to J.E. Littlewood, E.C. Titchmarsh and others, see [19]) |ζ(1 + it)| < (2eγ + ǫ) log log t

(6)

17

(due to J.E. Littlewood, see [19]. Littlewood’s method shows that if θ defined as the least upper bound of the real parts of the zeros of ζ(s) is less than 1, then (6) would follow with some positive constant in place of 2eγ . Here as elsewhere we denote by γ the Euler’s constant). If we compare (1), (2), (3) with (4), (5), (6) we see how much has been achieved in the direction of Lindel¨of hypothesis (L. H.) (which is a consequence of R.H.) with states that in (1) we can take µ = 0. A consequence of L.H. is that we can take µ(σ) = 0 in (2′ ). The L.H. has also remained unsolved for a long time. We do not know whether we can take µ(σ) = 0 for any value of σ in 1/2 6 σ < 1. These results seem to be out of reach for many centuries to come. Also we do not know whether the results (4), (5), (6) can be improved on the assumption of R.H.. However we can show that in (6) 2eγ cannot be replaced by any constant less than eγ . The corresponding results regarding (4) and (5) are not so satisfactory. In (4), we can show that we cannot replace 1 the right hand side by Exp((log t) 2 −ǫ ) and that the right hand side in (5) cannot be replaced by Exp((log t)1−σ−ǫ ) in 1/2 + δ 6 σ 6 1 − δ. These results (which are called Ω results) are due to J.E. Littlewood and E.C. Titchmarsh. Littlewood generally assumes R.H. and Titchmarsh’s 15 results are independent of any hypothesis. For references to the work of Littlewood and Titchmarsh see [19].

THE PROBLEM. Let σ be fixed in σ > 1/2 (σ may depend on T and H to follow). Let I denote an interval of length H contained in [T, 2T ], where H > 1000. (We may also make σ depend on T and H; for example, we can take σ = 1 + 1/ log H). In the first two papers max [11, 12] of the series max second author investigated |ζ(σ + it)| t in I max max max and also |ζ(α + it)| and other problems like |ζ(σ + it)| α > σ(1)t in I t in I where the minimum is taken over all intervals I of length H contained in [T, 2T ]. (He has also improved Theorem ?? of [11] as follows. Let ∞ an (K) P ). Then the RHS in Theorem ?? can be replaced (ζK (s))1/2 = s n=1 n

18

2

KEY RESULT AND ITS APPLICATIONS.

P |an (K)|2 (log n)l with the condition 1000 log log X 6 U 6 X). 2σ n6U n 0 These and other problems were studied further in papers [2, 1, 3, 4, 13]. by U ×

2 Key result and its applications. The method employed in the first paper of series was systematized by the second of us in [14]. This was improved by us in [5]; but this improvement (though a significant progress) does not give any new Ω results. The net result is as follows. Theorem 1. Let {an } be a sequence of complex numbers satisfying a1 = 1 and for n > 2, |an | < (n(H + 2))A , where H > 0 and A is an arbitrary ∞ P positive constant. Let 0 < H 6 T and F(s) = an n−s be analytically n=1

continuable in σ > 0 and continuous in σ > 0. Then     Z max  1  2  |F(σ + it)| dt >   σ>0 H I

1 log n CAΣ |an |2 1 − + H n 6 100 + 1 log H log log H

!

where C A > 0 is effective and depends only on A. 1 R H max , then |F(it)|2 dt > |F(σ+it)| < Exp Exp σ > 0, t in I 100 HI ! P 1 1 log n CA + . |an |2 1 − 2 n6 H +1 log H log log H

Corollary. If

100

16

Proof of the corollary. One method of deduction of the corollary is by Gabriel’s two variable two variable convexity theorem coupled with the kernel Exp .(sin(z/100)2 ), (see the appendix to [15]). For another method see [14]. Note that this kernel decays in | Re z| 6 1/2 uniformly at most like a constant multiple of (Exp Exp(c| Im z|))−1 where c > 0 is a constant. Given faster decaying kernels we can deduce the Corollary

19

with more relaxed conditions. The same applies to all applications of the key theorem. It may be remarked that we do not know any kernel which decays (uniformly in |z| 6 1/2) at most like a constant multiple of (Exp Exp(c| Im z|))−1 where c is any large positive constant. We begin the applications by the following remark, which follows from The theorem by putting F(s) = (ζ(α + s))k where α > 1/2 (we may assume without loss of generality that H exceeds a large positive constant) and k is a positive integer which may depend on H. Theorem 2. We have, for α > 1/2,   !1/2k  2 X  1 (dk (n)) log n  max  , + 1 − |ζ(σ+it)| > C A σ > α, t in I log H log log H   n2α H n6 100

where k 6 log H so that the condition dk (n) 6 (n(H+2))A is satisfied. (In fact it may be noted that the maximum of RHS is attained in k 6 log H itself). Remark. We may also state a similar theorem for max |ζ(σ + it)|−1 σ > α, t ∈ I

where α > 1/2 + δ and (σ > α − δ/2, t in I) is free from zeros of ζ(s). Here δ is an arbitrary positive constant. max It is not very difficult to investigate the order of magnitude of k>1 (R.H.S.) in Theorem 2. By a very ingenious argument, R. Balasubramanian has shown (see [1]) that its logarithm is asymptotic to C0 (log H/ log log H)1/2 where C0 = 0.75 . . . , when α = 1/2. This gives the best known Ω result   1/2  max |ζ(σ + it)| > Exp 34 logloglogHH . 1 σ > 2 , t in I Earlier in [2], we had obtained a small positive constant in place of 17

20

2

KEY RESULT AND ITS APPLICATIONS.

3/4. Balasubramanian’s asymptotic formula shows that it is not possible to replace 3/4 by even 0.76 by our method. Earlier to our result [2], nearly around the same time, H.L. Montgomery [9] had obtained the constant 1/20 (in place of 3/4) on the assumption of Riemann hypothesis. We would also like to remark that in order to obtain the maximum order (of RHS in Theorem 2) as k varies, we have to take (in case α = 1/2) a large number of terms of the sum and ignore the rest. One the contrary, when α > 1/2 + δ it is enough to take a particular term, namely, the maximum term of the sum. If 1/2 + δ 6 α 6 1 − δ then it is enough to take n to be the biggest square-free product of the first k primes, which does not exceed H. If α = 1, then for each p we select that prime power pm for which (dk (pm ))2 p−2mα is the largest and then take n to be the product of the first k prime powers pm which does not exceed H (for details see [2] or [17].) If α = 1 + (1/ log H) for instance we may take out from (dk (n))2 n−2α the portion n−(2/ log H) (which certainly exceeds e−2 for n 6 H). In [2] the first of us has shown that even if we take all the terms of the sum we do not get a better result. In fact following the method of [1] we may show that  !  2 1/2k   max [d (n)]  Σ k log   (log x)α−1 (log log x) k > 1 n 6 x n2α tends to a positive constant if 1/2 + δ 6 α 6 1 − δ. These results show the limitations of our method. However out net result are Theorem 3(A). We have, if 1/2 + δ 6 α 6 1 − δ, ! C(log H)1−α max . |ζ(σ + it)| > Exp σ > α, t in I log log H Theorem 3(B). We have, if 1 −

1 1 6α 61+ , log H log H

max |ζ(σ + it)| > eγ (log log H − log log log H + O(1)). σ > α, t in I

21

Remark 1. H. L. Montgomery [9] has shown that if 1/2 + δ 6 α 6 1 − δ then |ζ(a + it)| exceeds Exp[C(log t)1−α ]/(log log t)α ] for a sequence of values of t tending to infinity by a different method. But this method does not enable one to conclude that the maximum of |ζ(α+it)| as t varies for example, over [T, 2T ] exceeds Exp[C(log T )1−α ]/(log log T )α ]. The lower bound Exp[C(log T )1−α /(log log T )] given by our method seems to be the best known till today. Remark 2. N. Levinson [8] has shown by a different method that |ζ(1 + 18 it)| exceeds eγ log log t + O(1) for a sequence of values of t tending to infinity. But, for short intervals like [T, 2T ], ours is the only result known. Remark 3. Let 1/2 + δ 6 α 6 1 − δ. I suspected that if we take F(s) = (log ζ(α + s))k then we might get a better result Theorem 3(A). But it ∞ P was shown by H. L. Montgomery [10] that if (log ζ(s))k = ak (n)n−s , n=1

then

1/2k   max X 2 −2α   (ak (n)) n  k > 1 n6x

lies between two constant multiples of (log x)1−α (log log x)−1 . Remark 4. H.L. Montgomery has conjectured [9] that if 1/2 6 α 6 1−δ then |ζ(α + it)| does not exceed Exp([C(log t)1−α ]/[log log t]α )

3 Further study of the maximum in 1/2 +δ 6 α 6 1 − δ by other methods. In the second paper of the series [12], the second of us proved that if 1/2 + δ 6 α 6 1 − δ and I runs over all intervals, of fixed length H, contained in [T, 2T ] then ! min max |ζ(α + it)| ∼ (1 − α) log log H, log log I t in I provided C < 100 log log T 6 H 6 Exp[D log T ]/[log log T ]) where C is a large positive constant and D a positive constant depending only on

22

4 STUDY OF THE MAXIMUM ON σ = 1.

α. This aspect of the problem has been studied further by us in [2]. The method is very closely related to a principle which we formulated and employed in [6]. The main result of [3] is as follows: Theorem 3. let! α be as above, E > 1 an arbitrary constant, C 6 H 6 D log T Exp where C is a large positive constant and D an arbitrary log log T positive constant. Then there are > T H −E disjoint open intervals I (of fixed length H) all contained in [T, 2T ], such that, (log H)1−α (log log H)1−α ≪ max | log ζ(α + it)| ≪ . α t in I (log log H) (log log H)α Here log ζ(s) is the analytic continuation in t > 2 along lines parallel to the real axis (and free from zeros of ζ(s)) from σ > 1. The Vinogradov symbol ≪ means “less than a positive constant times”.

4 Study of the maximum on σ = 1. 19

As a corollary to Theorem 3, we deduced in [4] the following Theorem 4. Let J denote the interval I (of Theorem 3) with intervals of length (log H)2 removed from both extremities. Then max |ζ(1 + it)| 6 eγ [log log H + log log log H + O(1)]. t in J Note that LHS is > eγ (log log H − log log log H + O(1)) by applying the Corollary to the key theorem. (The conditions for deriving this lower bound from the Corollary to the key theorem are satisfied in the course of the proof of Theorem 3). The key result of §2 can be used to obtain lower bounds for max |ζ(σ + it)| and also a similar result (the lower bound gets σ > 1, t in I multiplied by the factor (6/π2 ) for |ζ(σ + it)|−1 . But to obtain lower

23

max max |ζ(1 + it)|−1 we need conditions |ζ(1 + it)| and for t in i t in I looking like 100 log log log T 6 H 6 T . But by a somewhat complicated application of the key result and other techniques the second of us [13] proved the following theorem. To state the theorem it is better to introduce some notation. The letter θ will, as before, denote the least upper bound of the real parts of the zeros of ζ(s) (we do not know whether θ < 1 or not). For x > 1 we define log1 x = log x and for n > 2 we define logn x to be log(logn−1 x); similarly we define for real x, Exp1 (x) = Exp(x) and for n > 2 we define Expn (x) = Exp(Expn−1 (x)). bounds for

Theorem 5. Consider for open intervals I (for t, of length H > 100) contained in [T, 2T ] where T > T 0 , a large positive constant, the inequality max |ζ(1 + it)| > eγ (log log H − log log log H − ρ), t in I

(∗)

where ρ is a certain real constant which is effective. Then we have the following four results: (1) (∗) holds for all I for which T > H > A1 log4 T (2) If θ < 1 then (∗) holds for all I for which T > H > A2 log5 T . (3) Let now H < A1 log4 T . Consider a set of disjoint intervals I (of fixed length H) for which (∗) is false. Then the number of such intervals I does not exceed T X1−1 where X1 = Exp4 (βH) where β is a certain positive constant less than A−1 1 . (4) Let now H < A2 log5 T . Consider a set of disjoint intervals I (of fixed length H) for which (∗) is false. Then the number of such 20 intervals I does not exceed T X2−1 where X2 = Exp5 (β′ H) where β′ is a certain positive constant which is less than A−1 2 .

5 An announcement In this section the length of the interval will not be denoted by H. We wish to announce a result [16] due to the second author which is ob-

24

REFERENCES

tained by quite a different method. Theorem 6. Let ǫ be a constant satisfying 0 < ǫ !< 1, T > T 0 (ǫ), a log T constant depending only on ǫ, X = Exp . If from the interlog log T vals T 6 t 6 T + eX we exclude certain (boundedly many depending on ǫ) disjoint open intervals I each of length at most X −1 , then in the remaining portions of the interval, we have, | log ζ(1 + it)| 6 ǫ log log T. Further put β0 = A(log T )−mu (log log T )−2µ where µ = 2/3 and A is any positive constant. Consider the rectangle R defined by σ > 1 − β0 , T 6 t 6 T + eX . Let I denote an open interval for t of length 1/X and let J denote the corresponding rectangle σ > 1 − β0 , t in I. Then with the exception of certain boundedly many (depending on ǫ and A) disjoint rectangles J we have for s in R, | log ζ(s)| 6 ǫ log log T where T > T 0 (ǫ, A) Remark . The first result can be proved without assuming the Vinogradov’s zero free region. But if we assume the Vinogradov’s zero free region, we get a better upper bound for the number of intervals which have to be excluded. However, for the proof of the second part, the Vinogradov zero free region is essential.

References [1] R. Balasubramanian : On the frequency of Titchmarsh’s phenomenon for ζ(s)-IV, Hardy-Ramanujan J., Vol. 9(1986), 1-10. 21

[2] R. Balasubramanian and K. Ramachandra : On the frequency of Titchmarsh’s phenomenon for ζ(s)-III, Proc. Indian Acad. Sci., 86 (A) (1977), 341-351.

REFERENCES

25

[3] R. Balasubramanian and K. Ramachandra : On the frequency of Titchmarsh’s phenomenon for ζ(s)-V, Arkiv f¨or Mathematik 26(1) (1988), 13-20. [4] R. Balasubramanian and K. Ramachandra : On the frequency of Titchmarsh’s phenomenon for ζ(s)-VI, (to appear). [5] R. Balasubramanian and K. Ramachandra : Progress towards a conjecture on the mean-value of Titchmarsh series-III, Acta Arith., XLV (1986), 309-318. [6] R. Balasubramanian and K. Ramachandra : On the zeros of a class of generalised Dirichlet series-III, J. Indian Math. Soc., 41 (1977), 301-315. [7] E. Bombieri and H. Iwaniec : On the order of ζ(1/2 + it), Ann. Scoula Norm. Sup. Pisa. 13 no. 3 (1986), 449-472. [8] N. Levinson : Ω theorems for the Riemann zeta-function, Acta Arith, XX (1972), 319-332. [9] H. L. Montgomery : Extreme values of the Riemann zeta-function, Comment. Math. Helv., 52 (1977), 511-518. [10] H. L. Montgomery : On a question of Ramachandra, HardyRamanunan J., 5 (1982), 31-36. [11] K. Ramachandra : On the frequency of Titchmarsh’s phenomenon for ζ(s)-I, J. London Math. Soc., (2) 8(1974), 683-690. [12] K. Ramachandra : On the frequency of Titchmarsh’s phenomenon for ζ(s)-II, Acta Math. Acad. Sci. Hungaricae, Tomus 30 (1-2), (1977), 7-13. [13] K. Ramachandra : On the frequency of Titchmarsh’s phenomenon for ζ(s)-VII, (to appear). [14] K. Ramachandra : Progress towards a conjecture on the meanvalue of Titchmarsh series-I, Recent Progress in Analytic Number

26

REFERENCES

Theory (Edited by H. Halberstam and C. Hooley) Vol. I, Academic Press (1981), 303-318. 22

[15] K. Ramachandra : A brief summary of some results in the analytic theory of numbers-II Addendum, Number Theory. Proceedings, Mysore (1981), Edited by K. Alladi, Lecture Notes in Mathematics, 938, Springer Verlag, 106-122. [16] K. Ramachandra : A remark on ζ(1 + it), Hardy-Ramanujan J., 10 (1987) (to appear). [17] K. Ramachandra and A. Sankaranarayanan : Omega theorems for the Hurwitz zeta-function, (to appear). [18] H. E. Richert : Zur Absch¨atzung der Riemannschen Zetafunktion in der N¨ahe der Verti¨calen σ = 1, Math. Ann., 169 (1967), 97-101. [19] E. C. Titchmarsh : The theory of the Riemann zeta-function, Clarendon Press, Oxford (1951). [20] A. Walfisz : Weylsche exponential Summen in der neueren Zahlentheorie, VEB Deutscher Verlag der Wiss., Berlin (1963) . * Institute of Mathematical Sciences Madras - 600 113, Tamil Nadu India ** School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Bombay-400 005 India

RAMANUJAN’S FORMULAS FOR EISENSTEIN SERIES By Bruce C. Berndt*

As is customary, N denotes the set of positive integers, Z denotes the 23 ring of rational integers, H = {τ : Im τ > 0}, and Γ0 (n) =

(

! ) a b : a, b, c, d ∈ Z, ad − bc = 1, c ≡ 0(mod n) , c d

where n ∈ N. If n = 1, Γ0 (1) is the full modular group Γ(1). Let ∞ X kq2k E2 (τ) = 1 − 24 1 − q2k k=1 and Fn (τ) = E2 (τ) − nE2 (nτ), where q = eπiτ , τ ∈ H , and n ∈ N. Although E2 (τ) is not a modular form, it can be easily shown that Fn (τ) is a modular form of weight 2 and trivial multiplier system on Γ0 (n). In a very famous paper [8, pp. 23-39], Ramanujan gave formulas for Fn when n = 2, 3, 4, 5, 7, 11, 15, 17, 19, 23, 31, 35. However, no proofs are indicated. Furthermore, in Chapter 21 of his second notebook [9], Ramanujan offers, without proofs, formulas for Fn when n = 3, 5, 7, 9, 11, 15, 17, 19, 23, 25, 31, 35. In contrast to [8] where only one formula is given for each value of n, in [9] several formulas are stated for most values of n. * Research

partially supported by a grant from the Vaughn Foundation.

27

28

Part of Ramanujan’s motivation in calculating Fn arose from its appearance in certain approximations to π found by Ramanujan [8]. J.M. and P.B. Borwein [6] have extensively developed Ramanujan’s ideas. Using their work, we shall very briefly indicate how these approximations are obtained. Let K denote the complete elliptic integral of the first kind associated with the modulus k, where 0 < k < 1, and let E ′ denote the complete elliptic integral √ of the second kind associated with the complementary modulus k′ = 1 − k2 . For r > 0, define α(r) = 24

√

E′ π , − K 4K 2 √

where k = k(r) = θ22 (e−π r )/θ32 (e−π r ), where θ2 and θ3 are the classical theta-functions, usually so denoted. Put αm = α(n2m r), where m ∈ N ∪ {0} and n ∈ N. There exists a recursion formula for αm in terms of Fn [6, p. 158]. This leads to an approximation of for 1/π given by √ m√ 0 < αm − 1/π < 16nm re−n rπ provided that rn2m > 1 [6, p. 169]. For complete details, see [6]. The Borweins leave the calculation of Fn for n = 2, 3, 4 as exercises [6, p. 161]. In fact, they [6, 9. 158] state that “The verification... is tedious but straightforward for small n. For larger n, we rely on Ramanujan.” The Surpose of this paper is to indicate how Ramanujan’s formulas for Fn can be proved. Complete proofs for all of Ramanujan’s formulas for Fn can be found in the author’s forthcoming book [2]. We offer two general approaches. The first is probably similar to that employed by Ramanujan, while the second depends upon the theory of modular forms. The first method rests upon modular equations. Thus, we need to give the definition of a modular equation, as understood by Ramanujan. Definition . Let K, K ′ , L, and L′ denote complete elliptic integrals of the first kind associated with the moduli k, K ′ , l, and l′ , respectively. Suppose that the equality K ′ L′ = (1) n K L

RAMANUJAN’S FORMULAS FOR EISENSTEIN SERIES

29

holds for some n ∈ N. Then a modular equation of degree n is a relation between the moduli k and l which is implied by (1). Ramanujan sets α = k2 and β = l2 . If q = exp(−πK ′ /K) and ϕ(q) =

∞ X

2

qj

j=−∞

then it is well known that K=

π 2 ϕ (q). 2

Furthermore, set zn = ϕ2 (qn ). Definition . The multiplier m for a modular equation of degree n is defined by K z1 ϕ2 (q) m= = 2 n = . L ϕ (q ) zn In his notebooks [9], Ramanujan devotes more space to modular 25 equations than to any other topic. Despite this, Ramanujan never published any of his work on modular equations, except for the aforementioned formulas for Eisenstein series in [8]. For an expository account of Ramanujan’s discoveries on modular equations, see our paper [1]. Some of Ramanujan’s modular equations have been proved in three papers [3], [4], [5] that we have coauthored with A. J. Biagioli and J. M. Purtilo. For proofs of all of Ramanujan’s modular equations, see the author’s forthcoming book [2]. We now state perhaps the primary formula that Ramanujan employed in establishing formulas for Fn (τ). He has not stated this formula in either [8] or [9]. However, some cryptic remarks on p. 253 of his second notebook [9] point to a result such as that given below. Theorem 1. Let q, Fn , α, β, m, and z1 be as given above. Then ! β(1 − β) 2 d Log 6 . Fn (τ) = −α(1 − α)z1 dα m α(1 − α)

30

We now sketch proofs for three of seven formulas for F3 (τ) found in Entry 3 of Chapter 21 in Ramanujan’s second notebook [9]. Theorem 2. Let ϕ, α, and β be as given above. Put ψ(q) =

∞ X

q j( j+1)/2 .

j=0

Then ( 4 )2 ϕ (q) + 3ϕ4 (q3 ) 1 S 3 (τ) : = − F3 (τ) = 2 4ϕ(q)ϕ(q3 ) = ϕ2 (q)ϕ2 (q3 ) − 4qψ2 (−q)ψ2 (−q3 ) o n p p 1 = ϕ2 (q)ϕ2 (q3 ) 1 + αβ + (1 − α)(1 − β) . 2

(2) (3) (4)

The last formula was stated by Ramanujan in [8], [10, p. 33]. Proof. Letting n = 3 in Theorem 1, we find that ! 1 β(1 − β) 2 d S 3 (τ) = α(1 − α)z1 Log 6 . 2 dα m α(1 − α)

(5)

We need to determine the interdependence of α, β and m in order to calculate the derivative above. From our work [2] on modular equations of degree 3 in Section 5 of Chapter 19 in Ramanujan’s second notebook [9], (m2 − 1)(9 − m2 )3 , 256m6 m4 (m2 − 1)2 β(1 − β) , = α(1 − α) (9 − m2 )2

α(1 − α) =

26

(6) (7)

and dm 16m4 = . dα (9 − m2 )2

(8)

RAMANUJAN’S FORMULAS FOR EISENSTEIN SERIES

31

Substituting (6)-(8) into (5) and employing the chain rule, we deduce that ! m2 − 1 (m2 − 1)(9 − m2 ) 2 d Log S 3 (τ) = z1 (9) dm 16m2 m(9 − m2 ) z2 = 1 3 (m2 + 3)2 . 16m If we now use the definition of m, we find that (2) readily follows. Using again the definition of m, we may rewrite (9) in the form ! (9 − m2 )(m2 − 1) (10) S 3 (τ) = z1 z3 1 − 16m2 = z1 z3 (1 − {αβ(1 − α)(1 − β)}1/4 ), where we have employed (6) and (7). Now in Chapter 17 of his second notebook [9], Ramanujan offers a “catalogue” of evaluations of thetafunctions in terms of q(qn ), α(β), and z1 (zn ). In particular, from Entry 11, 1 ψ(−q) = ( z1 )1/2 {α(1 − α)/q}1/8 2 and 1 ψ(−q3 ) = ( z3 )1/2 {β(1 − β)/q3 }1/8 . 2 Solving these two equalities for α(1 − α) and β(1 − β), respectively, and substituting them in (10), we immediately deduce (3). The simplest modular equation of degree 3 is given by (αβ)1/4 + {(1 − α)(1 − β)}1/4 = 1.

(11)

This was first discovered by Legendre and may be found in Cayley’s book [7, p. 196], for example. Ramanujan [9, chpater 19, Entry 5(ii)] rediscovered (11). If we square both sides of (11) and substitute in (10), we immediately deduce (4). Unfortunately, we have been unsuccessful in using Theorem 1 to establish certain formulas of Ramanujan for Fn (τ). We thus have had

32

to invoke the theory of modular forms in these cases. In order to offer 27 one such example, we need to make an additional definition. Let, in the notation of Ramanujan, f (−q) =

∞ Y (1 − q j ), j=1

where, as above, q = eπiτ . Note that f (−q2 ) = q−1/12 η(τ), where η denotes the Dedekind eta-function. We now state Entry 8(i) in Chapter 21 of Ramanujan’s second notebook [9].  Theorem 3. Let ϕ, ψ, and f be defined as above. Then 1 (12) − F11 (τ) = 5ϕ2 (q)ϕ2 (q11 ) − 20q f 2 (q) f 2 (q11 ) 2 + 32q2 f 2 (−q2 ) f 2 (−q22 ) − 20q3 ψ2 (−q)ψ2 (−q11 ). We now briefly describe how the theory of modular forms can be used to prove Theorem 3. The functions ϕ(q), ψ(q), and f (−q) are associated with modular forms of weight 1/2 on ! a b Γ(2) = { ∈ Γ(1) : a ≡ d ≡ 1(mod 2), b ≡ c ≡ 0(mod 2)}. c d Thus, (12) is first converted into an equality relating modular forms. Each of the five expressions in (12) is a modular form of weight 2 on Γ(2) ∩ Γ0 (11). We have already mentioned that the multiplier system of F11 (τ) is trivial. By employing the multiplier system of η(τ), we can show that each of the four expressions on the right side of (12) also has a trivial multiplier system. Let Γ = Γ(2) ∩ Γ0 (p), where p is an odd prime. Let F be a fundamental set for Γ. If F is a nonconstant modular form of weight r on Γ, then the valence formula X

z∈F

OrdΓ (F; z) =

1 r(p + 1) 2

(13)

REFERENCES

33

is valid, where OrdΓ (F : z) is the invariant order of F at z. Suppose that we can show that the coefficients of q0 , q1 , q2 , . . . , qµ in F are equal to 0, i.e. OrdΓ (F; ∞) > µ + 1. Suppose furthermore that µ + 1 > 21 r(p + 1). Then if OrdΓ (F : z) > 0 for each z ∈ F , X

z∈F

1 OrdΓ (F; z) > OrdΓ (F; ∞) > µ + 1 > r(p + 1). 2

Hence F(τ) ≡ 0, for otherwise, we could have a contradiction to the valence formula (13). Now write the proposed identity (12) in the form 28 F := F1 + . . . + F5 = 0.

(14)

We have shown that F is a modular form of weight 2 and trivial multiplier system on Γ = Γ(2) ∩ Γ0 (11). Moreover, OrdΓ (F : z) > 0 for each z ∈ F . Since (1/2)r(p + q) = 12, it suffices to show that the coefficients of q j , 0 6 j 6 12, in F are equal to 0 in order to prove (14), and hence also (12). Using MACSYMA, we have indeed done this, and so the proof of Theorem (3) has been completed. More complete details on the use of modular forms and MACSYMA in proving modular equations may be found in [2] and [4]. We are grateful to A.J. Biagioli and J.M. Purtilo for their collaboration on modular forms and MACSYMA, respectively.

References [1] B.C. Berndt : Ramanujan’s modular equations, Ramanujan Revisited, Academic Press, Boston 1988, 313-333. [2] B.C. Berndt : Ramanujan’s Notebooks, Part III, Springer Verlag, New York, to appear. [3] B.C. Berndt, A.J. Biagioli and J.M. Purtilo : Ramanujan’s “mixed” modular equations, J. Ramanujan Math. Soc. 1(1986), 46-70.

34

REFERENCES

[4] B.C. Berndt, A.J. Biagioli, and J.M. Purtilo : Ramanujan’s modular equations of “large” prime degree, J. Indian Math. Soc., 51 (1987), 75-110. [5] B.C. Berndt, A.J. Biagioli, and J.M. Purtilo : Ramanujan’s modular equations of degrees 7 and 11, Indian J. Math., 29 (1987). 215-228. [6] J.M. and P.B. Borwein : Pi and the AGM, John Wiley, New York, 1987. [7] A. Cayley : An Elementary Treatise on Elliptic Functions, Second Ed., Dover, New York, 1961. 29

[8] S. Ramanujan : Modular equations and approximations to π, Quart. J. Math. 45(1914), 350-372. [9] S. Ramanujan : Notebooks (2 Volumes), Tata Institute of Fundamental Research, Bombay, 1957. [10] S. Ramanujan : Collected Papers, Chelsea, New York, 1962. Departement of Mathematics University of Illinois 1409 West Green street Urbana, Illinois 61801 U.S.A.

ON THE PROOF OF ANDREWS’ q-DYSON CONJECTURE By D. M. Bressoud* This is a breif sketch of work done by Doron Zeilberger, Ian Goulden 31 and myself in late 1983 and early 1984 which settled in the affirmative a conjecture made by George Andrews [1] as well as more detailed conjectures made by Kevin Kadell [12]. The problem has its origins in the evaluation of a definite integral which arose in a physical problem [5], its solution has given evaluations for other definite integrals arising in physics [4]. The original integral was discovered by Freeman Dyson [5]: −n

I(n, z) = (2π)

Z2π

...

0

Z2π

|∆n (eiθ )|2z dθ1 . . . dθn ,

(1)

0

where ∆n (eiθ ) = Π(eiθ j − eiθk ), 1 6 j < k 6 n. Dyson conjectured that I(n, z) =

Γ(nz + 1) , Γn (z + 1)

(2)

a conjecture which was simultaneously and independently proved by Gunson [7] and Wilson [23]. It is sufficient to prove that conjecture for positive integral integral z. In this case, we can use the following equality: |∆n (eiθ )|2 = Π(eiθ j − eiθk )(e−iθ j − e−iθk ). * Partially

supported by N.S.F. grant no. DMS.-8521580

35

(3)

36

= Π(1 − ei(θ j −θk ) )(1 − ei(θk −θ j ) ). If we set x j = eiθ j , then the integral picks out the constant term in a −1 polynomial in x1 , x−1 1 , . . . , xn , xn . Given a monomial, M, in the xi ’s, let [M] denote the coefficient of M in the succeeding polynomial. Let x0 denote the monomial in which each xi appears to the power 0. Equation (2) for z ∈ N can be restated as [x0 ]Π(1 − x j /xk )z (1 − xk /x j )z =

(nz)! , 1 6 j < k 6 n. (z!)n

(4)

32

Dyson discovered that more was probably true, and actually stated his conjecture in the following form: [x0 ]Π(1 − x j /xk )ak (1 − xk /x j )a j (a1 + . . . + an ) = a1 ! . . . an !

(5)

In 1975, Andrews [1] noted that equation (5) seemed to have a nice generalization in which the product (1 − x)a could be replaced by (x)a = (1 − x)(1 − xq)(1 − xq2 ) . . . (1 − xqa−1 ) Specifically, Andrews conjectured the following: [x0 ]Π(x j /xk )ak (qxk /x j )a j , 1 6 j < k 6 n, (q)a1 +...+an . = (q)a1 . . . (q)qn

(6)

On reason for the interest in equations (5) and (6) is the intractability of the blunt approach. If one expands the binomials in equations (5), the constant term is a simple summation when n = 3, and Dyson’s conjecture is the classical identity: ! ! ! X a1 + a3 a + a2 a2 + a3 (−1)i 1 (7) i i − a2 + a3 i + a1 − a2 i

ON THE PROOF OF ANDREWS’ q-DYSON CONJECTURE a2

= (−1)

37

! a1 + a2 + a3 . a1 , a2 , a3

  For larger n, however, the constant term is an n−1 2 -fold summation, and virtually nothing is known about such non-trivial multiple summations. The same situation applies to Andrews’ conjecture, except that instead of multiple hypergeometric series we get multiple basic hypergeometric series. To understand how equation (6) was first proved, one must understand an ingenious combinatorial proof of Dyson’s equation (5) which was found by Zeilberger [24] a few years earlier. Equation (5) is equivalent to [(xa11 . . . xann )n−1 ]Π(x j − xk )a j +ak (a1 + . . . + an )! . = (−1)a2 +2a3 +...+(n−1)an a1 ! . . . an !

(8)

We can formally expand the product of binomials in equation (8): 33 X (9) (−1)u(T ) xw1 1 (T ) . . . xwn n (T ) Π(x j − xk )a j +ak = T ∈T ∗

where T ∗ is the set of “multi-tournaments” in which each pair of players, say j and k, meet a total of a j + ak times and the “winner” of each game is recorded. The exponent wi (T ) is the number of games won by player i, and u(T ) records the number of “upsets” : k > j and k beats j. If we let T ⊆ T ∗ be the subset of multi-tournaments in which each player j wins (n − 1)a j games, then equation (8) can be restated as: ! X u(T ) a2 +...+(n−1)an a1 + . . . + an (−1) = (−1) . (10) a1 , . . . , an T ∈T

The right side of equation (10) involves the multinomial coefficient which counts the number of “words” which can be constructed with a1 1′ s, a2 2’ s, . . . , an n′ s. Each such word corresponds to a multi-tournament in a natural way. Given j and k, remove the subword of length

38

a j + ak in the letters j and k. The winners in order are read off left to right. As an example, if n = 4, a1 = a2 = a3 = a4 = 2, the word 32114243 corresponds to the multi-tournament: 2112 3113 1144 3223 2424 3443 We observe that the number of upsets is always a2 + 2a3 + . . . + (n − 1)an . If we let T ′ ⊆ F be the subset of multi-tournaments which do not correspond to a word, then equation (10) can be further simplified to X (−1)µ(T ) = 0. (11) T ∈T ′

34

Zeilberger showed how to prove this by establishing a bijection between the set of T ∈ T ′ for which u(T ) is even and the set of T ∈ T ′ for which u(T ) is odd. We shall demonstrate the bijection with an example. Let T be 2111 3133 1144 2232 2424 3443 Inspection immediately shows us that while this element is in T , it cannot be the two letter subwords of a single word. Nevertheless, we shall attempt to construct a word to which this multi-tournament corresponds.

ON THE PROOF OF ANDREWS’ q-DYSON CONJECTURE

39

The leading entries of each row define a tournement: 2 beats 1, 3 beats 1, 1 beats 4, etc. Schematically, this tournament is given by: 1G W✳ ✏ ✳ ✏✏✏ ✳✳✳ ✏✏  ✳✳ ✏✏✏ @ 4 ^❂ ✳✳✳ ✏ ✁✁✁ ❂❂❂ ✳✳ ❂❂ ✳ ✏✏✏✁✁✁ ❂❂✳ ✏✁✁ /3 2

We call a tournament “transitive” if it contains no cycles, “non-transitive” otherwise. If our multi-tournament arose from a single word, then this tournament is transitive and the player beating everyone else is the first letter of the word. Since our tournament is transitive, it is possible at this stage that it comes from a single word. We record the first letter : 2, and modify the tournement by looking at the next outcome of the games of player 2: 1 beats 2, 2 beats 3, 4 beats 2. The tournament becomes : 1

✏ W✳✳ ✏✏ ✏ ✳✳✳ ✏  ✳ ✏✏ ✏ 4 ✳✳✳ ✏ ✁ ^❂❂❂ ✳✳ ❂❂ ✳ ✏✏ ✏ ✁✁✁ ❂❂✳✳ ✁ ❂ ✏✏✁✁✁ /3 2

Our tournament is now non-transitive which will eventually happen if and only if T is in T ′ . Every non-transitive tournament contains a 3-cycle and reversing the arrows in a 3-cycle will change the parity of the number of upsets in the tournament. We have two 3-cycles in this tournament. which one we choose to reverse is significant. 35 If we reverse 2 → 3 → 4 → 2 and then restore the first letter, 2, we get the multi-tournament 2111

40

3133 1144 2332 2224 4443 But the leading entries of this multi-tournament give us a non-transitive tournament: G1W✳ ✏ ✳ ✏✏✏ ✳✳✳ ✏✏  ✳✳ ✏✏✏ @ 4 ❂ ✳✳✳ ✏ ✁✁✁ ❂❂❂ ✳✳ ❂❂ ✳ ✏✏✏✁✁✁ ❂❂✳ ✏✁✁  /3 2

An iteration of our procedure would not take us back to the original multitournament. If no letters of the word have been recorded, then it doesn’t matter which 3-cycle we reverse as long as we are consistent. If at least one letter has been recorded, then we are in a peculiar situation. Let v1 be the last letter recorded. Since we have only changed the arrows connected to v1 , all cycles of the non-transitive tournament include vertex v1 . Let the remaining vertices be labelled v2 , v3 , . . . , vn where v2 beats v3 beats . . . beats vn , and choose the smallest i for which v1 beats vi and vi+1 beats v1 . It is the 3-cycle v1 → vi → vi+1 → v1 that we reverse. it is exactly this procedure that was used to prove Andrews’ conjecture, except that the details are more complicated because the parameter q introduces an additional weight on the multi-tournaments. The proof first demonstrates that [x0 ]Π(x j /xk )ak (qxk /x j )a j X (−1)µ(T ) qwt(T ) , = (−1)a2 +...+(n−1)an

(12) (13)

t∈F

where wt(T ) is the sum of the “Major Indices” of all the two letter words in the multi-tournament. The Major Index of a word is the sum of the

ON THE PROOF OF ANDREWS’ q-DYSON CONJECTURE

41

number of letters to the left of each “descent” in the word. Thus

36

32114243 has four descents : (32, 21, 42, 43) , and its major index is 1 + 2 + 5 + 7 = 15. On the other hand, if we sum the major indices of the two letter subwords of 32114243, we get 1 + 1 + 0 + 1 + 2 + 3 = 8. This sum of Major Indices is called the Z-statistic, denoted Z(T ). The second part of the proof involves showing that the sum of qZ(T ) over all multitournaments corresponding to a single word is equal to (q)a1 +...+an (q)a1 . . . (q)an Equation (6) now reduces to verifying that X (−1)u(T ) qwt(T ) = 0.

(14)

T ∈T ′

The bijection given above does not preserve weights. The last and most elaborate part of the proof involves finding and verifying a bijection which does. It is curious that this combinatorial approach is still the only known proof of equation (6). Goulden and I[3] generalized this proof of yield a more useful identity. In the following we let A be an arbitrary set of unordered pairs ( j, k), 1 6 j , k 6 n, χ(S ) is 1 if S is true, 0 otherwise, SA is the set of permutations of {1, . . . , n} for which j > i and σ−1 (i) implies (i, j) < A, and wt(σ) is the sum over all j of a j times the number of k < j for which σ−1 ( j) < σ−1 (k). [x0 ]Π(x j /xk )a j (qxk /x j )ak −χ(( j,k)∈A) (q)a1 +...+an X wt(σ) 1 − qaσ( j) q Πj = (q)a1 . . . (q)an σ∈S 1 − qaσ(1) +...+aσ( j)

(15) (16)

A

This identity implies several conjectures of Kadell [12] and has had applications in studying the characters of S L(n, C)[21] and in evaluating definite integrals arising in statistical mechanics [4].

42

37

REFERENCES

The theorems first conjectured by Dyson and Andrews are only the tip of the iceberg of a very extensive theory. These identities are related to the Vandermonde determinant formula which is Weyl’s denominator formula for the root system An . Macdonald [15] conjectured the appropriate generalizations to arbitrary root systems and he and W.G. Morris [16] gave conjectures and some proofs for the basic analogs. Macdonald’s conjecture for the root system BCn was discovered to be equivalent to a multi-dimensional beta integral evaluation of Selberg [18, 19]. A basic analog of this was conjectured by Askey [2]. Habsieger [8] and Kadell [13] independently proved Askey’s conjecture and then Habsieger [9] and Zeilberger [25] showed that this integral evaluation implied some of Morris’ conjectures. Most recently, Kadell [14] has proved Macdonal’s conjecture for the basic analog of the BCn conjecture, Garvan [6] has done the same for F4 , and E.M. Opdam [17] has proved the original Macdonald conjecture for arbitrary root systems. Only the basic analogs for the special root systems E6 , E7 and E8 are unproven at the moment. Stembridge [22] has found a strikingly simple proof of Andrews’ conjecture in the case where the parameters are equal. He has also found formulas for some of the non-constant terms [21]. Connections with representation theory can be found in an article by Stanley [20]. Hanlon has pursued the connections between these identities and cyclic homology [10, 11].

References [1] G. E. Andrews : Problems and prospects for basic hypergeometric functions, in Theory and Application of Special Functions, ed. R. Askey, Academic Press, New York, 1975, 191-224. [2] R. A. Askey : Some basic hypergeometric extensions of integrals of Selberg and Andrews, SIAM J. Math. Anal. 11 (1980), 938-951. [3] D. M. Bressoud and I. P. Goulden : Constant term identities extending the q-Dyson Theorem, Trans. Amer. Math. Soc. 291

REFERENCES

43

(1985), 203-228. [4] D. M. Bressoud and I. P. Goulden : The generalized plasma in 38 one dimension : evaluation of a partition function, Commun. Math. Phys. 110(1987), 287-291. [5] F. J. Dyson : Statistical theory of the energy levels of complex systems, J. Math. Physics 3(1962), 140-156. [6] F. Garvan : Personal communication. [7] J. Gunson : Proof of a conjecture by Dyson in the statistical theory of energy levels, J. Math. Physics 3 (1962) , 752-753. [8] L. Habsieger : Une q-integrale de Selberg-Askey, SIAM J. Math, Anal., to appear. [9] L. Habsieger : La q-conjecture de Macdonald-Morris pour G2 , C. R. Acad. Sc. Paris 302 (1986), 615-618. [10] P. Hanlon : The proof of a limiting case of Macdonald’s root system conjecture, Proc. London Math. Soc. 49 (1984), 170-182. [11] P. Hanlon : Cyclic homology and the Macdonald conjectures, Invent. Math. 86 (1986), 131-159. [12] K. Kadell : Andrews’ q-Dyson conjecture : n = 4, Trans. Amer. Math. Soc. 290 (1985), 127-144. [13] K. Kadell : A proof of Askey’s conjectured q-analog of Selberg’s integral and a conjecture of Morris, SIAM J. Math. Anal., to appear. [14] K. Kadell : Personal communication. [15] I. G. Macdonald : Some conjectures for root systems. SIAM J. Math. Anal. 13 (1982), 988-1007. [16] W. G. Morris : Constant term identities for finite and infinite root systems, Ph. D. thesis, University of Wisconsin, Madison, 1982.

44

REFERENCES

[17] E. M. Opdam : Doctoral thesis, University of Leiden, Netherlands. [18] A. Selberg : Uber einen Satz von A. Gelfond, Arch. Math. Naturvid. 44 (1941), 159-170. [19] A. Selberg : Bemerkninger om et multiplelt integral, Norsk Mat. Tidsskr. 26, (1944), 71-78. 39

[20] R. Stanley : The q-Dyson conjecture, generalized exponents and the internal product of Schur functions, in “Combinatorics and Algebra” ed. Curtis Greene, Amer. Math. Soc., Providence, 1984, 81-94. [21] J. Stembridge : First layer formula for the characters of S L(n, C), Trans, Amer. Math. Soc. 299 (1987), 319-350. [22] J. Stembridge : A short proof of Macdonald’s conjecture for the root systems of type A. Preprint. [23] K. Wilson : Proof of a conjecture by Dyson, J. Math. Physics 3 (1962) 1040-1043. [24] D. Zeilberger : A combinatorial proof of Dyson’s conjecture, Discrete Math. 41 (1982), 988-1007. [25] D. Zeilberger : A proof of the G2 case of Macdonald’s root system – Dyson conjecture, SIAM J. Math Anal. 18 (1987), 880-883. [26] D. Zeilberger and D. Bressoud : A proof of Andrews’ q-Dyson conjecture, Discrete Math. 54 (1985), 201-224. Penn State University, University Park, PA 16802

WEYL’S INEQUALITY, WARING’S PROBLEM AND DIOPHANTINE APPROXIMATION By D. R. Heath-Brown

For fixed positive integers s and k, we define R s,k (N) = #{(n1 , . . . , n2 ) ∈ N s :

s X

41

nkj = N}

1

One central question in Waring’s problem is to prove the HardyLittlewood asymptotic formula R s,k (N) =

Γ(1 + 1/k) s S(N)N (s/k)−1 + O(N (s/k)−1−δ ) Γ(s/k)

(1)

for as large a range of s as possible. To tackle this, one uses an exponential sum P X S (α) = e(αnk ), n=1

where P = [N 1/k ]. One then has R s,k (N) =

Z1

S (α)s e(−αN)dx.

(2)

0

The trivial bound for S (α) is |S (α)| 6 P. However, one can improve on this for suitable α, by using the following estimate. 45

46

Weyl’s inequality. Let |α − a/q| 6 q−2 , with (a, q) = 1. Then 1−k

S (α) ≪ǫ P1+ǫ (q−1 + P−1 + qP−k )2 , for any ǫ > 0. Thus if α can be approximated with P 6 q 6 Pk−1 one has 1−k +ǫ

S (α) ≪ P1−2

,

(3)

and the corresponding contribution to (2) is 1−k +ǫ)

≪ P s(1−2

42

≪ N s/k−1−δ ,

provided that s > 2k−1 k. Those α which have no useable approximation produce the main term of (1). Thus one obtains (1) for s > 1 + 2k−1 k. One can improve on this agrument by using an average bound. Hua’s inequality. For any ǫ > 0, one has Z1

k

k

|S (α)|2 dα ≪ǫ P2 −k+ǫ .

0

This leads to (1) for s > 1 + 2k . Until recently, this was the best known range for (1), for small k > 3. The sum S (α) may also be used in Diophantine approximation problems. It was shown by Danicic [2] that if ǫ > 0 and k ∈ N are given, then there exists P(ǫ, k) as follows. For any P > P(ǫ, k) and any α ∈ R, one can find n 6 P with 1−k

||αnk || 6 Pǫ−2 .

(4)

This generalizes Dirichlet’s approximation Theorem, when k = 1, and a result of Heilbronn (4), for k = 2. To prove Danicic’s theorem one can use a result of Montgomery (see Baker [1, Theorem 2.2]): If ||an || > ∆ for 1 6 n 6 P, then X X | e(han )| > P/6. 16h6∆−1 n6P

WEYL’S INEQUALITY, WARING’S PROBLEM AND DIOPHANTINE 47 APPROXIMATION

We therefore wish to estimate

X

|S (αh)|.

(5)

h6∆−1

As with Weyl’s inequality, this can be done satisfactorily, with a 1−k relative saving of P−2 +ǫ , unless α has an approximation |α − a/q| 6

∆ , q 6 P. qPk−1

(6) 1−k

Thus Montgomery’s result allows us to take ∆ ≈ Pǫ−2 . Of course, if (6) holds then ||αqk || 6 ∆, and q 6 P. A sharpening of Weyl’s inequality has recently been obtained (HeathBrown [3]). Theorem 1. Let |α − a/q| 6 q−2 with (a, q) = 1, and suppose that k > 6. Then −k S (α) ≪ǫ P1+ǫ (Pq−1 + P−2 + qP1−k )(4+3)2 for any ǫ > 0. Thus

43

S (α) ≪ P

1−(4/3)21−k +ǫ

,

if P3 6 q 6 Pk−3 . One therefore has a sharper bound than (3), but for a shorter range of q, (and only for k > 6). Closely related to Theorem 1 is an improvement on Hua’s Inequality (Heath-Brown [3]). Theorem 2. Let k > 6 and ǫ > 0. Then Z1

k

k −k+ǫ

|S (α)|(7/8)2 dα ≪ P(7/8)2

.

0

As before one may deduce : Corollary. The Hardy-Littlewood asymptotic formula [1] holds for k > 7 6 and s > 1 + 2k . 8

48

One may also try to sharpen Danicic’s result. One obtains a saving in (5) of 1−k P−(4/3)2 +ǫ relative to the trivial estimate, unless |α − a/q| 6

∆ , q 6 P3 . qPk−3

Unfortunately in this latter case, one gets no useable bound for ||aqk ||. The attempt to improve on (4) therefore fails. However, if one starts with an approximation |α − a/q| 6 q−2 and fixes P = [q(1/3) ], for example, one is led to an “unlocalized” result (Heath-Brown [3]). Theorem 3. Let α ∈ R and ǫ > 0 be given. For any integer k > 6, there are infinitely many n ∈ N with 1−k .

||αnk || 6 nǫ−(4/3)2

Let us now look at the proof of Theorem 1. One uses Weyl’s “square and difference” trick, but with the symmetric difference (∇h )(x) = (x + h) − (x − h) in place of the forward difference. After j steps, one has X j j |S (α)|2 ≪ P2 − j−1 |R(α)|,

(7)

h1 ,...,h j

44

where |hi | < P/2 and R(α) = R(α; h1 , . . . , h j ) =

X

e(α∇h1 . . . ∇h j (nk )).

n∈I

Here I is a subinterval of [1, P], depending on h1 , . . . , h j . As a function of n, the polynomial ∇h1 . . . ∇h j (nk )

WEYL’S INEQUALITY, WARING’S PROBLEM AND DIOPHANTINE APPROXIMATION 49

has degree k − j. An appropriate version of Weyl’s Inequality would therefore give 1−(k− j) +ǫ R(α) ≪ P1−2 , (8) for suitable α. in conjunction with (7) we would then obtain 1−(k− j)+ǫ

j

j

|S (α)|2 ≪ P2 − j−1 · P j · P1−2 whence

1−k +ǫ

S (α) ≪ P1−2

,

,

for suitable α. One thus merely recovers Weyl’s Inequality again. To improve on this, we replace (8) by a mean-value bound, where one averages over the parameters hi . If one takes j = k − 1 or k − 2 then R(α) is a linear or quadratic sum, and the bound (8) is essentially best possible. Thus nothing can be gained by averaging. One therefore chooses j = k − 3, in which case R(α) is a cubic sum of the form X R(α) = e(An3 + Bn). n∈1

Here the interval I and the coefficients A and B depend on the hi . In fact, A takes the form A=

k! k−3 2 h1 . . . hk−3 . 6

Had one used forward differences in deriving (7) rather than symmetric differences, there would have been a term in n2 appearing in R(α), and so one would have to average over three coefficients, rather than two. With j = k − 3, the Weyl bound now takes the form |R(α)| ≪ P3/4+ǫ ,

(9)

for suitable α, whereas one would conjecture that |R(α)| ≪ P1/2+ǫ , in general. In fact, one can easily prove that

45

50

REFERENCES

Z1 Z1 X | e(An3 + Bn)|6 dAdB ≪ P3+ǫ , 0

0

(10)

n6P

by counting the number of solutions of the simultaneous equations n31 + n32 + n33 = n34 + n34 + n36 n1 + n2 + n3 = n4 + n5 + n6 .

(1 6 ni 6 P)

To pass from the sum on the right hand side of (7) to the mean value (10), one uses the bound ′ X

6

|R(α)| ≪ P

hi ...,hk−3

4+ǫ

Z1 Z1 X N | e(An3 + Bn)|6 dAdB, 0

0

n6P

where N = max #{(h1 , . . . , hk−3 ) : || A∈[0,1]

k! k−3 2 h1 . . . hk−3 α − A|| 6 P−3 }. 6

Here we exclude the possibility that any hi vanishes, both in the sum and in the maximum occurring in the definition of N . It is apparent that there is a loss of a factor P in passing from the discrete average of R(α) over the hi to the mean-value (10). Nonetheless, one finds that R(α) is O(P2/3+ǫ ) on average, and this is a sufficient improvement on (9) for the proof of Theorem 1.

P′

References [1] R. C. Baker : Diophantine Inequalities (Oxford Science Publications, 1986). [2] I. Danicic : Contributions to Number Theory (Ph.D. Thesis, London, 1957). [3] D. R. Heath-Brown : Weyl’s inequality, Hua’s inequality, and Waring’s problem, J. London Math. Soc. (2), to appear.

REFERENCES 46

51

[4] D. R. Heath-Brown : The fractional part of αnk , Mathematika, to appear. [5] H. Heilbronn : On the distribution of the sequence n2 θ(mod 1), Quart. J. Math. Oxford Ser., 19 (1948), 249-256. Magdalene College, Oxford OX1 4AU, United Kingdom.

THE CIRCLE METHOD AND THE FOURIER COEFFICIENTS OF MODULAR FORMS By Henryk Iwaniec To the memory of Srinivasa Ramanujan

1 Introduction. 47

The circle method was first used in number theory by G. Hardy and S. Ramanujan [2] to establish an asymptotic formula for the partition function (see also [7]) and it was applied extensively in the series of papers under the common title : Some problems of “Partitio Numerorum” by G. Hardy and J.E. Littlewood to study additive problems such as the Waring problem or the Goldbach problem (see for example [1]). The method was particularly interesting for additive problems with many summands. Yet at that time the important results were conditional subject to sharp estimates for the relevant extponential sums. Perhaps the most ambitious are the binary problems, i.e. the problems of evaluating the number of solutions to the equation a + b = n, where a, b range over finite sets of integers A, B respectively and n is a fixed integer. Clearly, the number of solutions is given by the integral (Vinogradov’s modification)    Z1  X  X e(−αn)  e(αa)  e(αb) dα. 0

a∈A

b∈B

52

53

The Hardy-Littlewood arguments fail to handle the binary problem for a fundamental reason–the use of Parseval’s identity Z1 X | e(αa)|2 dα = |A|. 0

a∈A

It is evident that when dealing with a binary problem one cannot ignore a cancellation in the integration over any set of positive constant measure. Taking this into account in 1926, H. D. Kloosterman [4] introduced a brilliant refinement which is described by Yu. V. Linnik in [6] as the 48 process of levelling (a sophisticated partition of the segment 0 < α < 1 by means of Farey’s points). Kloosterman’s method was originally used for the binary problem in which a and b assume values of some quadratic forms. The important point should be mentioned that the exponential sum X e(αa) a∈A

is evaluated precisely enough to control the oscillatory behaviour of the remainder term which is usually of the order of magnitude |A|1/2 . Both the partition of the segment 0 < α < 1 and the nature of the remainder term comprise the appearance of the Kloosterman sums ! X md + nd¯ , S (m, ˙ n : c) = e c d(mod c) P where ∗ means that the summation ranges over d prime to c and d¯ is the multiplicative inverse to d(mod c). Then a non-trivial bound for S (m, n; c) yields a cancellation of the remainder terms and consequently one breaks the barrier set by the use of Parseval’s identity. The Kloosterman device enables one to handle a large class of binary problems. Moreover it turns out to be successful in answering various questions about the Fourier coefficients of modular forms (see for example [5]). Kloosterman did not exploit a cancellation of terms of summation over the moduli c that exists due to the variation of sign of the Kloosterman sum S (m, n; c). In this connection Linnik [6] was led to formulate

54

1

the following hypothesis X c6C

49

INTRODUCTION.

c−1 S (m, n; c) ≪ C ǫ

and he said : “This hypothesis can be considered as a certain analogy to the well-known hypothesis of Hasse on the behaviour of congruence zeta-functions arising by the reduction of a given curve with respect to all prime moduli.” A somewhat stronger statement was expressed by A. Selberg [8] in the context of estimating the Fourier coefficients of modular forms. The recent developments in the spectral theory of automorphic functions brought a remarkable progress towards the LinnikSelberg hypothesis. If one sequence A or B in the binary problem has no reference to the modular forms, then naturally other exponential sums emerge in place of the Kloosterman sums. For example see the paper by C. Hooley [3] in which the Kloosterman refinement is applied to advance in the Waring problem for cubes under the Riemann hypothesis for certain Hasse-Weil L-Functions. In this paper we elaborate the Kloosterman ideas in a general context. We shall express the distribution     1 if n = 0 δ(n) =    0 if n , 0 in terms of the Ramanujan sums

S (n; c) =

∗ X

d(mod

d e n c c)

!

and of new sums of type S v (n; c) =

∗ X

d(mod c)

d+v c

!!

! d e n , c

where ((ζ)) = ζ − [ζ] − 1/2. We shall establish a formula for the Fourier coefficient of a cusp form in terms of the Kloosterman sums S (m, n; c)

55

and of the new Kloosterman type sums S v (m, n; c) =

∗ X

d+v c

d(mod c)

!!

! md + nd¯ e . c

These sums are closely related. Indeed, by the Fourier expansion (boundedly convergent) X (ζ)) = (2πih)−1 e(ξh) + O[(1 + ||ζ||H)−1 , 0 4. ZT

1 |ζ( + it)8 dt ≪ T (log T )16+B , T → +∞. 2

0

58

From various consequences of this estimate, one can derive the conclusion: there is a fixed constant B so that 1 |ζ( + it)| ≪ |t|1/8 (log |t|)B , |t| → ∞, t is real. 2

Part I. Sums of Kloosterman sums 1.1 The Lobatcevskii plane. This plane will be considered as the upper half plane H of the complex variable z = x + iy, x, y ∈ R, y > 0, with the metric ds2 = y−2 (dx2 + dy2 ),

(1.1)

dµ(z) = y−2 dx dy

(1.2)

measure and with the corresponding Laplace operator 2

L = −y

! ∂2 ∂ + . ∂x2 ∂y2

(1.3)

1.2 The appearance of Kloosterman sums.

65

The full modular group acts on this plane in the natural way: z 7→ γz =

ax + b , a, b, c, d ∈ Z, ad − bc = 1. cz + d

(1.4)

Most of the results may be developed for certain Fuchsian groups but there are no essentially new ideas for these cases; so, I restrict myself to the full modular group Γ only. 1.2 The appearance of Kloosterman sums. Their appearance is inescapable, if one calculates the Fourier coefficients of an automorphic function which is defined as a sum over a gorup. For example, let us define the classical Poincar´e series by Pn (z; k) =

(4πn)k−1 X −k j (γ, z)e(nγz), n > 1 Γ(k − 1) γ∈Γ /Γ

(1.5)

∞

(where Γ∞ is generated by the translation z 7→ z + 1, j(γ, z) = cz + d if
the transformation γ is defined by a matrix ∗c d∗ and we assume that k is an even integer and k > 4). Then, for the m-th Fourier coefficient fo this series we have an almost obvious formula (the so-called “Petersson formula”): pn,m (k) :=

=

Z1

Pn (z, k)e(−mx)dx e2πmy

(1.6)

0

√ ! X1 4π nm (4π nm)k−1 (δn,m + 2πi−k S (n, m; c)Jk−1 ), n, m > 1, Γ(k − 1) c c c>1 √

where Jk−1 is the Bessel function of the order k − 1 and S is the Kloost- 59 erman sum ! X nd + md′ e S (n, m; c) := . (1.7) c 16d6|c|,(d,c)=1 dd′ ≡1(mod c)

66

0

INTRODUCTION.

We have a similar (but more complicated) representation, for the Fourier coefficients of the non-holomorphic Poincar´e-Selberg series which, for Re s > 1, is defined by X Un (z, s) := (Im γz)s e(nγz), n > 1, (1.8) γ∈Γ∞ /γ

(For n = 0, it is the Eisenstein series.) For the Kloosterman sums, we have the famous estimate due to A. Weil: |S (n, m; c)| 6 (2n, 2m, c)d2 (c)c1/2 . (1.9)

But, for the applications, we need estimates for the averages of these sums. Yu V. Linnik was the first to conjecture that Kloosterman sums oscillate regularly; his conjecture is that X 1 (1.10) S (n, m; c) ≪n,m X ǫ c c6X

for every ǫ > 0 as X → +∞. It is obvious that A-Weil’s estimate give only O(X (1/2)+ǫ ) on the right side and A. Selberg destroyed the hopes of any near progress in this conjecture when he constructed the counterexamples of groups for which Linnik conjecture is not valid (1963). At this point, there was a nice result from my first paper on this subject (1977): for every fixed ǫ > 0, we have X 1 (1.11) S (n, m; c) ≪n,m X (1/6)+ǫ c 16c6X

At the same time, for the “smoothing” average, we have a stronger estimate : if ϕ ∈ C ∞ (0, ∞), ϕ = 0 outside! the interval (a, 2a) and if, for r ∂ every fixed integer r > 0, we have ϕ(x) ≪ a−r , then, for every ∂x fixed A > 0, the following estimate is valid: X ϕ(c)S (n, m; c) ≪ a−A , a → +∞. (1.12) a6c62a Thus it is a confirmation of the Linnik conjecture.

1.3 The eigenfunctions of the automorphic Laplacian.

67

1.3 The eigenfunctions of the automorphic Laplacian. As a generalization of the classical cusp forms of even integral weight 60 k (which are regular functions on the upper half plane such that f (γz) = jk (γ, z) f (z) for any γ ∈ Γ and yk/2 | f (z)| is bounded for y > 0, the Poincar´e series Pn (z, k), n > 1, being an example of a cusp form of weight k with respect to full modular group), Maass introduced the nonholomorphic cusp forms (the so called Maass waves). The Laplace operator L in L2 (Γ/H) has a continuous spectrum on the half axis λ > 14 and a discrete spectrum λ0 = 0, 0 < λ1 < λ2 6 . . . with limit point at ∞ (note that λ1 ≃ 91.07 . . .). For the case of the full modular group, there are no exceptional eigenvalues in the interval (0, 41 ); (Huxley proved that the same is true for any congruence subgroup with the level 6 19). So L2 (Γ/H) decomposes as L2eis (Γ/H) ⊕ L2cusp (Γ/H) where L2eis is the continuous direct sum of E(z, 21 + it), t ∈ R (E being the Eisenstein series) and L2cusp is spanned by the eigenfunctions u j (z) given by  2 2   ∂ u j ∂ u j  2 (1.13) Lu j ≡ −y  2 + 2  = λ j u j , ∂x ∂y Z u j (γz) = u j (z), γ ∈ Γ; (u j , u j ) = |u j |2 dµ(z) < ∞. Γ/H

Any f ∈ L2 (Γ/H) which is smooth enough can be expanded into eigenfunctions of L and we have ∞

Z X 1 1 1 f (z) = ( f, u j )u j (z) + ( f, E(., + ir))E(z, + ir)dr 4π 2 2 j>0

(1.14)

−∞

if we choose u j so that we have an orthonormal basis {u j } j>0 . Note that the Eisenstein series has the Fourier expansion ξ(1 − s) 1−s y + (1.15) E(z, s) = y2 + ξ(s) √ 2 yX + τ s (n)e(nx)K s−1/2 (2π|n|y), ξ(s) := π−s Γ(s)ζ(2s), ξ(x) n,0

68

0

INTRODUCTION.

where K s−1/2 (.) is the modified Bessel function of order s − τ s (n) = |n|

s−1/2

σ1−2s(n)

X |n| ! s−1/2 = d2 d|n

1 2

and (1.16)

d>0

61

The eigenfunctions of a point λ j of the discrete spectrum have a similar Fourier expansion X √ uj = ρ j (n)e(nx) yKiχ j (2π|n|y) (1.17) n,0

with χ j =

q

λ j − 41 , λ j > 41 .

1.4 The Hecke operators. The ideas behind Hecke operators go back to Poincare and Mordell used them to prove that Ramanujan’s τ-function was multiplicative. The main observation is a simple fact that if H is a subgroup if Γ with S finite index, so that Γ is a finite coset union Hγ j , and f is automorphic j P on H, then f (γ j z) is automorphic on Γ. j

By appropriately choosing the set of representatives, we can define the n-th Hecke operator as the average ! az + b 1 X X (1.18) f (T n f )(z) = √ d n ad=n b(mod d) d>0

For this normalization, we have X Tn Tm = T nm/d2

(1.19)

d|(n,m)

and all these operators commute. Now we have a set of commuting Hermitian operators, with the same set of eigenfunctions that arose for the Laplace operator. Thus

1.5 The sum formulae for Kloosterman sums.

69

we can choose the eigenfunctions of the Laplace operator, so that in the basis which was constructed from these, each Hecke operator has a diagonal form. Then these eigenfunctions will be called “Maass waves”. For that, choose T n u j = t j (n)u j , n > 1, j > 0,

(1.20)

T n E(., s) = τ s (n)E(., s).

(1.21)

The eigenvalues t j (n) of the discrete spectrum of the n-th Hecke operator T n are connected with the Fourier coefficients of u j by the equalities ρ j (1)t j (n) = ρ j (n), n > 1, j > 1. (1.22) It is convenient to choose the eigenfunctions so that they will be eigenfunctions of the operator T −1 : (T −1 f )(z) = f (−¯z). Then T −1 u j = ǫ j u j with ǫ j = ±1 and we have ρ j (−n) = ǫ j ρ j (1)t j (n), n > 1, j > 1.

62

(1.23)

1.5 The sum formulae for Kloosterman sums. The natural generalization of the classical Petersson formula Z ( f, Pn ) = f (z)Pn (z, k)yk dµ(z)

(1.24)

Γ/H

= a f (n)(= n − th Fourier coefficient) for any f from the space Mk of the regular cusp forms of weight k is the same formula for the inner product ( f, Un (., s¯)) for an automorphic f from the space of cusp forms M0 of weight zero. It is not hard to show that the non-holomorphic Poincar´e series Un (z, s) may be continued analytically (with its Fourier expansion) in the half plane Re s > 43 (this being based on A. Weil’s estimate for

70

0

INTRODUCTION.

the Kloosterman sum). So, for Re s1 , ℜs2 > 34 , the inner product (Un (., s1 ), Um (., s2 )) is well-defined. Since Un may be expressed as a sum over a group, this inner product is a sum of Kloosterman sums. On the other hand, the inner product (u j , Un ) may be calculated explicitly in terms of Γ-functions and the n-th Fourier coefficient of j-th eigenfunction u j of the automorphic Laplacian, Hence the bilinear form of n-th Fourier coefficients of the eigenfunctions X ρ j (n)ρ j (m)h(χ j ) j>0

for a certain test function h, may be expressed as a sum of Kloosterman sums. Of course, we have, for the case (Un (., s1 ), Um (., s¯2 )), two free variables s1 , s2 and we can try to construct an arbitrary test function in our bilinear form by integration with respect to these variables. This plan was fulfilled in my first paper and in this way, we have following sum formula (referred to by some authors as the “Kuznetsov trace formula”).

63

Theorem 1. Let us assume that the function h(r) of the complex variable r is regular in the strip | Im r| 6 δ with some δ > 12 and |h(r)| ≪ |r|−B for some B > 2 when r → ∞ in this strip. Then, for any integers n, m > 1, we have ∞ X

1 α j t j (n)t j (m)h(χ j ) + 4π j=1 × δn,m = 2 π

Z∞

−∞

Z∞

τ(1/2)+ir (n)τ(1/2)+ir (m)×

(1.25)

−∞

h(r)dr = |ζ(1 + 2ir)|2

r th(πr)h(r)dr +

X1 c>1

c

S (n, m; c)ϕ

√ ! 4π nm , c

where α j = (ch(πχ j ))−1 |ρ j (1)|2 ,

(1.26)

1.5 The sum formulae for Kloosterman sums.

71

ζ is the Riemann zeta function and for x > 0, the function ϕ(x) is defined in terms of h by the integral transform 2i ϕ(x) = pi

Z∞

J2ir (x)

rh(r) dr. ch(πr)

(1.27)

−∞

Identity (1.25) is modified in the following manner, if the integers, n, m on the right-hand side have different signs. Theorem 2. Assume that the function h satisfies the conditions of the preceding theorem. Then, for any integers n, m > 1, we have 1 ǫ j α j t j (n)t j (m)h(χ j ) + 4π j>1

X

=

X1 c>1

c

Z∞

τ(1/2)+ir (n)τ(1/2)+ir (m)

−∞

S (n, −m; c)ψ

√

4π nm c

h(r)dr = |ζ(1 + 2ir)|2 (1.28)

!

where ψ(x), for x > 0, is defined in terms of h by the integral 4 ψ(x) = 2 π

Z∞

K2ir (x)h(r), sh(πr)dr.

(1.29)

−∞

We can invert identities (1.25) and (1.28) and we shall assume that the sum of Kloosterman sums is given rather than the bilinear form in the Fourier coefficients. Theorem 3. Assume that to a function ψ : [0, ∞) → C, the integral 64 transform Z∞ dx h(r) = 2ch(πr) K2ir (x)ψ(x) (1.30) x 0

associates the functions h(r) satisfying the conditions of Theorem 1. Then, for this ψ and for integers n, m > 1, identity (1.28) is satisfied, where h is defined by the integral (1.30).

72

0

INTRODUCTION.

Theorem 4. Let ϕ ∈ C 3 (0, ∞), ϕ(0) = ϕ′ (0) = 0 and assume that ϕ(x), together with its derivatives up to the third order, is O(x−B ) for some B > 2, as x → ∞. Then, for any integers n, m > 1, we have X1 c>1

+

X

c

S (n, m; c)ϕH

√ ! Z∞ δn,m 4π nm J0 (x)ϕ(x)dx+ =− c 2π

(1.31)

0

α j t j (n)t j (m)h(χ j ) +

j>1

1 4π

Z∞

τ(1/2)+ir (n)τ(1/2)+ir (m)

−∞

h(r)dr , |ζ(1 + 2ir)|2

where the functions ϕH (x) and h(r) are defined in terms of ϕ by the integral transforms ∞

Z ∞ X dy ϕH (x) = ϕ(x) − 2 (2k − 1)J2k−1 (x) J2k−1 (y)ϕ(y) . y k=1

(1.32)

0

h(r) =

iπ 2sh(πr)

Z∞

(J2ir (x) − J−2ir (X))ϕ(x)

dx . x

(1.33)

0

It should be useful to note that the transformation ϕ → ϕH in (1.32) is a projection by which, to a given ϕ, one associates its component orthogonal on the semiaxis x > 0 (with respect to the measure x−1 dx) to all the Bessel functions of odd integral order. Together with (1.32), this projection can be defined by the equality ϕH (x) = ϕ(x) − x

Z∞

Z1 ϕ(u)( ξJ0 (xξ)J0 (uξ)dξ)du

= ϕ(x) − x

Z∞

ϕ(u)

0

0

65

(1.34)

0

xJ0 (u)J1 (x) − uJ0 (x)J1 (u) du x2 − u2

and any sufficiently smooth ϕ admits a decomposition ϕ = ϕH + (ϕ − ϕH )

(1.35)

1.6 Some relations with Bessel functions.

73

where ϕ − ϕH is a combination of the Bessel functions defined by (1.32) while ϕH is equal to integral (1.27), in which by h one means the integral transform (1.33) of the function ϕ. The classical Petersson formula vk X j=1

k

|| f j,k ||−2 t j,k (n)t j,k (m) =

= i δn,m + 2π

∞ X 1 c=1

c

S (n, m, c)Jk−1

(1.36) √ ! 4π nm c

(where f j,k form an orthogonal basis in the space Mk of cusp forms of weight k, || f j,k ||2 = ( f j,k, f j,k ) and vk = dim Mk ) allows us to represent the sum √ ! X1 4π nm S ((n, m; c))ϕ (1.37) c c c>1 as a bilinear form in the eigenvalues of the Hecke operators for the case when ϕ may be represented by the Neumann series of the Bessel functions of odd order. Together with (1.31) this gives a representation of the sum (1.37) as a bilinear form of the eigenvalues of the Hecke operators (in all Mk with even k and M0 ) for an arbitrary “good” function ϕ. 1.6 Some relations with Bessel functions. The special case of the following expansion in terms of Bessel functions is the crucial key to prove the identities of the preceding theorems (Really our identities are consequences of a suitable averaging of the initial identity which results from a comparison of two different expressions for the inner product (Un (., 1 + it), Um (., 1 − it)), t ∈ R). Theorem 5.Let f ∈ C 2 (0, ∞), f (0) = 0 and

2 P

r=0

| f (t) (x)| ≪ x−B for some

B > 2, as x → +∞. Let α ∈ R and F(x, t; α) be defined by the equality π π F(x, t; α) = Jit (x) cos (α − it) − J−it (x) cos (α + it). 2 2

(1.38)

74

0

INTRODUCTION.

Then we have the representation f (x) = −

Z∞

F(x, t; α) fˆ(t; α)

0

+

X

t dt + sh(πt)(ch(πt) + cos(πα))

(1.39)

J2n+1−α (x)hn ( f )

n>(α−1)/2

66

where fˆ(t; α) =

Z∞

F(x, t; α) f (x)

dx , x

(1.40)

0

hn ( f ) = 2(2n + 1 − α)

Z∞

J2n+1−α (x) f (x)

dx . x

(1.41)

0

1.7 Some consequences. We have an explicit from of the connection between ρ j (n) and the sum of Kloosterman sums. For this reason, we can transform the information about the Kloosterman sums into information about the Fourier coefficients of the eigenfunctions and vice versa. The first example is the confirmation of the Linnik conjecture. The second is Theorem 6. For any n > 1, as T → +∞, we have p T2 2 α j t (n) = 2 + O(T (log T + d2 (n))) + O( nd3 (n) log2 n) (1.42) j π χ 6T X j

P n d d d1 d2 d3 =n d|n The following (indirect) consequence is due to V. Bykovskij: X d(n2 − D) = T (e1 (D) log T + c0 (D)) + OD ((T log T )2/3 ) (1.43)

where α j = (ch(πχ j ))−1 |ρ j (1)|2 , d3 (n) =

P

=

n6T

where D is a fixed non-square and c1 , c0 are constants.

1.8 The Hecke series.

75

H. Iwaniec proved the excellent estimate for the number πΓ (X) of the conjugate primitive hyperbolic classes {P0 } with NP0 < X: πΓ (X) = liX + O(X (35/48)+ǫ ) for any ǫ > 0.

(1.44)

The proof is based essentially on the sum formulae for Kloosterman sums. We have some progress in the additive divisor problem (H. Iwaniec and J.-M. Deshoouillers and myself): X d(n)d(n + N) = T P2 (log T, N) + ON (T log T )2/3 ) (1.45) n6T

where P2 (z, N) is a polynomial in z of degree 2. 1.8 The Hecke series. To each eigenfunction of the ring of Hecke operators (regular in the case of Mk with k > 0 and real analytic in the case of M0 ), we associate 67 the Dirichlet series whose n-th coefficient is the eigenvalue of the n-th Hecke operator. As we have relations connecting the spectra of the Hecke operators with the Fourier coefficients of the eigenfunctions, these series differ, only upto normalization, from the series associated by Hecke to regular parabolic cusps by means of the Mellin transform. We set H j,k (s; x) = H j (s; x) =

∞ X

n=1 ∞ X

e(nx)n−s t j,k (n),

(1.46)

e(nx)n−s t j (n),

n=1

and we denote by Lv (s; x) the Hecke series associated with the EisensteinMaass series E(z, v), X Lv (s; x) = e(nx)n−s τv (n). (1.47) n>1

76

0

INTRODUCTION.

For x = 0, these series are denoted by H j,k (s), H j (s), Lv (s) respectively. Theorem 7. Let x be rational, x =

d c

with (d, c) = 1, c > 1. Then

(1) H j,k (s, d/c), H j (s, d/c) are entire functions of s, (2) for v , 12 , the only singularities of Lv (s, d/c) are simple poles at the point s1 = v + 21 and s2 = 23 − v with residues c−2v ζ(2v) and c2v−2 ζ(2 − 2v); the function ((S − 1)2 − (v − 12 )2 )Lv (s, d/c) is an entire function of s. For what follows, it is convenient to set 1 1 22µ−1 Γ(u + v − )Γ(u − v + ); (1.48) π 2 2 as a consequence of the functional equation for the gamma function, this function for any u, v ∈ C satisfies the relation γ(u, v) =

γ(u, v)γ(1 − u, v) = −(cos2 πu − sin2 πv)−1 .

(1.49)

Theorem 8. The Hecke series have functional equations of the Riemann type; moreover 1) for even integers k > 12 and for (d, c) = 1, c > 1, we have H j,k (s, d/c) = −(4π/c)2s−1 γ(1 − s, k/2) cos(πs)H j,k (1 − s, −d′ /c) (1.50) where d′ is defined by the congruence dd′ ≡ 1(mod c) 2) with the same d′ , Lv (s, d/c) = (4π/c)2s−1 γ(1 − s, v){− cos(πs)Lv (1 − s, −d/c)+ (1.51) + sin(πv)Lv (1 − s, d′ /c)}, 1 H j (s, d/c) = (4π/c)2c−1 γ(1 − s, + iχ j ) (1.52) 2 {− cos(πs)H j (1 − s, d′ /c) + ǫ j ch(πχ j )H j (1 − s, d/c)}. 68

1.9 The spectral mean of Hecke series.

77

We conclude with the simple but important consequence of the multiplicative relations (1.19) for Hecke operators : for Re s > 1+| Re v− 21 |, we have ∞ X τv (n)t j (n) 1 1 1 = H j (s + v − )H j (s − v + ). (1.53) s n ζ(2s) 2 2 n=1 If we replace t j (n) by τµ (n) (that corresponds to the continuous spectrum of the Hecke operators) then the well-known Ramanujan identity will arise instead of (1.53): ∞ X τv (n)τµ (n) n=1

=

ns

=

(1.54)

1 ζ(s + v − µ)ζ(s + µ − v)ζ(s − v − µ + 1)ζ(s + v + µ − 1). ζ(2s)

For this reason, equality (1.53) is a direct generalization of the Ramanujan identity; both will be essential for the estimate of the eighth moment of the Riemann zeta-function. 1.9 The spectral mean of Hecke series. Let N > 1 be an integer and let s, v be complex variables. We set X 1 1 ZN(d) (s, v; h) = α j t j (N)H j (s + v − )H j (s − v + )h(χ j ) (1.55) 2 2 j>1 X 1 1 ZN(d) (s, v; h) = ǫ j α j t j (N)H j (s + v − )H j (s − v + )h(χ j ) (1.56) 2 2 j>1 (with α j = (ch(πχ j ))−1 |ρ j (1)|2 ). Here the summation is over the positive discrete spectrum of the automorphic Laplacian and one assumes that its eigenfunctions have been selected in such a manner that they are at the same time eigenfunctions of the ring of Hecke operators and of the reflection operator T −1 (ǫ j = ±1 are the eigenvalues of T −1 ). Further, we define the square mean of the Hecke series over the continuous spectrum by the equality ZN(c) (s, v; h) =

(1.57)

78

0 1 = π

Z∞

−∞

ζ(s + v −

1 2

INTRODUCTION.

+ ir)ζ(s + v − 21 − ir)ζ(s − v + 21 + ir)ζ(s − v + ζ(1 + 2ir)ζ(1 − 2ir)

1 2

− ir)

× τ(1/2)+ir (N)h(r)dr 69

with the stipulation that, by means of integral (1.57), the function ZN(c) is defined under the conditions 1 1 Re(s + v − ) < 1, Re(s − v + ) < 1. 2 2 If any one of the points s ± (v − 21 ) lies to the right hand side of the unit line, then the integral (1.57) defines another function, connected with ZN(c) by the Sokhotskii formulae. Fro example, if by Z˜ N(c) , we denote the function which is defined by (1.57) with Re s > 1, Re v = 12 , then a simple computation gives 1 Z˜ N(c) (s, v; h) = ZN(c) (s, v; h) + 4ζN (s, v)h(i(s − v − ))+ 2 3 +4ζN (s, 1 − v)h(i(s + v − )) 2 where we have introduced the notation ξN (s, v) =

ζ(2s − 1)ζ(2v) τ s−v (N) ζ(2 − 2s + 2v)

and the regularity strip of h is assumed to be sufficiently wide for the right hand side to make sense. Now we need the mean with respect to the weights of the Hecke series associated with regular cusp forms. For an integer k > 1, we set v

ZN,k (s, v) = 2(−1)k

2k Γ(2k − 1) X |α j,2k (1)|2 t j,2k (N) (4π)2k j=1

(1.58)

1 1 ×H j,2k (s + v − )H j,2k (s − v + ) 2 2 where t j,2k (N) is the eigenvalue of the N-th Hecke operator in the space M2k of regular cusp forms of weight 2k, v2k = dim M2k ; the empty sum for 1 6 k 6 5 and k = 7 is assumed to be equal to zero.

1.10 The convolution formula.

79

Assume now that h∗ = {h2k−1 }∞ k=1 is a sequence of sufficiently fast decreasing numbers; we define the mean of the Hecke series with respect to weights by the equality X (p) ZN (s, v; h∗ ) = h2k−1 ZN,k (s, v) (1.59) k>6

1.10 The convolution formula. Some of the consequences of the algebra of modular forms are the so 70 called “exact formulae”, an example of which is the identity N−1 X n=1

σ3 (n)σ3 (N − n) =

X 1 (σ7 (N) − σ3 (N)), σa (n) = d! 120 d|n

(1.60)

A source of similar identities is the obvious assertion that the product of modular forms of weight k and l is a modular form of weight k + l. There are analogues of these identities for the real analytic Eisenstein series of weight zero. For an integer N > 1, we associate to a pair of series E(z, s) and E(z, v) the expression of convolution type WN (s, v; w0 , w1 ) = N s−1

∞ X n=1

p τv (n)(σ1−2s (n − N)w0 ( n/N)

(1.61)

p +σ1−2s (n + N)w1 ( n/N)) where σ1−2s (0) means ζ(2s−1) and w0 , w1 are assumed to be sufficiently smooth and sufficiently fast decreasing for x → +∞. Theorem 9. Assume that the functions w0 , w1 are continuous on the semiaxis x > 0 together with derivatives up to the fourth order, w j (0) = w′j (0) = 0 for j = 0, 1 and that, for x → +∞, the functions w j (x) as well as their derivatives up to the third order are O(x−B ) for some B > 4. Then, for any integer N > 1 and s, v ∈ C satisfying Re v = 21 , 12 < Re S < 1, we have WN (s, v; w0 ; w1 ) = ZN(d) (s, v; h0 ) + ZN(d) (s, v; h1 )+

(1.62)

80

0

INTRODUCTION.

1 (p) ZN(c) (s, v; h0 + h1 ) + ZN (s, v; h∗ ) + ζN (s, v)V( , v)+ 2 1 ζN (s, 1 − v)V( , 1 − v) + ζN (1 − s, v)V(s, v) + ζN (1 − s, 1 − v)|× 2 V(s, 1 − v) where ζN (s, v) = V(s, v) = 2

Z∞

ζ(2s)ζ(2v) τ s+v (N), ζ(2s + 2v)

(|1 − x2 |1−2s )w0 (x) → (1 + x2 )1−2s w1 (x))x2v dx

(1.63) (1.64)

0

! ! h0 w0 and the column vector h(r; s, v) = is defined in terms of w = h1 w1 by the integral transform h(r; s, v) = π

Z∞ 0

!  0 k0 (x, 21 + ir) x 2s−1 × 1 4π 0 k1 (x, 2 + ir)

Z∞

! k0 (xy, v) k1 (xy, v) w(y)ydydx k1 (xy, v) k0 (xy, v)

(1.65)

0

71

with the kernels J2v−1 (x) − J1−2v (x) , 2 cos(πv) 2 k1 (x, v) = sin(πv)K2v−1 (x). π k0 (x, v) =

(1.66)

Finally, the coefficients of the mean of the regular forms ZN(ρ) are given by the relations

×

Z∞ 0

h2k−1 = 2(2k − 1)×  x 2s−1 Z∞ (k0 (xy, v)w0 (y) + k1 (xy, v)w1 (y))ydy dx. J2k−1 (x) 4π 0

1.11 Some consequences of the convolution formula.

81

1.11 Some consequences of the convolution formula. The first example of the use of (1.62) is the additive divisor problem; if we choose s √= v = 1/2, w1 = √0 and w0 so that it is close to 1 in the interval (0, T/N) (so that w0 ( n/N) will be close to 1 for n 6 T ), then the left hand side of (1.62) gives the sum on the left side of (1.45). Terms with the integral (1.64) are leading terms and all other terms give the remainder term. Of course, the asymptotic formula for the additive divisor problem is crucial for the investigation of the fourth power moment of the Riemann zeta-function. A consequence of (1.62) in this direction is the following Theorem. (N. Zavorotnyi, 1987). Let T → +∞; then, for any ǫ > 0, we have ZT 1 (1.67) |ζ( + it)|4 dt = T P4 (log T ) + O(T 2/3+ǫ ) 2 0

where P4 (z) is a polynomial in z of the fourth degree with constant coefficients. We can consider the functions h0 and h1 in (1.62) as given; the fol- 72 lowing unusual integral transform is useful to invert (1.62). Let us define the matrix kernel K(x, v) by the equality k (x, v) k1 (x, v) K(x, v) = 0 k1 (x, v) k0 (x, v)

!

(1.68)

with k0 , k1 from (1.66). Now we shall consider the matrix equation

w(x) =

Z∞

√ K(xy, v)u(y) xydy

0

! ! u0 w0 (x). (x), u = where w = u1 w1

(1.69)

82

0

INTRODUCTION.

Theorem 10. Let Re v = 12 and w ∈ L2 (0, ∞) in the sense that w0 , w′1 ∈ L2 (0, ∞). Then there exists a unique solution u in L2 (0, ∞) of the equation (1.69) and this solution is given by the formula

u(x) =

Z∞

√ K(xy, u)w(y) xydy

(1.70)

0

where the integral is understood in the mean-square sense. Now, as a special case of the convolution formula (1.62), we have the following asymptotic formulae. Theorem 11. Let T → +∞. Then for a fixed σ and t ∈ R with 1, we have ζ(2 − 2σ) T 2 T2 α j |H j (σ + it)| = 2 (ζ(2σ) + 2(1 − σ) 2π π χ 6T X

2

j

while, for σ =

1 2

!1−2σ

1 2

1, (2π)2s−1 A00 (r, y; s, v) = ! ( 1 iy1−2s Γ(a)Γ(b) F a, b; c; 2 cos π(s + ir)− = 2sh(πr) yc−1 Γ(c) y ! ) ′ ′ Γ(a )Γ(b ) 1 − c′ −1 ′ F a′ , b′ ; c′ ; 2 cos π(s − ir) y Γ(c ) y

(1.77)

At the same time, for all y > 0, we have (2π)2s−1 A00 (r, y; s, v) = sin(πs)Γ(2s − 1)y2v |1 − y2 |1−2s × Γ(a)Γ(a′ )Γ(b)Γ(b′ ) × F(1 − b, 1 − b′ ; 2 − 2s; 1 − y2 ) + 2πΓ(2s) cos(πs) (ch2 πr + sin2 πv − sin2 πs)y2v F(a, a′ ; 2s; 1 − y2 )

(1.78)

where, in the first term, the absolute value sign combines the two cases y < 1 and y > 1. Proposition 2. With the same parameters, we have

74

84

0 2s−1

(2π)

INTRODUCTION.

Z∞

x 1 + ir)k1 (xy, v)( )2s−1 dx (1.79) 2 2 0 ! ( iy1−2s sin(πv) Γ(a)Γ(b) 1 = F a, b; c; − 2 2sh(πr) yc−1 Γ(c) y !) Γ(a′ )Γ(b′ ) 1 − c′ −1 ′ F a′ , b′ ; c′ ; − 2 y Γ(c ) y

A01 (r, y; s, v) := πy

k0 (x,

Proposition 3. The kernel A10 is defined by the relation 2s−1

(2π)

A10 (r, y; s, v) := πy

Z∞

k1 (x,

x 1 + ir)k1 (xy, v)( )2s−1 dx (1.80) 2 2

0

Γ(a)Γ(a′ )Γ(b)Γ(b′ ) = ch(πr) sin(πv)y2v F(a, a′ ; 2s; 1 − y2 ). πΓ(2s) Proposition 4. With the parameters (1.74)-(1.75), we have (2π)2s−1 A11 (r, y; s, v) := πy

Z∞

k1 (x,

x 1 + ir)k0 (xy, v)( )2s−1 dx = 2 2

0

( Γ(a)Γ(a′ ) 2v ch(πr) y F(a, a′ ; 2v; −y2 )− = 2 cos(πv) Γ(2v) ) Γ(b)Γ(b′ ) 2−2v ′ 2 − y F(b, b ; 2 − 2v; −y ) Γ(2 − 2v)

(1.81)

and, at the same time, (2π)2s−1 A11 (r, y; s, v) = ( ! 1 iy1−2s Γ(a)Γ(b) F a, b; c; − = cos π(s + ir)− 2sh(πr) yc−1 Γ(c) y2 ! ) 1 Γ(a′ )Γ(b′ ) ′ ′ ′ − c′ −1 ′ F a ; b ; c ; − 2 cos π(s − ir) . y Γ(c ) y

(1.82)

1.12 The explicit formulae for the transformation (1.65)

85

Proposition 5. Let us write the quantities h2k−1 in (1.62) as h2k−1 (s, v) =

Z∞

(A0k (y; s, v)w0 (y) + A1k (y; s, v)w1 (y))dy.

(1.83)

0

Then, for 0 < y < 1,

75

(1.84) (2π)2s−1 A0k (y; s, v) = ( 2k − 1 Γ(s + v − 1 + k) y2v F(k + s + v − 1, s + v − k; 2v; y2 )− cos(πv) Γ(2v)Γ(1 − s − v + k) ) Γ(k + s − v) 2−2v 2 − y F(k + s − v, s + 1 − v − k; 2 − 2v; y ) Γ(2 − 2v)Γ(v − s + k) and, for y > 1, (2π)2s−1 A0k (y; s, n) =

(1.85)

(−1)k−1

sin(πs) Γ(k + s + v − 1)Γ(k + s − v) × Γ(2k − 1) ! 1 F k + s + v − 1, k + s − v; 2k; 2 . y

πy2k+2s−2

For the second kernel in (1.83), we have (1.86) (2π)2s−1 A1k (y; s, v) = sin(πv) Γ(k + s + v − 1)Γ(k + s − v) = − 2k+2s−2 × Γ(2k − 1) πy ! 1 F k + s + v − 1, k + s − v; 2k; − 2 y

Part II. The eighth moment of the Riemann zeta-function.

86

0

INTRODUCTION.

2.1 The result and a rough sketch of the proof. Since the question about the true order of zeta-function on the critical line is open even today - and it will be so in the foreseeable future-, a sizeable part of the theory of the Riemann zeta-function is an attempt to present the asymptotic mean value 1 T

ZT

1 |ζ( + it)|2k dt, k = 1, 2, . . . 2

(2.1)

0

The case k = 4 will be investigated here; for this case, the following new estimate will be given. 76

Theorem 12. There is an absolute constant B such that ZT

1 |ζ( + it)|8 dt ≪ T (log T )B 2

(2.2)

0

when T → +∞. Furthermore, the same estimate is valid for the fourth power moment of the Hecke series of the discrete spectrum of the automorphic Laplacian. Namely, we have Theorem 13. For every fixed j > 1 with the same B as in (2.2), we have αj

ZT

1 |H j ( + it)|4 dt ≪ T (log T )B−6 , T → +∞. 2

(2.3)

0

To give these estimates we shall consider the fourth spectral moment of the Hecke series over the discrete and continuous spectrum. The one over the discrete spectrum is defined by ! dis s, v Z h = (2.4) ρ, µ X H j (s + v − 1 )H j (s − v + 1 )H j (ρ + µ − 1 )H j (ρ − µ + 1 ) 2 2 2 2 h(χ j ) αj ζ(2s)ζ(2ρ) j>1

2.1 The result and a rough sketch of the proof.

87

with α j = (ch(πχ j ))−1 |ρ j (1)|2 .

(2.5)

Of course, this function results from the following summation (see the generalized Ramanujan identity (1.53)):   !  X τ (n) τµ (m) X s, v v h =  α j t j (n)t j (m)h(χ j ) (2.6) Z dis  ρ, µ ns mρ  j>1 n,m>1 when Re s > Re v − 21 | + 1, Re ρ > | Re µ − 12 | + 1. The function ! dis s, v ζ(2s)ζ(2ρ)Z h ρ, µ

is regular in some domain of the kind Re s > s0 ’ Re ρ > ρ0 , where s′0 ρ0 depend on the order of decay of the !function |h(r)| for |r| → +∞. s, v h the expression obtained on replacWe shall denote by Z˜ dis ρ, µ ing α j by ǫ j α j in (2.6):   !  X τ (n) τµ (m) X s, v v h =  ǫ j α j t j (n)t j (m)h(χ j ) (2.7) Z˜ dis  ρ, µ n2 mρ  j>1 n,m>1

In the same manner as in (2.6), we define the fourth spectral moment 77 over the continuous spectrum by Z0con

! Z∞ 1 1 h(r)dr 1 s, v h = Z(s; v, + ir)Z(ρ; µ, + ir) ρ, µ 4π 2 2 ζ(1 + 2ir)ζ(1 − 2ir) −∞

(2.8) where we assume Re(s ± (v − > 1, Re(ρ ± (µ − > 1 and the notation z is introduced for the right side of the well-known Ramanujan identity: 1 2 ))

z(s; v, µ) =

∞ X τv (n)τµ (n) n=1

ns

1 2 ))

(2.9)

88

0

INTRODUCTION.

1 ζ(s + v − µ)ζ(s − v + µ)ζ(s + v + µ − 1)× ζ(2s)

=

ζ(s − v − µ + 1). Finally, for the given sequence h∗ = {h2k−1 }∞ k=1 we define the fourth moment of the Hecke series over regular cusp forms ! ∞ X cusp s, v ∗ Z h =2 (−1)k/2 hk−1 (2.10) ρ, µ k=2 k≡0(mod 2)

×

vk X j=1

α j,k

X τv (n)τµ (m) t j,k (n)t j,k (m), n s mρ n,m>1

where vk = dim Mk is the dimension of the space Mk of the regular cusp forms of weight k and t j,k (n) are the eigenvalues of the n-th Hecke operators in this space Mk . The quantities α j,k are the normalized coefficients; if the functions f j,k form an orthonormal basis in Mk , then α j,k =

Γ(K − 1) |a j,k (1)|2 , (4π)k

(2.11)

where a j,k (1) is the first Fourier coefficient of f j,k . Together with (2.10), we have ! vk X X cusp s, v ∗ h =2 (−1)k/2 hk−1 α j,k × (2.12) Z ρ, v H j,k (s + v −

78

k>2 k≡0(mod 2) 1 1 2 )H j,k (s − v + 2 )H j,k (ρ +

ζ(2s)ζ(2ρ)

j=1

µ − 12 )H j,k (ρ − µ + 21 )

;

this equality is quite similar to (2.4). Now we shall describe the main idea. We shall consider the double Dirichlet series ! ∞ X τv (n)τµ (m) (±) (±) s, v (2.13) Km,n (ϕ), L ϕ = ρ, v n s mρ n,m=1

2.1 The result and a rough sketch of the proof.

89

where the coefficients are sums of Kloosterman sums with the smooth “test” function ϕ: √ ! X1 4π nm (±) S (n, ±m; c)ϕ . (2.14) Km,n (ϕ) = c c c>1 Since there is a functional equation of the Riemann type for the Dirichlet series ∞  a X a = n−s τv (n)e(n ), e(x) := e2πix , Lv S , c c n=1 when a, c ∈ Z are coprime and this equation connects Lv (S , ac ) with ′ Lv (1 − s, ± ac′ ), aa′ ≡ 1(mod c), a functional equation has to exist for the ! s, v (±) functions L ϕ ; it will connect this function with the functions ρ, µ ! ρ, v (±) of the same kind L ϕ for an appropriate ϕ. As a consequence s, µ of the sum formula for Kloosterman sums, it means that there is a functional equation for the function ! ! ! s, v dis s, v con s, v Z h =Z h +Z h ; (2.15) ρ, µ ρ, µ ρ, µ roughly speaking, this equation (which will be written below in detail) is the result of the exchange of s and ρ and the replacement of h by some integral transform. Now, on the left side of this functional equation, for the special case 1 1 , ρ = v = + it, t ∈ R is large and positive, 2 2 we have in the continuous spectrum the product s=µ=

(2.16)

1 1 1 1 ζ 3 ( + it + ir)ζ 3 ( + it − ir)ζ( − it + ir)ζ( − it − ir) (2.17) 2 2 2 2 and the other product will be on the right side; namely, we have therein 1 1 1 1 ζ 3 ( + ir)ζ 3 ( − ir)ζ( + 2it + 2ir)ζ( + 2it − 2ir). 2 2 2 2

(2.18)

90

0

INTRODUCTION.

After this specialization, we shall choose the special function h essentially as exp(−αr2 ) with a fixed positive α. Then the essential part of the interval of the integration is |r| ≪ (log t)(1/2) . The length of this interval is small in comparison to the large t and for this reason, (using the Riemann functional equation), we can reduce the main term of our product on the left side to the form 1 1 |ζ 4 ( + it + ir)ζ 4 ( + it − ir)|. 2 2 79

It means that we may hope to estimate the integral Zǫ Z∞

−ǫ 0

1 1 ωT (t)|ζ 4 ( + it + ir)ζ 4 ( + it − ir)|dtr2 dr 2 2

(2.19)

for arbitrary small positive ǫ, if ωT (t) is the smooth function which is not zero for t ∈ (T, 2T ) only and close to 1 when 54 T 6 t 6 74 T (see picture). figure 79 page The main term will be close to the integral ǫ

3

Z∞

1 ωT (t)|ζ( + it)|∞ dt 2

0

if ǫ ≪ (log T )−2 (see subsection (2.4)); so we have the eighth moment of the Riemann zeta-function here. At the same time, the contribution of the discrete spectrum is positive too. Hence the desired conclusion follows if the integrals on the right side can be estimated with sufficient accuracy. But the integrand for these integrals contains one Hecke series only; so the integration may be done asymptotically. As a result, we shall reduce the problem of the estimate of the eighth power moment to the problem of the estimate for the fourth spectral moment. It is sufficient to prove our theorem for the latter.

2.2 The first functional equation

91

In the conclusion of the introduction of the second part, we shall note that the estimate 1 |ζ( + it)| ≪ |t|1/8 (log |t|)B1 , , t → ±∞, 2 follows from (2.2), B1 = 81 B + 21 . It is preceptibly better than the last achievement in the long chain of the results of the kind |ζ( 12 + it) ≪ |t|γ . 2.2 The first functional equation Since the sum Z dis + Z con is connected with the sum of Kloosterman sums, we shall consider the triple sum L

(±)

! ∞ X τv (n)τµ (m) (±) s, v ϕ = Kn,m (ϕ). ρ, µ n s mρ n,m=1

(3.1)

Here the notation (2.14) is used and we assume that a function ϕ is “good” namely, the Mellin transform ϕ(u) ˆ

ϕ(u) ˆ =

Z∞

ϕ(x)xu−1 dx,

(3.2)

0

is regular in the strip −α0 6 Re u 6 α1 with positive α0 , α1 and |ϕ(u)| ˆ decreases sufficiently rapidly in this strip. For this reason, we can write 1 ϕ(x) = iπ

Z∞ Z∞

−2u ϕ(2u)x ˆ du, x > 0,

(3.3)

iπ (α)

where

R∞

stands for the integral over the line Re u = α. As we have

(α)

|S (n, m; c)| ≪ c1/2 (n, m, c)d(c),

80

92

0

INTRODUCTION.

the triple sum (3.1) converges absolutely if α < − 14 and both Re(s + α), Re(ρ + α) are larger than 1. For this case, we have, for the sum (3.1), the following expression ! 1 X1 (±) s, v × (3.4) L ϕ = ρ, µ iπ c>1 c Z X a d Lv (s + u, )Lµ (ρ + µ, ± )(c/4π)2u ϕ(2µ)du. ˆ c c ad≡1(mod c) (α)

Now we shall integrate over the line Re u = α0 where α0 will be chosen so that both Re(s + α0 ), Re(ρ + α0 ) are negative. Taking into account the contribution from the poles, we have (±)

L

! s, v ϕ ρ, µ X X

(3.5)

(4π)2s−1 ζ(2v) 1 ˆ + 1 − 2s)+ L (ρ − s + v + , ±d/c)ϕ(2v 2s 2v µ 2 c (4π) c>1 ad≡1(mod c) ! 3 ζ(2 − 2v) L (ρ − s + + − v, ±d/c) ϕ(3 ˆ − 2v − 2s) + µ (4π)2−2v 2 ! 1 (4π)2ρ−1 ζ(2µ) Lv (s − ρ + µ + , a/c) × + 2 c2ρ (4π)2µ !) 3 ζ(2 − 2µ) Lv (s − ρ + − µ, a/c)ϕ(3 ϕ(2µ ˆ + 1 − 2ρ) + ˆ − 2µ − 2ρ) + 2 (4π)2−2µ Z X c 1 X1 ˆ Lv (s + u, a/c)Lµ(ρ + u, ±d/c)( )2u ϕ(2u)du. + iπ c>1 c ad≡1(mod c) 4π =2

(

α0

81

In the last term, we shall use the functional equation (1.51). If the sign “plus” is taken, then it gives for our sum the expression √ ! X 1 X τv (n)τµ (m) 4π nm (S (n, m; c)Φ0 + c n,m>1 nρ m s c c>1 √ !! 4π nm +S (n, −m; c)Φ1 c

(3.6)

2.2 The first functional equation

93

where Φ0 = Φ0 (x; s, v; ρ, µ) = Z 1 2s+2ρ−2 γ(1 − s − u, v)γ(1 − ρ − u, µ)× = x iπ

(3.7)

(α0 )

(cos π(s + µ) cos π(ρ + u) + sin(πv) sin(πµ))x2u ϕ(2u)du, ˆ Φ1 = Φ1 (x; s, v; ρ, µ) = =−

1 2s+2ρ−2 x iπ

Z

(3.8)

γ(1 − s − u, v)γ(1 − ρ − u, µ)(sin(πµ)×

(α0 )

cos π(s + u) + sin(πv) cos π(ρ + µ)x2u ϕ(2u)du, ˆ Of course, when the sign “minus” is taken in (3.5), then the same function s Φ0 and Φ1 are the coefficients, but Φ0 will occur with S (n, −m; c) and Φ1 with S (n, m; c). We have Φ j (x) = O(x2 min(Re s,Re ρ) ) as x → 0+ and these functions are bounded when x is large. For this reason, the triple sums in (3.6) converge absolutely and we can again interchange the order of the summations. Hence we have, in (3.7), the sum ! ! (+) ρ, v (−) ρ, v L Φ +L Φ (3.9) s, µ 0 s, µ 1

for the case “+” on the left side (3.5) and ! ! (+) ρ, v (−) ρ, v L Φ +L Φ s, µ 1 s, µ 0

(3.10)

for the other case. Now we are ready to give the first functional equation.

Theorem 14. Let Re v = Re µ = 12 and let, for some positive δ < 14 , the variables s, ρ satisfy 45 < Re s, Re ρ < 54 +δ. Let ϕ : [0, ∞) → C have the Mellin transform ϕ(u) ˆ such that ϕ(u) ˆ is regular for − 32 − 2δ 6 Re u 6 2. Then we have ! ! ! (+) s, v (+) ρ, v (−) ρ, v L ϕ =L ϕ L ϕ + (3.11) ρ, µ s, µ 0 s, µ 1

82

94

0

INTRODUCTION.

2(4π)2s−1 ζ(2v) z(ρ + v; s, µ)ϕ(2v ˆ + 1 − 2s)+ ζ(2s) (4π)2v ! ζ(2 − 2v) + z(ρ + 1 − v; s, µ) ϕ(3 ˆ − 2v − 2s) + (4π)2−2v 2(4π)2ρ−1 ζ(2(µ) z(s + µ; ρ, v)ϕ(2µ ˆ + 1 − 2ρ)+ + ζ(2ρ) (4π)2µ ! ζ(2 − 2µ) z(s + 1 − µ; ρ, v)ϕ(3 ˆ − 2µ − 2rho) , + (4π)2−2µ

+

where Φ0 , Φ1 are defined by the following integral transformations 2s+2ρ−2

Φ0 (x) ≡ Φ0 (x; s, v; ρ, µ) = x

k1 (ξ, v)k1 (η, µ))ϕ(ξη/x)ξ

"∞

0 1−2s 1−2ρ

η

Φ1 (x) ≡ Φ1 (x; s, v; ρ, µ) = x2s+2ρ−2

(k0 (ξ, v)k0 (η, µ)+

(3.12)

dξdη

"∞

(k0 (ξ, v)k0 (η, µ)+

(3.13)

0

k1 (ξ, v)k0 (η, µ))ϕ(ξη/x)ξ 1−2s η1−2ρ dξdη. Of course, it is the same as what we have in (3.5). If Re ρ > Re s, then in the first term on the right side of (3.5), one has the sum ∞ X 1 X τµ (n) S (0, n; c) = 2s ρ−s+v+(1/2) c n=1 n c>1

= 83

(3.14)

∞

1 X τµ (n)σ1−2s (n) z(ρ + v; s, µ) = ζ(2s) n=1 nρ−s+v+(1/2) ζ(2s)

On the right side, we have a meromorphic function of ρ in the halfplane Re ρ > 21 ; so this equality holds for the analytic continuation of the initial sum ! X 1 X 1 d (3.15) Lµ ρ − s + v + , 2 c c2s (d,c)=1 c>1

2.3 The main functional equation: the preparations.

95

if we can be sure that this function is meromorphic not only for Re ρ > Re s. It is sufficient for this to know that Lµ (w, dc ) as a function of c is bounded in the mean when Re w > 21 (except at the poles). But this fact is a consequence of the Bombieri-Vinogradov inequality which asserts that 2 P+Q X X P+Q X X nd ) b(n)e( ) ≪ max(Q, M 2 ) |b(n)|2 (3.16) n=P c n=P 16c6M (d,c)=1 for an arbitrary sequence of complex numbers b(n). Now one can check that relations (3.12) - (3.13) and (3.7) - (3.8) are idential. We have the tabular integrals Z∞  πw  w  3 w−1 , 0 < Re w < , , v cos (3.17) k0 (x, v)x dx = γ 2 2 2 0

Z∞ 0

k1 (x, v)xw−1 dx = γ

w  , v sin(πv), Re w > 0, 2

(3.18)

After writing ϕ in (3.12)-(3.13) as the Mellin integral, !2u  ξη  1 Z x ϕ = ϕ(2u) ˆ du, x iπ ξη we shall come to an absolutely convergent triple integral if ! 3 3 max − Re s, − Re ρ < Re u < min(1 − Re s, 1 − Re ρ). 4 4 Hence there is a non-empty strip where we can integrate in any order; this gives our relations for Φ0 and Φ1 . 2.3 The main functional equation: the preparations. For the given function h(r) and the sequence h∗ := {h2k−1 }∞ k−1 , we shall consider the function ! ! ! ! s, v cusp s, v ∗ con s, v dis s, v h . h +Z h + Z0 h =Z Z ρ, µ ρ, µ ρ, µ ρ, µ (4.1)

96 84

0

INTRODUCTION.

When Re s, Re ρ > 1, this function is equal to Z∞ ∞ X τv (n)τµ (m) (+) δn,m {Kn,m (ϕ) + 2 rth(πr)h(r)dr} = n s mρ π n,m=1

(4.2)

−∞

= L(+)

! Z∞ 1 s, v ϕ + 2 z(s + ρ; v, µ) rth(πr)h(r)dr ρ, µ π −∞

where ϕ corresponds to h and h∗ in the sense of Theorems 1 and 4. Our intention must be clear now; we shall use the first functional equation for L(+) and after this, the analytic continuation of both sides will be carried out. Firstly, it is convenient to write the analytic continuation for the function Z0con . Let us denote by Z con the integral in which (under the usual conditions Re v = Re µ = 12 ) we have Re s < 1, Re ρ < 1. Then Z0con and Z con are connected by the following relation. Proposition 6. Let h be a regular function on the sufficiently wide strip | Im r| 6 ∆, ∆ > 21 . Then for Re s > 1, Re ρ < 1, the meromorphic ! s, v con continuation of Z0 h is given by the equality. ρ, µ ! ! s, v con s, v h −Z h + ρ, µ ρ, µ ) ( 1 ζ(2s − 1) ζ(2v)z(ρ; µ, 1 − s + v) h(i(s − v − )) + + ζ(2s) ζ(2 + 2v − 2s) 2 ζ(2 − 2v)z(ρ; µ, 2 − s − v) 3 + h(i(s + v − ))+ ζ(4 − 2v − 2s) 2 ( ζ(2ρ − 1) ζ(2µ)z(s; v, 1 − ρ + µ) 1 + h(i(ρ − µ − ))+ ζ(2ρ) ζ(z + 2µ − 2ρ) 2 ) 3 ζ(2 − 2µ)z(s; v, 2 − ρ − µ) h(i(ρ + µ − )) . + ζ(4 − 2µ − 2ρ) 2

Z0con

(4.3)

Really Z0con is a Cauchy integral, because ζ has only a simple pole.

2.3 The main functional equation: the preparations.

97

so the poles of z(s; v, 21 + ir) are the points r j , 1 6 j 6 4, with ir1 =

3 1 − s + v, ir2 = − s − v, r3 = −r1 , r4 = −r2 . 2 2

When Re s > 1 the points r1 , r2 are lying above the real axis and if Re s < 1, they are below the same. Now one can deform the path of integration (see picture; the deformation must be so small that the functions ζ(1 + ±2ir) have no zeros inside the lines; it is possible, since the 85 Riemann zeta-function has no zeros on the line Re s = 1) and the desired conclusion is the result of the direct calculation of the residues. figure 85 page ! ρ, v The next step is the representation of the functions Φ s, µ j as a bilinear form in the eigenvalues of the Hecke operators. For this, we need to consider the integral transforms of Theorems (4) and (1). The situation now is the following : for a given h, we define ϕ by the transformation (1.27), or what is the same, by the equality L(±)

1 ϕ(x) = π

Z∞

−∞

k0 (x,

1 + iu)uth(πu)h(u)du, 2

(4.4)

and thereafter, we should calculate the integrals Φ0 and Φ1 in (3.12) and (3.13) with this ϕ and finally the integral transformations h0 (r) ≡ h0 (r; s, v; ρ, µ) = π

Z∞

k0 (x,

dx 1 + ir)Φ0 (x) , 2 x

(4.5)

h1 (r) ≡ h1 (r; s, v; ρ, µ) = π

Z∞

k1 (x,

1 dx + ir)Φ1 (x) . 2 x

(4.6)

0

0

In order to obtain an asymptotic estimate, it is preferable to diminish the length of the sequence of these integral transformations; we give the results in the following

98

0

INTRODUCTION.

Proposition 7. Assume that the function h(u) is even and regular in the strip | Im u| 6 32 and h has zeros at u = ± 2i . Let |h| decrease as O(|u|−B ) for some B > 4 when |u| → ∞ with | Im u| 6 32 . Then the function h0 is given by the integral transform 2 h0 (r) = 2 π

Z∞

B0 (r; u; ρ, v, µ; s)u th(πu)h(u)du

(4.7)

−∞

where, with the notation from (1.78), (1.79), we have, for Re v = Re µ = 1 1 2 , 2 6 Re s, Re ρ < 1 B0 (r, u; ρ, v, µ; s) =

Z∞

1 (A00 (r, ξ, ρ, v)A00 (u, ; 1 − ρ, µ)+ ξ

(4.8)

0

1 A01 (r, ξ; ρ, v)A01 (u, ; 1 − ρ, µ))ξ 2ρ−2s−1 dξ ξ 86

and here B0 (r, u; ρ, v, µ; s) = B0 (r, u; s, µ, v; ρ).

(4.9)

Proposition 8. Under the same conditions 2 h1 (r) = 2 π

Z∞

B1 (r, u; s, µ, v; ρ)u th(πu)h(u)du

(4.10)

−∞

where, with the notation (1.78), (1.80), B1 (r, u; s, µ, v; ρ) = B1 (r, u; ρ, v, µ; s) = Z∞ 1 = (A10 (r, ξ, s, µ)A00 (u, ; 1 − s, v)+ ξ

(4.11)

0

1 + A11 (r, ξ, s, µ)A01 (u, ; 1 − s, v))ξ 2s−2ρ−1 dξ. ξ Both the propositions result from term-by-term integration in the corresponding multiple integrals; it is sufficient to consider the first relation (4.7).

2.3 The main functional equation: the preparations.

99

First of all, the function ϕ in (4.4), for our case, is O(x3 ) when x → 0 and O(x−(1/2) ) for x → +∞. Furthermore, the Mellin transform of this function, which is defined by the integral Z∞ ϕ(w) ˆ := ϕ(x)xw−1 dx = (4.12) 0

 πw  2 = cos π 2

Z∞

−∞

! w 1 γ , + iu u th(πu)h(u)du 2 2

is regular for Re w > −3 and |ϕ(w)| ˆ may be estimated as O(|w|Re w−1 ) when |w| → ∞ and Re w is fixed. For this reason, the integrals (3.7) - (3.8) are absolutely convergent if α0 < Re(s + ρ) − 1. At the same time, both the integrals with k j (x, 12 + ir)× x2s+2ρ+2u−3 for j = 0, 1 are absolutely convergent for 1−Re(s+ρ) < Re u < 54 − Re(s + ρ). If Re(s + ρ) > 12 , we can choose α0 in such a manner that the term-by-term integration would be valid in the integrals which will arise on replacing Φ j in (4.5) and (4.6) by the representations (3.7) and (3.8). In this way, we have Z∞ 2i uth (πu)h(u)× (4.13) h0 (r) = 2 π −∞ Z 1 1 γ(s + ρ + w − 1, + ir)γ(1 − s − w, v)γ(1 − ρ − w, µ)γ(w, + iu)× × 2 2 (α0 ) × cos(πw) sin π(s + ρ + w)(cos π(s + w) sin(πv)+ + cos π(ρ + w) sin(πµ))dwdu.

After this, it is sufficient to check that two representations are identical 87 for Re s, Re ρ < 1; but this results immediately from the explicit formulae for the Mellin transforms of the kernels k j and the definitions of the kernels Ak,l . To finish the preparations, it remains to write the coefficients in the sum over the regular cusps for the sum of Kloosterman sums with weight function Φ0 and, finally, to !consider the analytic continuation of the ρ, v h + h1 . function Z0con s, µ 0

100

0

INTRODUCTION.

The first is not difficult; it is sufficient to do the formal substitution r = i(k − 21 ) in the expression for h0 (r) and to note the well-known limiting case Γ(a + 2k + 1)Γ(b + 2k + 1) × Γ(2k + 2) ×z2k+1 F(a + 2k + 1, b + 2k + 1; 2k + 2; z)

lim (Γ(c))−1 F(a, b; c; z) =

c→−2k

(4.14)

which holds for a positive integer k. The analytic continuation is given by the same kind of relation as in (4.3); so it is sufficient to calculate the values (h0 + h1 )(i(s − 1) ± (µ − 21 )) and (h0 + h1 )(i(ρ − 1) ± (v − 12 )). Proposition 9. Let h be the same function as in Proposition (7); then, for 21 < Re s, Re ρ < 1, Re v = Re µ = 12 , we have 1 2 (h0 + h1 )(i(ρ − v − )) = 2 2 π

Z∞

−∞

u th (πu)h(u)B˜ 0 (u; ρ, µ, s − v)du (4.15)

where B˜ 0 (u; ρ, µ, w) = (2π)1−2ρ sin(πρ)Γ(2ρ − 1)× Z∞ × (|1 − ξ 2 |1−2ρ A00 (u, 1/ξ; 1 − ρ, µ)+ 0 2 1−2ρ

+ (1 + ξ )

(4.16)

A01 (u.1/ξ; 1 − ρ, µ))ξ 2ρ−2w−1 dξ.

88

This relation is a consequence of the explicit formulae for the kernels Ak,l . If r = i(ρ − v − 12 ), then, in these formulae, we have a = 2v, b = 1, c = 2 − 2ρ + 2v; a′ = 2ρ − 1, b′ = 2ρ − 2v, c′ = 2ρ − 2v.

Now, we have, for the special case of the hypergeometric functions, F(0, b; c; z) ≡ 1 and F(a, b; b; z) = (1 − z)−a and as a result, we have the following equalities 1 1 A00 (i(ρ − v − ), ξ; ρ, v) + A10 (i(ρ − v − ), ξ; ρ, v) 2 2

(4.17)

2.4 The main functional equation and the specialization.

101

= (2π)1−2ρ sin(πρ)Γ(2ρ − 1)|ξ 2 − 1|1−2ρ ξ 2v and 1 1 A01 (i(ρ − v − ), ξ; ρ, v) + A11 (i(ρ − v − ), ξ; ρ, v) 2 2 1−2ρ 2 1−2p 2v = (2π) sin(πρ)Γ(2ρ − 1)|ξ + 1| ξ ; our proposition follows from these expressions. 2.4 The main functional equation and the specialization. Theorem 15. Assume that the even function h(r) is regular in the strip | Im r| 6 23 , decreases as O(|r|−B ), B > 4, when r → ∞ in this strip and has zeros at r = ± 2i . Then we have, for Re v = Re µ = 12 , 12 < Re s, Re ρ < 1, the following functional equation Z dix =

z(s + ρ; v, µ) π2

s, ρ, Z∞ −∞

! ! v s, v h + Z con h = µ ρ, µ

r th(πr)(h(r) − h0 (r; s, v; ρ, µ))dr+

! ρ, v ρ, +Z h0 + Z˜ dis s, µ s, ! ρ, v +Z con h + h1 + Z cusp s, µ 0 dis

(5.1)

! v h + µ 1 ! ρ, v ∗ h . s, µ

+Φϕ (s, v; ρ, µ) + Φϕ (s, 1 − v; ρ, µ) + Φϕ (ρ, µ; s, v)+

Φϕ (ρ, 1 − µ; s, v) + ϑh (s, v; ρ, µ) + ϑh (s, 1 − v; ρ, µ)+

+ϑh (ρ, µ; s, v) + ϑh (ρ, 1 − µ; s, v) + ϑh0 +h1 (s, v; ρ, µ)+

+ϑh0 +h1 (s, 1 − v; ρ, µ) + ϑh0 +h1 (ρ, µ; s, v) + ϕh0 +h1 (ρ, 1 − µ; s, v) where h0 and h1 are defined in terms of h by (4.7) and (4.10), the se- 89 quence h∗ is the result of the formal substitution of i(k − 12 ) in place of r

102

0

INTRODUCTION.

in the expression for h0 (r) and ϑn (s, v; ρ, µ) =

ζ(2s − 1)ζ(2v) 1 z(s; µ, 1 − ρ + v)h(i(ρ − v − )) ζ(2ρ)ζ(2 + 2v − 2s) 2 (5.2)

Φϕ (s, v; ρ, µ) = 2

(4π)2s−2v−1 ζ(2v) z(ρ + v; s, µ)ϕ(2v ˆ + 1 − 2s) ζ(2s)

(5.3)

with ϕ from (4.4). This functional equation follows simply on putting together the preceding considerations. Of particular interest is the special case when s, v, ρ, µ are chosen as in (2.16) and the function h is positive for real r and decreases very rapidly; namely, we choose 2 9 25 1 h(r) = (r2 + )2 (r2 + )(r2 + )e−αr , α > 0. 4 4 4

(5.4)

Now we have only one variable t and, for brevity, we shall introduce new notation. Let, for s = µ = 12 , ρ = µ = 21 + it and further, for the function h from (5.4), let ! con s, v Zc (t) = ζ(2s)ζ(2ρ)Z h (5.5) ρ, µ Then we have

Zc (t) = (5.6) Z∞ 3 1 ζ ( 2 + it − ir)ζ 3 ( 21 + it − ir)ζ( 12 − it + ir)ζ( 21 − it − ir) 1 h(r)dr, 4π |ζ(1 + 2ir)|2 −∞

where the main contribution is determined by the interval |r| ≪ (log t)1/2 (we assume that t is a positive large number). For this reason, we can write 1 1 1 1 ζ( + it + ir)ζ( + it − ir) = ζ( − it + ir)ζ( − it − ir)χ(t, r), (5.7) 2 2 2 2

2.4 The main functional equation and the specialization.

103

where χ(t, r) = π2it

Γ( 14 −

it 2 it 2

+ ir2 )Γ( 14 −

it 2 it 2

− ir2 )

Γ( 14 + + ir2 )Γ( 14 + − ir2 ) !2it !! r2 + 1 2π 2it e 1+O . =i t t

(5.8)

Now we have

90

χ(t, 0)Zc (t) = (5.9) ! Z∞ 4 1 |ζ ( 2 + it + ir)ζ 4 ( 12 + it − ir)| 1 1 + r2 )h(r)dt (1 + O 4π t |ζ(1 + 2ir)|2 −∞

and the main term in the integrand is positive. If we estimate the integral Z∞

ωT (t)χ(t, 0)Zc (t)dt, T → +∞,

(5.10)

0

then the desired estimate for the eighth moment will be a consequence of the following simple statement. Proposition 10. For T → +∞, we have, with a fixed positive integer k > 1 and for every fixed δ > 0, Z2T

2TZ(1+δ)

1 |ζ( + it)|2k dt 2

(5.11)

To prove this inequality, one can see firstly that X 1 1 , s = 1 + it |ζ ′ ( + ut)| ≪ log T max s x>1 2 n 2 n6t

(5.12)

1 |ζ ( + it)|2k dt ≪ (log T )4k 2 ′

T

T (1−δ)

and we have with 4M = T 2/3 and ǫ = (log T )−1

ǫ+iM Z X 1 dw 1 ζ(s + w)xw + O(1), x 6 t. = 2 2πi w n n6x ǫ−iM

(5.13)

104

0

INTRODUCTION.

As a consequence of (5.12), (5.13) and Holder’s inequality, we have  M 2k 2T Z2T Z+M Z  dη 1 1   |ζ ′ ( + it)|2k dt ≪ (log T )2k |ζ( + ǫ + it)|2k dt   |ǫ + iη|  2 2 T −M

T

≪ (log T )4k ≪ (log T )4k

−M

(5.14)

2T Z+M

1 |ζ( + ǫ + it)|2k dt 2

T −M 2T Z+M

1 |ζ( + it)|2k dt, M = T 2/3 , 2

T −M

91

since the last integral is non-increasing as a function of ǫ. Now, for every positive ǫ ∈ (0, 1) we have Zǫ 1 1 |ζ 4 ( + it + ir)ζ 4 ( + it − ir)|r2 h(r)dr |χ(t, 0)Zc (t)| ≫ 2 2 1 ≫ ǫ 3 |ζ 8 ( + it)| − 2

Zǫ

−ǫ

−ǫ

(ǫ − r)r2 h(r)

1 ∂ 4 1 |ζ ( + it + ir)ζ 4 ( + it − ir)|dr, ∂r 2 2 (5.15)

so that Z

Z

1 |ζ 8 ( + it)|ωT (t)dt− (5.16) ωT (t)χ(t, 0)Zc (t)dt ≫ ǫ 2 Z Z 1 1 −ǫ 4 ( ωt (t)|ζ 8 ( + it)|dt)7/8 ( (ωT (t)|ζ ′ ( + it)|8 dt)1/8 2 2 3

If ǫ = A(log T )−2 with some sufficiently small constant A, then the last term of the right side of (5.16) is of a lower order than the first term and so we have the inequality Z Z 1 (5.17) ωT (t)|ζ ∞ ( + it)|dt ≪ (log T )6 | ωT (t)χ(t, 0)Zc (t)dt|. 2 For this reason, an estimate for the integral in (5.10) is sufficient for our purpose.

2.5 The functions h j : the non-essential terms.

105

2.5 The functions h j : the non-essential terms. To carry out the non-trivial integration over t, we must know the asymptotic behaviour of the functions h0 and h1 in the special case (2.16), where ρ = v = 12 + it with some large positive t. The plan is simple: instead of the hypergeometric functions we shall use the corresponding asymptotic formulae (these will be written by the asymptotic integration of the differential equation with a large parameter) and after this, using the saddle-point method, we shal integrate over ξ. 2.5.1 The integral with A01 in (4.8) 92

We have (see (1.79)) ! 1 i(ξ/2π)1−2 ρ 2iu 1 ξ × = A01 u, ; 1 − ρ, ξ 2 2sh(πu) Γ2 (1 − ρ + iu) F(1 − ρ + iu, 1 − ρ + iu; 1 + 2iu; −ξ 2 )+ Γ(1 + 2iu) + {the same with u → −u} , i(2πξ)1−2ρ sin(πρ)Γ(2ρ −

1 2

(6.1)

+ ir)Γ( 21 + ir)

× (6.2) 2ξ 2ir sh(πξ)Γ(1 + 2ir) ! 1 1 1 F 2ρ − + ir, + ir; 1 + 2ir; − 2 + {the same with r → −r} , 2 2 ξ

A01 (r, ξ, ρ, ρ) =

Later, we shall use the following method of considering our integrals. It is well-known that the function w = zc/2 (1 ∓ z)(a+b+1−c)/2 F(a, b; c; ±z) is a solution of the differential euqation   1 2   c(a + b + 1 − c) − ab c(2 − c) 1 − (a + b − c)  2  w = 0. w′′ +  + ± z(1 ∓ z) 4z2 4(1 ∓ z)2 (6.3) As a consequence (using an appropriate transformation of the varialbe), we see that the function W(η) = (tgη/2)(1/2)+2iu (cos η/2)2ρ−2 ×

(6.4)

106

0

INTRODUCTION.

F(1 − ρ + iu, 1 − ρ + iu; 1 + 2iu; −tg2 η/2) satisfies the differential equation (for ρ =

1 2

+ it): ! 1

u2 d2 w 2 + −t + + W = 0. dη2 sin2 η/2 4 sin2 η

(6.5)

Hence, for large t and 0 < η < π − δ for any fixed δ > 0, we have w=

93

p

η/2I2iu (tη)

Γ(1 + 2iu) 1 (1 + O( )). 2iu t t

(6.6)

This consequence is the distinctive feature of our method of considereing the asymptotic behaviour for all hypergeometric functions here. This method is based on the principle: “neighbouring equations have neighbouring solutions”; the method of estimation for the corresponding closeness is routine today (see, for example, [5], where the estimates are written for similar equations). Now the following statment would be obvious for the reader: the contribution of the term with the kernels A01 from (4.8) is negligible for large values of t. Indeed, |t−2iu Γ2 (1 − ρ + it)| ≪ e−πt and the part of the integral with ξ 6 At(log t)−1 with some (small) fixed A is small. But, for large ξ, we have an additional resource. We shall assume that parameter α in the definition of the initial function h(r) will be small; then we can move the path of the integration over u in (4.7) and (4.10) and the factor of the type ξ 2iu for ξ ≫ t(log t)−1 and Im u > +∆ will give O(t2∆ (log t)2∆ ). Here ∆ is defined by the width of the strip where (ch(πu))−1 h(u) is regular; for the function (5.4), one can choose ∆ = 72 . For the same reason, one can reject the term with A01 (u, 1ξ ; 1 − ρ. 21 ) in the expression (4.10), (4.11) for the function h1 . Furthermore, we have A10 (r, ξ, ρ, ρ) =

Γ(2ρ −

1 2

+ ir)Γ(2ρ − Γ(2ρ)

1 2

− ir)

×

(6.7)

2.6 The integral with A00 .

sin(πρ)ξ 2ρ F(2ρ −

107

1 1 + ir, 2ρ − − ir; 2ρ; 1 − ξ 2 ) 2 2

and the hypergeometric function with these parameters is a solution of the differential equation   2+ 1   r ρ(1 − ρ)  4   w = 0 w′′ +  2 ± 2 z(1 ± z)  z (1 ± z)

(6.8)

if w = zρ (1 ± z)ρ F(2ρ − 12 + ir, 2ρ − 12 − ir; 2ρ; ∓z). For the upper sign (which corresponds to ξ > 1), all solutions are oscillating; at the same time, we have Γ(2ρ − 12 + ir)Γ(2ρ − 21 − ir) (6.9) sin(πρ) ≪ Γ(2ρ) π ≪ exp(− (|2t + r| + |2t − r| − 3t) 2 π ≪ exp(− (max(t, 2t − 3t)) 2 So, if ξ > 1, the kernel (6.7) is exponentially small. For the case ξ < 1 (which corresponds in (6.8) to the case z ∈ (0, 1) pand the sign “minus”), the solution (6.8) does not exceed exp(rarcsin 1 − ξ 2 ) and so we have the factor eπr/2 for ξ = 0 only. But the contribution of the interval with small ξ, ξ ≪ t−1 log t, is small (for the same reason - one 94 can move the path of the integration over u and to render the factor ξ 2iu small). 2.6 The integral with A00 . 2.6.1 The explicit form. The unique essential term is the first integral in (4.8) and we shall consider this term in greater detail; in passing, we shall give some examples of the asymptotic integration of the differential equations with a large parameter.

108

0

INTRODUCTION.

First of all, we shall write the result of substituting the special values for our parameters. Let us introduce the notation 1 1 v ≡ v(z; ρ, r) = |z|1−ρ (1 + z)ρ F( + ir, − ir; 2 − 2ρ; −z) 2 2

(7.1)

w = w(z; ρ, u) = |z|ρ (1 + z)(1/2)+iu F(ρ + iu, ρ + iu; 2ρ; −z)

(7.2)

and

(here z is a real variable and z > −1). Then we have, for all ξ > 0, (2π)2ρ−1 A00 (r, ξ; ρ, ρ) = 2

(7.3) 2

= sin(πρ)Γ(2ρ − 1)(v(ξ − 1; ρ, r) + Av(ξ − 1; 1 − ρ, r)) where Γ(2ρ − 12 + ir)Γ(2ρ − 21 − ir) ch(πr) . A= Γ(2ρ)Γ(2ρb − 1) sin(2πρ)

(7.4)

This relation is a consequence of (1.78) and the simple relationship F(a, b; c; z) = (1 − z)c−a−b F(c − a, c − b; c; z). The representation (7.3) is very convenient for r < 2t, because, in this case we have |A| ≪ e−π(2t−r) . Hence with exponential accuracy, we can retain just the first term on the right side (7.3) for r 6 2t(1 − δ) with some fixed (small) δ > 0. The representation (1.78) will be used for the other kernel too; here, z ) and as a we shall use the relation F(a, b; c, z) = z−a F(a, c − b; c; z−1 consequence we shall obtain the equality 1 1 (7.5) (2π)1−2ρ A00 (u, ; 1 − ρ, ) = ξ 2 = sin(πρ)Γ(1 − 2ρ)(w(ξ 2 − 1; ρ, u) − Bw(ξ 2 − 1; 1 − ρ, u)) where B=

ch2 πu + cos2 πρ Γ2 (1 − ρ + iu)Γ2 (1 − ρ − iu) · π sin(2πρ) Γ(2 − 2ρ)Γ(1 − 2ρ)

(7.6)

2.6 The integral with A00 .

109

Now, after the change of the variable of integration ξ 2 − 1 7→ z, we have a representation for the essential part of the function (h0 + h1 ): h(0) (r, u; t) =

Z∞

1 1 A00 (r, ξ; ρ, ρ)A00 (u, ; 1 − ρ, )ξ 2ρ−2 dξ ξ 2

(7.7)

0

π tg(πρ) = 8 2ρ − 1

Z∞

(v(z; ρ, r) + Av(z; 1 − ρ, r))(w(z; ρ, u)−

−1

−Bw(z; 1 − ρ, u))

dz |z|(1 + z)3/2

and for r 6 2(1 − δ)t with fixed δ > 0, we can reject the term with A. 95 If r > 2t, then the representation (1.78) is not convenient: the bounded function is expressed here as a linear combination of exponentially large terms. The relations (1.76) and (1.77) are more suitable in this case (One can see that (1.78) is the consequence of the preceding equalities and the Kummer relations, which connect the hypergeometric function in z with the functions of the argument 1 − z). To write the explicit form of the obtained equality for h(0) , we shall introduce the additional notation 1 1 + ir, + ir; 1 + 2ir; z), 2 2 w(z; ˜ ρ, u) = zρ (1 − z)1/2 F(ρ + iu, ρ − iu; 2ρ; z). v˜ (z; ρ, r) = |z|(1/2)+ir (1 − z)ρ F(2ρ −

(7.8) (7.9)

Then, for the function h(0) (r, u; t), we have the representation (0)

h (r, u; t) = C1 (r, ρ)

Z1

v(−z; 1 − ρ, r)(w(z − 1; ρ, u)−

(7.10)

0

dz −Bw(z − 1; 1 − ρ, u)) 3/2 + z (1 − z)

Z1

(C2 (r, ρ)˜v(z; ρ, r)+

0

+C2 (−r, ρ)˜v(z; ρ, −r))(w(1 ˜ − z; ρ, u) − Bw(1 ˜ − z; 1 − ρ, u))

dz z3/2 (1 − z)

110

0

INTRODUCTION.

where Now we have 10 integrals V j , 0 6 j 6 9; we shall enumerate these integrals so that h(0) is equal to the sum V0 + AV1 − BV2 − ABV3

or

C1 V4 − C1 BV5 + C2 (r, ρ)V6 + C2 (−r, ρ)V7 − BC2 (r, ρ)V8 − BC2 (−r, ρ)V9 . 96

Later, we shall see that the essential contribution will arise only from the integrals V0 and V4 . 2.6.2 The Liouville-Green transformation There is a clear method worked out for the asymptotic integration of the differential equations of the second order with a large parameter. This method is based on the Liouvill-Green transformation. Assume we have a differential equation of the kind v + (t2 p0 (z) + p1 (z))v = 0, · :=

d , dz

(7.13)

where t is a large positive parameter and p0 , p1 are real functions. then we can transform the independent variable and the unknown function by the relation v = ξ˙ −(1/2) (z)W(ξ(z)), ξ˙ :=

dξ . dz

(7.14)

The formal differentiation gives, for the function W the equation d2 W ˙ −2 2 1 + ξ (t p0 + p1 − {ξ, z})W = 0. 2 2 dξ

(7.15)

where {ξ, z} denotes the Schwarzian derivative, ... ξ 3 ξ¨ 2 . {ξ, z} = − ξ˙ 2 ξ˙ 2 If one can choose the function ξ so that the new equation is close to the equation with the known solution, then we shall be successful in

2.6 The integral with A00 .

111

finding the desired asymptotic approximation. The possibility of getting the known functions is explained by the vast set of the investigated equations for the special functions. The simplest case is one when p0 has no zeros and p1 is smooth and bounded. Then we can choose ξ so that ξ˙−2 p0 = ±1. If p0 has a zero of the first order, then we can transform our equation, choosing ξ˙ −2 p0 = ξ (so that ξ˙ −1 will be smooth); the Airy function will arise as the main term of the asymptotic formula. For the case when p0 has two simple zeros nearby, one transforms the initial equation to the Weber equation; if p0 has a zero and a pole (both simple), then the transformation to the Whittaker equation will be useful and so on. For the purpose of giving asymptotic forumlae for the four functions v, w, v˜ , w˜ in the integrals V j , it is sufficient to use the inequalities from [5]. The initial differential equations for these functions have the form 97 (7.13); the coefficients p0 , p1 are given in the following tables where the parameter α is equal to t−1 r and q(z) = z(1 + z): Function v w v˜ w˜

Coefficients: p0 p1 −2 2 q (z)(1 + α q(z)) (2q(z))−2(1 + q(z)) −1 (zq(z)) −u2 ((1 + z)q(z))−1 + +(2q(z))−2(1 + q(z)) (q(−z))−2 (α2 − α2 z + z2 ) (2q(−z))−2(1 + q(−z)) (−zq(−z))−1 u2 (q(−z))−1 + (2q(−z))−2(1 + q(−z))

2.6.3 The function v, the case z > 0 or the case z ∈ (−1, 0) and α < 2. The function v, the case z > 0 or the case z ∈ (−1, 0) and α < 2. For these cases, we shall use the transformation (7.14) by choosing 1 + α2 q , q = z(1 + z), ξ˙ 2 = q2

(7.16)

112

0

INTRODUCTION.

and therefore we can assume √ p 2 +α2 1 + α 4q + 1 ξ(z) = log |q| + α log 2+α q p 1 + α2 q + 4q + 1 −2 log ; 2

(7.17)

so ξ = log |z| + O(z) when z → 0. Now equation (7.15), for this case, has the form d2 W + t2 W = Q1 (ξ.α)W 2 dξ

(7.18)

where, with q = q(z(ξ)), we have Q1 = −

1 q(1 + α2 q)−3 (α2 (α2 − 16)q − 4(α2 + 2)). 16

(7.19)

It is essential that this function tends to zero both when q → 0 and q → ∞. Taking into account the fact v = |z|−it (1+O(z)) = e−itξ (1+O(eξ )) when q → 0 (it corresponds to ξ → −∞), we conclude that, for all z > 0, v has the asymptotic expansion ξ˙ 1/2 (z)v(z; ρ, r) = e−itξ 98

X a (ξ; Q ) n 1 n (−2it) n>0

(7.20)

Zξ

(7.21)

where

a0 = 1, a1 =

Zξ

−∞

Q1 (η)dη, . . . , an+1 =

a′n

+

Q1 (η)an (η)dη.

−∞

The polynomial 1 + α2 q(−2) = 1 − α2 z(1 − z) has no zeros in the interval z ∈ (0, 1) if α2 < 4; for this reason, we can use the same transformation and we have the same expansion (7.20) for −z ∈ (0, 1) if α2 6 4(1 − δ).

2.6 The integral with A00 .

113

2.6.4 The function w : z positive. For the case z > 0, we suppose ξ˙2 = z−2 (1 + z)−1 , so that z = sh−2 2ξ √ √ and ξ = 2 log((1/ z) + ( 1 + (1/z))). The transformed equation has the form ! d2 W 1 2 + t + 2 W = Q2 (ξ)W (7.22) dξ 2 4ξ with

u2

! 1 1 1 + − . Q2 = 2 ch ξ/2 4 ξ 2 sh2 ξ

(7.23)

when ξ → 0 (which corresponds to z → ∞), we have Γ(2ρ)z−(1/4) × (7.24) Γ(ρ + iu)Γ(ρ − iu) !! Γ′ log z Γ′ Γ′ log z + 2 (1) − (ρ + iu) − (ρ − iu) + O . Γ Γ Γ z ξ˙ 1/2 w(z; ρ, u) =

(Here the analytic continuation of the hypergeometric function is used in the logarithmic case). If z → 0, then ξ˙ 1/2 w = zit (1 + O(z)) = 22it e−itξ × (1 + O(e−ξ )) and for this reason, our solution must be pro√ portional to ξ × H0(2) (tξ) (it is the Hankel function). Finally, have the uniform asymptotic expansion  X b (ξ)  p iπΓ(2ρ) n ξ˙ 1/2 w(z; ρu) = − √ + ·  ξH0(2) (tξ) t2n 2Γ(ρ + iu)Γ(ρ − iu) n>0 (7.25)

 X Cn (ξ)  p (2)  ′ +( ξH0 (tξ)) 2n  t n>1

1 Rξ where b0 = 1, c1 = Q2 dη and for n > 1, 20 1 1 bn (ξ) = − cn (ξ) − 2 2

Zξ

Q2 (x)cn (x)dx,

(7.26)

114

1 1 cn+1 (ξ) = b′n (ξ) + 2 2

0

INTRODUCTION.

Zξ

1 Q2 (x)bn (x)dx − 4

0

Zξ

(x−1 cn (x))

dx . x

(7.27)

0

99

The same solution may be expanded again when ξ > ξ0 with some fixed ξ0 > 0; then we have ξ˙ 1/2 w(z; ρ, u) = 22it e−itξ

X an (ξ, Q˜ 2 ) n>0

(−2it)n

(7.28)

where a0 = 1 and the other coefficients are given by the recurrence relations (7.21) with the replacement of Q1 by Q˜ 2 = Q2 − ( 41 )ξ −2 . 2.6.5 The function w : z negative. In essence, there is no difference from the previous case. To get an asymptotic formula for w(−z; ρ, √ u) with z ∈ (0, 1), we choose the new √ variable ξ(z) = 2 log(1/ z + ( 1/z) − 1), so that ξ˙ 2 = z−2 (1 − z)−1 and z = (ch 2ξ )−2 ; z = 0 corresponds to ξ = +∞. The transformed equation for the function W = ξ˙1/2 W(−z; ρ, u) has the form ! d2 W u2 1 2 + t + 2 + W = 0. dξ 2 sh ξ/2 4sh2 ξ

(7.30)

As the initial condition at z = 0 is w(−z; ρ, u) = zρ (1 + O(z)), we have, for ξ > ξ0 with fixed ξ0 > 0, the expansion ξ˙ 1/2 w(−z; ρ, u) = 22it e−itξ

X a (ξ, Q ) n 3 , n (−2it) n>0

(7.31)

where a0 = 1 and an , n > 1, are defined by the relations (7.21) with Q3 = −u2 (shξ/2)−2 − (2shξ)−2 instead of Q1 .

2.6 The integral with A00 .

115

If ξ were small (which corresponds to a neighbourhood of z = 1), then we rewrite equation (7.30) as ! ! 1 1 d2 W 2 2 + t + 4u + W = Q4 W, 4 ξ2 dξ 2 ! ! (7.32) 1 1 1 1 2 4 + . Q4 = u − − 4 ξ 2 sh2 ξ 2 sh2 ξ/2 When z → 1, we have, as a consequence of the Kummer relation between the hypergeometric function in z and in (1 − z). F(ρ + iu, ρ + iu; 2ρ; z) = +

Γ(2ρ)Γ(−2iu) F(ρ + iu, ρ + iu; 1 + 2iu; 1 − z)+ Γ2 (ρ − iu)

Γ(2ρ)Γ(2iu) (1 − z)−2iu F(ρ − iu, ρ − iu; 1 − 2iu; 1 − z). Γ2 (ρ + iu)

(7.33) 100

It gives the initial condition at ξ = 0 for our function ξ˙ −(1/2) W(−z; ρ, u) : ! ξ 1/2 Γ(−2iu) ξ 2iu 2 ( ) (1 + O(ξ ) + W = Γ(2ρ)( ) 2 Γ2 (ρ − iu) 2 Γ(2iu) ξ −2iu + 2 ( ) (1 + O(ξ 2 ))). Γ (ρ + iu) 2

(7.34)

It means that this solution is a linear combination of solutions which √ are close to A(±) (ρ, u) ξJ±2iu (tξ) and we have Γ(2ρ)t−2iu iπ ξ˙1/2 w(−z; ρ, u) = × 2 sh(2πu)Γ2 (ρ − iu)    X b˜ (ξ) X c˜ (ξ)  p    p n n + ( ξJ2iu (tξ))′   +  ξJ2iu (tξ) 2n 2n  t t  n>0 n>1

(7.35)

+ {the same with u 7→ −u}

where b˜ 0 ≡ 1 and the coefficients are defined by relations which are similar to (7.26) and (7.27).

116

0

INTRODUCTION.

2.6.6 The function h(0) for α2 6 4(1 − δ). We shall use the standard formulae for the method of the stationary phase from [6]. The main principle (not an all-embracing one and nevertheless true for our integrals with hypergeometric functions) is the following statement: if one has an integral without points of the stationary phase, then this integral will be small in a suitable sense. It is easy to check that there is no point of the stationary phase in the integral with v(z; ρ, r)w(z; 1 − ρ, u). Furthermore, the coefficient A in (7.7) is exponentially small for α2 6 4(1 − δ). For these reasons, the function h(0) is defined by the integral V0 only. To distinguish the functions “ξ” in the asymptotic formulae for v and w we shall write ξv and ξw . With this agreement, the integral V0 is equal asymptotically to −1 2it

t 2

Z∞

−1

101

exp(−it(ξv (z) + ξ2 (z))) E (z, α)dz (1 + α2 q)1/4 (1 + z)3/4

(7.36)

where E is an asymptotic series in t−1 with smooth and bounded coefficients; the main term in E is equal to π/16. Now p 1 + α2 q 1 − √ ξ˙v + ξw = q z 1+z and the point of the stationary phase is equal to z0 = α−2 − 1

(7.37)

At this point, we have 2t log 2 − tξv (z0 ) − tξw (z0 ) = (2t − r) log(2t − r) − 2t log t + r log r (7.38) and

1 tξ¨v (z0 ) + tξ¨w (z0 ) = − tα5 . (7.39) 2 The other details may be omitted here; the methods explained in [6] give us the following

2.6 The integral with A00 .

117

Proposition 11. Let r 6 2t(1 − δ) with some fixed small δ > 0 and t be large. Then the function h(0) can be written as 1 h(0) (r, u, t) = √ eiψ(t,r) E (t, r, u) t r

(7.40)

where ψ(t, r) := (2t − r) log(2t − r) − 2t log t + r log r −

π 4

(7.41)

and E is a smooth non-oscillating function, |E | ≪ 1 and for any fixed integer n > 1, |(∂/∂t)n E | ≪ t−n . 2.6.7 The case r > 2(1 − δ)t. Now we shall use the representation (7.10). Here C2 (±r, ρ) is exponentially small for 2t − |r| ≫ 1. At the same time, for all α2 , there are not turning points in the equation for v˜ and this function has an oscillating nature. For the points of the stationary phase in the integrals with v˜ and w, ˜ we have the equation p √ (z(1 − z))−1 α2 (1 − z) + z2 = ((1 − z) z)−1 , (7.42)

or, what is the same, z0 = α2 . So, there are no such points in the interval (0, 1); for this reason, the last integral on the right side of (7.10) can be omitted. When α is close to 2, the full asymptotic investigation of the function v(−z; 1 − ρ, r) is very complicated. But due to a fortunate coincidence, the simplest case is sufficient for our purposes. The fact of the matter is given by (1) the exponentially small nature of the coefficient C1 (r, ρ) for r − 2t ≫ 1 and 2) the absence of points of 102 the stationary phase in the interval z > 12 for r > 2(1 − δ). Really, the equation for these points is p √ (z(1 − z))−1 1 − α2 z(1 − z) = ((1 − z) z)−1 , 1 − α2 z(1 − z) > 0, (7.43) and z0 = α−2 is the unique possible solution. For this reason, it is sufficient to know the exact asymptotic formulae for the function v(−z; 1 −

118

0

INTRODUCTION.

ρ, r) in the interval z 6 α−2 (1 + δ) only. But the turning points of 2 (±) = our equation √ (the zeros of the polynomial 1 − 1α z(1 − z)) are z −1 2 2(α(α ± α − 4)) ; these points are close to 2 when α is close to 2. So we have the interval (( 18 , 38 ), for example) where the stationary point lies and the positive polynomial 1 + α2 q(−z) is strongly separated from zero. For the last reason, we have, in this interval, an asymptotic expansion of the same kind as in (7.20). The unique natural difference is the exchange of the signs, because ρ is replaced by 1 − ρ here: ξ˙v1/2 v(−z; 1 − ρ, r) = eitξv

X an (ξv , Q1 ) n>0

(2it)n

.

(7.43)

Now one can see that at the stationary point z0 = α−2 , we have t(ξv (z0 ) − ξw (z0 )) = −(2t + r) log(2t + r) + 2t log(2t) + r log r (7.44) and ξ¨v − ξ¨w = − 12 α5 at this point. As a consequence of the Stirling expansion, in the case 2t − r ≫ 1, π exp i(2t + r) log(2t + t)+ 8t !!! 1 +(2t − r) log(2t − r) − 4t log(2t) + O 2t − r C1 (r, ρ) =

(7.45)

so that for r 6 2t(1 − δ) with δ > 0 at the point z0 = α−2 it(ξv −ξw )−iπ/4

C1 (r, ρ)e

iψ(t,r)

=e

π 1 · 1+O 8t t

!!

,

(7.46)

where ψ is the same phase as in (7.40). To estimate the contribution of the integration over the complement of the interval (α−2 − δ, α−2 + δ), especially in the transition region |α2 − 4| 6 δ, we shall use approximation by the Weber functions. Let, for definiteness, α > 2 and the quantity ǫ 2 = ( 41 − α−2 ) be small. Then the differential equation for v(−z; ρ, r) may be written in the form ! 2 2 1 3 + 4z2 1 ′′ 2 16(z − ǫ ) (7.47) v + r + v = 0, − < z < 2 2 2 2 2 2 (1 − 4z ) (1 − 4z )

2.6 The integral with A00 . 103

119

(here z is written instead of z − 21 in the initial equation). The corresponding Liouville-Green transformation is taken by choosing 16(z2 − ǫ 2 ) (7.48) ξ˙ 2 (ξ 2 − γ2 ) = (1 − 4z2 )2 with the conditions ξ(−ǫ) = −γ, ξ˙ > 0. The new parameter γ is chosen so that the equality ξ(+ǫ) = γ is fulfilled. This last condition gives p (7.49) γ2 = 2(1 − 1 − 4ǫ 2 ) = 4ǫ 2 (1 + ǫ 2 + . . .) If we denote ǫ −1 z and γ−1 ξ as x and y(x, ǫ), then for the Schwarzian derivative {ξ, z}, we have the expression ǫ −2 {y, x} and the function y(x, ǫ) is defined by the equation dy (y − 1) dx 2

!2

=

x2 − 1 16ǫ 4 . γ4 (1 − 4ǫ 2 x2 )2

(7.50)

Here the function on the right hand side is a power series in ǫ 2 with the leading term (x2 − 1). For this reason, we have a solution of the form y(x, ǫ) = x + ǫ 2 y1 (x) + ǫ 4 y2 (x) + . . .

(7.51)

Hence the Schwarzian derivative {y, x} is of order O(ǫ 2 ) (it being obvious that {x, x} = 0) and as a result, we have the boundedness of {ξ, z} in a certain interval ǫ 2 6 ǫ02 . Now we have the transformed equation for the function W = ξ˙ 1/2 v: d2 W + r2 (ξ 2 − γ2 )W = Q4 (ξ, ǫ)W dξ 2

(7.52)

where Q4 is bounded uniformly (in ǫ) for all ξ ∈ (−∞, ∞) and at the same time, this function tends to zero, for ξ → ±∞, as O(ξ −2 ). An estimate of the closeness of the solutions of this equation to the solutions of the equation with Q4 ≡ 0 (the Weber functions is given in [7]). We have useful inequalities for the Weber functions and the full asymptotic expansions due to F. Olver [8]. They allow us to given

120

0

INTRODUCTION.

˜ After that, the asymptotic formulae for v in the transition region α2 4. everyone who is a past master in integration by parts will be also to prove the smallness for all integrals, except in the case considered. As a result we have Proposition 12. For any r with the condition 1 ≪ r 6 2t + B0 log t, for fixed B0 > 1, we have ! 1 1 (0) (7.53) h (r, t) = √ C1 r, + it eiψ0 (t,r) E (t, r) 2 r where ψ0 (t, r) = −(2t + r) log(2t + r) + 2t log(2t) + r log r − 104

and E is a smooth non-oscillating function, !n ∂ E (t, r) ≪ t−n , n = 0, 1, . . . |E (t, r)| ≪ 1, ∂t

π 4

(7.54)

(7.55)

If r > 2t + B0 log t with fixed B0 > 1, then

|h(0) (r, t)|1 r−3B0 .

(7.56)

2.7 The integration over t 2.7.1 The summation formulae The next step is the calculation of the integrals over t, where the integrand contains the Hecke series (associated with the continuous or discrete spectrum) and the function h(0) (r, t). To do this, we need to approximate the corresponding Hecke series by a finite sum; it will be achieved by using the following summation formulae (using other forms of the functional equations for the Hecke series). Proposition 13. Assume that ϕ : [0, ∞) → C and its Mellin transform ϕ(s) ˆ satisfies the conditions: i) ϕ(2s) ˆ is regular in the strip α0 6 Re s 6 α1 with some α0 < 0 and α1 > 1;

2.7 The integration over t

121

ii) for σ ∈ [α0 , α1 ], the function ((1 + |t|)−1−2σ + 1)−1 |ϕ(2σ ˆ + 2it)| is integrable on the axis (−∞, +∞). Then, for any v with Re v = 12 and for any relatively prime integers c, d with c > 1, one has the identity ! √ ! 4π n ζ(2v) 4π X nd ϕ(2v ˆ + 1)+ (8.1) e τv (n)ϕ =2 c n>1 c c (4π)2v +2 +

X

τv (n)

n>1

Z∞

ζ(2 − 2v) ϕ(3 ˆ − 2v)+ (4π)2−2v

√ √ (e(−nd′ /c)k0 (x n, v) + e(nd′ /c)k1 (x n, v))ϕ(x)xdx

0

d′

where is defined by the congruence dd′ ≡ 1(mod c) and the kernels k0 , k1 are defined by the relations (1.66). Proposition 14. Let ϕ have the same properties as in (8.1) and let t j (n), n = 1, 2, . . ., be the eigenvalues of n-th Hecke operator. Let λ j > 14 and let the j-th eigenfunction of the automorphic Laplacian be even. Then, for any coprime integers c, d with c > 1, we have ! √ ! 4π X nd 4π n e t j (n)ϕ = (8.2) c n>1 c c X ∞ √ 1 = t j (n) inf e(−nd′ /c)k0 (x n, + iχ j )+ 0 2 n>1 ! !! ′ √ 1 nd +e k1 x n, + iχ j ϕ(x)x dx. c 2 105

2.7.2 The integration over t. Our next problem is the asymptotic calculation of the integral Z 1 J (T ) = ωT (t)H j ( + 2it)h(0) (χ j , t)χ(t, 0)dt 2

(8.3)

122

0

INTRODUCTION.

(where χ is defined by the equality (5.8)) and the similar integral Z 1 1 J (T, r) = ωT (t)ζ( + 2it + ir)ζ( + 2it − ir)h(0) (r, t)χ(t, 0)dt. (8.4) 2 2 We shall consider the second integral; the first one may be considered in the same manner. Let β : [0, ∞) → [0, 1] be the infinitely smooth monotone function with the conditions β(x) + β(1/x) ≡ 1, β(x) ≡ 0 for 0 6 x 6

1 2

(8.5)

(and for this reason β(x) ≡ 1 if x > 2). If Re s > 1, writing v instead of 12 + ir for brevity, we have, for any positive δ, ζ(s + v − 1/2)ζ(s − v + 1/2) = =

∞ X

n−s β(δn)τv (n) +

n=1

∞ X

∞ X τv (n) = ns n=1

(8.6)

n−s β(1/δn)τv (n).

n=1

Applying, to the second sum, the summation formula (8.1) with c = d = 1, we shall obtain the representation X ζ(s + v − 1/2)ζ(s − v + 1/2) = n−s β(δn)τv (n)+ (8.7) n>1

2s−1

+(4π)

X

τv (n)

n>1

Z∞

! 16π2 1−2s x dx− (k0 (x n, v) + k1 (x n, v))β δx2 √

√

0

−

δs−v−(1/2)ζ(2v) s−v−

1 2

Z∞

β′ (x)xs−v−(1/2) dx−

0

δs+v−(3/2) ζ(2 − 2v) − s + v − 32

Z∞ 0

β′ (x)xs+v−(3/2) dx

2.7 The integration over t 106

123

(using integration by parts in the terms with the Mellin transform of the function β). The series on the right side of (8.7) are convergent absolutely for all values of Re s (as will be obvious after some integrations by parts); so this identity gives the meromorphic continuation of the function on the left hand side in the critical strip 0 < Re s < 1. If we calculate the integrals in (8.7) (using the asymptotic formulae for the Bessel functions of large order) then the so-called “shortened functional equation” will be the result. But there is no need for an explicit asymptotic form, for the purpose of integration over t in our case. We can do the first integration over t: in the case 2t − r ≫ 1, we have the inner integral Z dt 1 ωT (t)eiψ(r,t)−4it log(x/4π)+2it log(t/2π)−2it E (r, t) (8.8) √ t r with ψ and E from (7.40). Here the point of the stationary phase is defined by the equation x2 . (8.9) 2t − r = 8π Note that t ∈ (T, 2T ) and x2 > 8δ−1 π2 in the integrand. So, for π , the derivative of the function in the exponent is ≪ − log x ≪ δ = 8T − log T . Hence we have the possibility of integrating by parts any number of times. Each integration by parts will give the additional factor O(t−1 )n in the integrand. After multiple integration by parts over t, we shall do the integration by parts over x (to obtain the absolutely convergent series summed over n) As a consequence, we can reject the second series in the representation (8.7) and this rejection does not affect any remainder terms in the integral (8.4) There are some differences in the case when the quantity 2t − r may 107 be small, i.e. when 2T 6 r 6 4T (note that ωT (t) ≡ 0 for t 6 T and t > 2T ). For this case, we have a slightly different expression (7.53) instead of (7.40). Using the Stirling expansion for Γ( 12 + 2it + ir) and the Binet representation for log Γ( 12 +2it−ir), we can rewrite this expression

124

0

INTRODUCTION.

in the form 1 h(0) (r, t)e2it log(t/2π)−2it = √ eiψ(r,t) E1 (r, t), t r

(8.10)

where E1 has the same properties as E and iπ 1 (8.11) ψ1 (r, t) = (2t − r)(log( + i(2t − r)) − ) − 2t log(4π)− 2 2 ! Z∞ dv 1 1 −2t + r log r − i (ev − 1)−1 − − e−v((1/2)+i(2t−r)) v v 2 0

The zero of the function ∂ψ1 x − 4 log ∂t 4π lies to the right of the interval of integration. So we can carry out integration by parts and this gives us Z2T

iG(t) dt

ωT (t)E1 e

t

=−

Z2t T

T

ωT (t) E1 t

!′ Z t

eiG(τ) dτ dt.

(8.12)

1

x where G(t) = ψ1 (r, t) − 4t log . Now in the inner integral, we have an4π alytic functions ans for this reason, we can use the saddle-point method. We shall integrate over the curve τ = τ(y; r, t), which is parametrized by the real positive new variable y; this curve is defined by the condition that imaginary part of the function in the exponent is constant: iG(τ) = iG(t) − y, y > 0. Then the inner integral has the form   r  Z  −y ′ ′ −1 −r  iG(t)    e  e (G (τ(y))τ (y)) dy + O(e ) 0

(8.13)

(8.14)

2.7 The integration over t

125

(Note that Re(iψ1 (r, τ)) < π2 (r−2τ) if τ is small in comparison with r/2). It means that there is the possibility of repeating the integration by parts and so on. 108 Hence, the contribution of the second sum on the right side of (8.7) in the integral over t is negligible in any case. In the last terms on the right side of (8.7), the integrand is not zero for 12 6 x 6 2 only. For this reason, we have the same integral (with the replacement of x/(4π) by (xδ)−(1/2) and with the additional factor (i(2t−r)− 12 )−1 ); these terms contribute to the integral over t, the quantity √ O((T r)−1 |ζ(1 + 2ir)|). Of course, the same considerations are applicable to the integral with H j ( 12 + 2it). Thus, to calculate the integrals (8.3) and (8.4), we can replace the corresponding Hecke series by the finite sum with the weight function π . β(δn) if we choose δ = 8T We have O(T ) members in these sums and its estimate gives us the following main inequality. Proposition 15. The integrals J (T, r).J j (T ) are exponentially small for r − 4T ≫ log T , χ j − 4T ≫ log T and for r, χ j ≤ 4(T + log T ), we have

|J (T, r)| ≪

|ζ(1 + 2ir)| + log T , r ≫ 1, √ r |J j (T )| ≪ 1/T

(8.15) (8.16)

First of all, we have the same integrals as in (8.8) with 2t log(4πn) x in the exponent. instead of 4t log 4π There is no large difference between the cases r < 2T and r ∈ (2T, 4T ) and the first case is typical. It is convenient to use transformation (of the variable of integration) t = 2r + 4πny (with the obvious intention of fixing the position of the

126

0

INTRODUCTION.

point of the stationary phase); then we have the integral √ β(δn) √ · 4π n · e−ir log(4πn)+ir log r−ir−(iπ/4) r

Z∞

g(ny, r)e4πin(y log y−y) dy

0

(8.17) with

ωT (2πx + r/2) E (r, 2πx + r/2). (8.18) 2πx + r/2 This integral is the classic example of an exercise in the method of stationary phase. It is essential that the derivatives with the respect to 109 y, of the function g(ny, r) are bounded by O(T −1 ) here. Really, we have n ≪ T and for l = 0, 1, . . . !l !l ∂ ∂ −1 E (r, t) ≪ T , ωT ≪ T −1 . ∂t ∂t g(x, r) =

Using the usual formulae, we shall obtain the expression 2πβ(δn) √ (4πn)−ir eir log r−ir × r ! ( !) 1 11 1 ′′ ′ g +g + g +O 2 g(n, r) + 4πn 12 n where

′

:=

(8.19)

d and g, g′ , g′′ are taken at y = 1; all these quantities are dy

O(T −1 ). The summation of the absolute values is not sufficient for our purposes but there is no problem in effecting this with the desired accuracy. Let us suppose that g˜ (s, r) =

Z∞

β(δx)g(x, r)xs−1 dx.

(8.20)

0

It is an entire function of s, if r < 2T and meromorphic with simple poles at s = 0, −1, −2, . . . , when r ∈ (2T, 4T ); if r > 4T , then this function is identically zero.

2.7 The integration over t

127

First of all, we can write g˜ (s, r) = T

s−1

Z∞

g0 (x, r)xs−1 dx

(8.21)

0

with g0 (x, r) = β(8πx)

ω1 (2πx + r/(2T )) E (r, T x + r/2) 2πx + r/(2T )

(8.22)

If, in the beginning, Re s > 0, then g˜ (s, r) = −(1/s)T

s−1

Z∞

xs g′0 dx

(8.23)

0

1 T s−1 = s(s + 1)

Z∞

xs+1 g′′ 0 dx

0

= ... This representation gives the meromorphic continuation on the whole 110 s-plane at the same time, we see that, for any fixed B > 1, g˜ (s, r) = O(T Re s−1 |s|−B ) when |s| → ∞ and Re s is fixed. Now, with this function, we have, for v = 12 + ir, Z X τ (n) 1 v ζ(s)ζ(s + 2ir)˜g(s, r)ds g(n, r) = 2πi nir n>1

(8.24)

(8.25)

(3/2)

where

R

denotes the integral over the line Re s = σ.

(σ)

Let us move the path of the integration and integrate over the line Re s = − 12 . The poles at s = 1, s = 1 − 2ir and s = 0 give the terms ≪ |ζ(1 + 2ir)| +

1 |ζ(2ir)| T

(8.26)

128

0

INTRODUCTION.

and the integral over the line Re s = − 21 contributes O(T −3/2 r). Since ζ(2ir) = O(r1/2 )ζ(1 − 2ir) and r ≪ T , it proves the inequality (8.15). In the second case, we have the same representation. Z X t j (n) 1 g(n, χ j ) = H j (s + iχ j )˜g(s, χ j )ds (8.27) 2πi niχ j n>1 (3/2)

but without the poles on the line Re s = 1. There is no pole at s = 0 also; really, from the functional equation for the Hecke series, we have H j (±iχ j ) = 0, if this series corresponds to the even eigenfunction (ǫ j = +1 in (1.52); note that H j ( 21 ) = 0 if ǫ j = −1, so the consideration of the case ǫ j = +1 is sufficient for our purpose). On the line Re s = −| 12 , for the case |s| = o(r), we have |H j (s + iχ j )| ≪ eπχ j |Γ(1 − s)Γ(1 − s − 2iχ j )H j (1 − s − iχ j )|

(8.28)

≪ χ j |s|e−(π/2)s

Together with (8.24), it gives the estimate O(χ j T −3/2 ) for the sum (8.27). The other terms from the asymptotic formula (8.19) contribute only smaller quantities and inequality (8.16) is proved. 2.8 The sum over cusps. 2.8.1 The explicit form. The next step will be to estimate the contribution of the sum ! 1 cusp 2 , ρ ∗ Z h ρ, 21 111

in the integral over t. Firstly, we shall write the explicit form of the coefficients h2k−1 in this sum; these coefficients h2k−1 (t), k = 6, 7, . . . , result from the analytic continuation of the integral 2(2k − 1)

Z∞ 0

J2k−1 (x)Φ0 (x; s, v, ρ, µ)

dx x

(9.1)

2.8 The sum over cusps.

129

at the point s = µ = 21 , ρ = v = 12 + it; here Φ0 is the same function as in (3.12) with ϕ from (4.4) (where h means the function (5.4)). Let us introduce some additional notation: wk (z) = z1/2 (1 − z)1/2 F(k, 1 − k; 1; z),

w˜ k (z) =

o ∂ n 1/2+ǫ . z (1 − z)1/2 F(k + ǫ, 1 + ǫ − k; 1 + 2ǫ; z) ǫ=0 ∂ǫ

(9.2) (9.3)

Further, let, for real z with z < 1, vk (z) and y(z; t, u) be defined by vk (z) = |z|k (1 − z)1/2 F(k, k; 2k; z), (9.4) 1 1 y(z; t, u) = |z|(1/2)+it (1 − z)1/2 F( + it + iu, + it − iu; 1 + 2it; z) (9.5) 2 2 Finally, it is convenient to suppose b(u, t) =

Γ( 21 + it + iu)Γ( 12 − it + iu) Γ(1 + 2iu)

(9.6)

With this notation, the coefficients h2k−1 (t) in the sum Z cusp in the case of our specialization are defined by the following equality. Proposition 16. We have h2k−1 (t) = 2(2k − 1)

Z∞

(k) (k) (B(k) 00 (u, t) + B01 (u, t) + B11 (u, t))

(9.7)

−∞

uth (πu)h(u)du,

where the kernels are given by B(k) 00 (u, t) = =

1 2π2

Z1 0

(w˜ k (x) + 2

(9.8)

! Γ′ Γ′ (k) − (1) wk (x))(b(u, t)y(x; u, t)+ Γ Γ

+b(−u, t)y(x; −u, t))x−it

dx , x3/2 (1 − x)

130

0

B(k) 01 (u, t)

INTRODUCTION.

(−1)k Γ2 (k) = 2π2 Γ(2k)

Z1

vk (x)(b(t, u)y(x; t, u)+

0

+b(−t, u)y(x; −t, u))xit B(k) 11 (u, t)

icth(πt)Γ2 (k) = 2π2 Γ(2k)

(9.9)

dx − x)′

x3/2 (1

Z+∞ 1 vk (− )x−it (b(u, t)y(−x; u, t)− x

(9.10)

0

−b(−u, t)y(−x; −u, t))

dx . + x)

x1/2 (1

112

2.8.2 The equations. To estimate the integrals (9.8)-(9.10), we shall use again the same method of asymptotic integration of the differential equation with a large parameter. This large parameter will be the integer k for the function wk , w˜k and vk and t for y(x; u, t), y(x; t, u). It is convenient to write here all equations to clarify the nature (oscillatory or monotonic) of the corresponding functions. The function y(x; t, u) is a solution of the equation ! 1 − x + x2 u2 t2 ′′ y = 0, (9.11) + − y + 2 x (1 − x) 4x2 (1 − x)2 x(1 − x) and so this function is oscillatory. When the parameters t and u are interchanged, we have for the function y(x; u, t) the equation ! u2 t2 1 − x + x2 ′′ + y˜ = 0; (9.12) y˜ + − + x(1 − x) 4x2 (1 − x)2 x2 (1 − x) it is obvious that, for large t, the solutions are non-oscillatory. Now both the functions wk (x) and w˜ k (x) are solutions of the equation     (k − 12 )2 1 ′′  w = 0  + 2 w +  (9.13) x(1 − x) 4x (1 − x)2 

2.8 The sum over cusps.

Finally, the function vk (x) is a solution of the equation   2   (k − 12 )2 1 − x + x  ′′ v + − 2 + 2 v = 0 2 x (1 − x) 4x (1 − x) 

131

(9.14)

2.8.3 The order of the integrals. From the Stirling expansion, it follows for a large positive t !! 1 + u2 2πt2iu −πt e 1+O b(u, t) = Γ(1 + 2iu) t

(9.15)

(noting that, in all integrals, we can assume |u| ≪ log t) and at the same time, !!  π 1/2 1 + u2 −2it−ipi/4 b(t, u) = e 1+O (9.16) t t So the integrals (9.8) and (9.10) are negligible and it is sufficient to 113 consider (9.9). For both the functions vk (x) and y(x; t, u) the transformation x = (ch (ξ/2))−2 , ξ ∈ (0, +∞)

(9.17)

of the variable is suitable. Then, for the functions Vk (ξ) = ξ˙1/2 vk (x(ξ)), Y(ξ; t, u) = ξ˙ 1/2 y(x(ξ); t, u) We have the differential equations   !2 1  d2 Vk  1 + + − k −  Vk = 0, 2 dξ 2  4sh2 ξ ! 1 u2 d2 Y 2 + t + − Y=0 dξ 2 4sh2 ξ ch2 ξ/2

(9.18)

(9.19) (9.20)

If we take into account the initial conditions, then we can conclude that Vk must be proportional to p 1 1 (k − )−1/2 ξK0 ((k − )ξ) 2 2

(9.21)

132

0

INTRODUCTION.

with an absolute coefficient (not depending on k). At the same time, Y must have the main term p (9.22) (constant) · t−1/2 ξH0(2) (tξ). For this reason, the main term of the asymptotic formula for the integral in (9.9) is defined by the integral 1 (t(k − ))−1/2 2

Z+∞ 1 ξg(ξ)K0 ((k − )ξ)H0(2) (tξ)dξ 2

(9.23)

0

where g(0) , 0 and g(ξ) in some neighbourhood of ξ = 0 is a power series in ξ 2 . All such integrals are O(t−2 ) and because of the factor (Γ(2k))−1 Γ2 (k) we have the following estimate. Proposition 17. Let h2k−1 be defined by the equality (9.7); then |h2k−1 (t)| ≪ 2−k t−5/2 .

(9.24)

It means that the trivial estimate for the integral over t is sufficient and we can omit the sum Z cusp . 2.9 The eighth moment. ˜ v; ρ, µ) the sum of all “main terms” on Let us for brevity, denote by M(s, the right side of (5.1) (excepting the functions Z dis , Z˜ dis , Z con and Z cusp ) for the case of the specialization (5.4). Further, let M(t) =

lim

s,µ→1/2 ρ,v⇒(1/2)+it

˜ v; ρ, µ). ζ(2s)ζ(2ρ) M(s,

(10.1)

114

Then a cumulative result of our preceding considerations is the inequality   2  X Z   χ j  1 4 α j ωT (t)|H j ( + it)| dt · 1 + O   h(χ j )+ (10.2) 2 T j>1

0

2.9 The eighth moment. −6

+(log T )

Z∞ 0

+

Z4T 0

133

1 X 1 1 α j |H j ( )|3 + ωT (t)|ζ( + it)|8 dt ≪ 2 T χ 64T 2 j

dr 1 |ζ( + ir)|6 √ + 2 r|ζ(1 + 2ir)|

Z∞

ωT (t)χ(t, 0)M(t)dt.

0

As |ζ( 21 + ir)| ≪ (|r| + 1)1/6 , the integral with the sixth power of zeta function is estimated by the quantity O(T 1−1/6+ǫ ) for any ǫ > 0. The sum over the discrete spectrum on the right side is not larger than  1/2  !1/2   X  1  X 1  (10.3) α j H j4   α j H j2   t χ 64T 2 ξ 64T j

j

Here the second sum is O(T 2 log T ). For the first sum, the estimate O(T 2+ǫ ) for any ǫ > 0 due to H. Iwaniec and J. M. Deshouillers is known. But this estimate may be elaborated; if, in the main functional equation, we choose the function h so that h is close to 1 for −T 6 r 6 T and s = µ = ρ = v = 21 , then we should get an asymptotic formula for this fourth spectral moment. A suitable h(r) may be taken, for example, −1 T  ∞  Z Z   (chδ(r − η))−1 dη (10.4) hδ (r) =  (ch(δr))−1 dr   −∞

−T

with a fixed small positive δ. Then the main term will be ! X 4 1 α jH j ≫ T2 2 χ 6T

(10.5)

j

and the contribution from the continuous spectrum is Z 1 2 ≪ (log T ) |ζ( + it)|8 dt ≪ T 5/3 . 2

(10.6)

0

On the right side of the functional equation, we have the same quantities H j4 ( 12 ) but with another weight function h0 (χ j )+h1 (χ j ) (note again 115

134

0

INTRODUCTION.

that H j ( 21 ) = 0 if ǫ j = −1). This function results from the integration of the initial h with an oscillatory kernel and for this reason, its order must be smaller; hence, an asymptotic formula must exist for the fourth spectral moment with the main term T 2 (log T )n0 . So the product (10.3) may be estimated as O(T (log T )B ) with some fixed B (positive, of course). For this reason, the proof will be complete, if we estimate the integral with the “main” terms; simultaneously, the estimate for the integrals with fourth power of the Hecke series follows. 2.9.1 The integral with h0 . First of all, we shall calculate the integral with h0 in the first term on the right side of (5.1). The key to do this is the equality Z∞

−∞

π h(r)rth(πr)dr = − 2

Z∞

J0 (x)ϕ(x) dx

(10.7)

0

if arbitrary “good” functions h and ϕ are connected by the transformation (1.27) (compare the coefficients before δn,m in (1.25) and (1.31); the proof of (10.7) is contained in [1]). Now, for our function h0 (r), we have the representation (4.7) and it is sufficient to calculate the integral Z∞

A00 (r; ξ; ρ, v)r th(πr)dr.

(10.8)

−∞

It may be interpreted as the main term in the summation formula for the sum of Kloosterman sums with weight function ϕ(x) = πξ(4π)1−2ρ k0 (xξ, v) (ξ being considered as the parameter). It allows us to use (10.7) and using the tabular integrals, we come to the relation Z∞

−∞

h0 (r)rth(πr)dr =

Z∞

−∞

h(u)uth(πu)b0 (u; s, v; ρ, µ)du

(10.9)

2.9 The eighth moment.

135

where, with the additional notation, ϕ0 (ξ; ρ, v) = 22ρ−1 Γ(v

(10.10) 1 v 2 )ξ

+ρ− 1 1 F(v + ρ − · v + ρ − ; 2v; ξ), cos(πv)Γ(2v)Γ(3/2 − v − ρ) 2 2 ϕ1 (ξ; ρ, v) = (10.11) !2ρ−1 1 1 1 1 1 1 2 Γ(ρ + v − )Γ(ρ − v + )F(ρ + v − , ρ − v + ; 1; ) = π |ξ| 2 2 2 2 ξ =

we have

116

(4π)1−2ρ b0 (u; s, v; ρ, µ) = − × (10.12) 2π   Z1    1  (ϕ0 (ξ 2 ; ρ, v) + ϕ0 (ξ 2 ; ρ, 1 − v))A00 (u, ; 1 − ρ, µ)ξ 2ρ−2s−1 dξ+    ξ   0

Z∞

1 ϕ1 (ξ 2 ; ρ, v)A00 (u, ; 1 − ρ, µ)ξ 2ρ−2s−1 dξ+ ξ 1  Z∞    1 2ρ−2s−1  2 . dξ  +2 ϕ1 (−ξ ; ρ, v)A01 (u, ; 1 − ρ, µ)ξ   ξ 

+2

0

2.9.2 The transition to the limit. No problems arises from terms which contain the values of the initial function at points ρ ± (v − 21 ) − 1 and s ± (µ − 12 ) − 1. Since h(± 2i ), h′ (± 2i ) are zeros, only one term is non-zero when s = µ = 12 . This term contains the value h(i(2it − 21 )) which is exponentially small. Further, taking into account the representations (4.15), (4.12) and (10.9), we can write the sum of all the other terms as Z∞

−∞

u th(πu)h(u){ζ(2s)ζ(2ρ)C0 (u; s, v, ρ, µ)}du.

(10.13)

136

REFERENCES

We know definitely that the function in the brackets must be finite for our specialization, because the left side of the main identity is finite (in addition, for an arbitrary function h). Firstly, we can take ρ = v; then we come to the limit µ → s, and finally, take the limiting case s → 21 . The requirement for the result to be finite in this limiting process will give us several identities for the integrals with some hypergeometric functions; after all, we shall come to a linear combination of some terms with the products of the derivatives ζ (k) (1 + 2it). The number of zeta-functions in each term does not exceed 6 and the order of the differentiation is not larger than 4. The coefficients of this linear combination are some integrals with hypergeometric functions and their derivatives 117 with respect to parameters. So these coefficients may be investigated in the same manner as before; for this reason, the integral with the “main” terms (over t) does not exceed O(T (log T )B ) with some fixed B. This would be the end of the proof. Of course, someone can say that, indeed, some final steps are not there. It is true; but I believe that there are no pressing reasons to extend this sufficiently long paper and it would be better to publish the details somewhere else.

References [1] N. V. Kuznetsov : Gipoteza Peterssona dlia form vesa nul i gipoteza Linnika I. Summa sum Kloostermana, Math. Sbornik, 111 (1980), 334-383. [2] N. V. Kuznetsov : Formula svertki dlia coeffisentov Fourier riadov Eisensteina, Zap. Naucn. Sem. Leningrad, LOMI, 129 (1983), 4384. [3] J. M. Deshouillers and H. Iwaniec : Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math., 70 (1982), 219288. [4] M. N. Huxley : Introduction to Kloostermania, Banach Centre Publ., Warsaw, 17 (1985), 217-306.

REFERENCES

137

[5] N. V. Kuznetsov : O sobstvennix functsnix odnogo integralnogo uravnenia, Zap. Naucn. Sem. Leningrad, LOMI, 17 (1970) 66-149. [6] A. Erdelyi : Asymptotic expansions, New York, 1956. [7] N. V. Kuznetsov : Asymptoticheskoi raspredelnie sobstvennix chastot ploskoi membrane, Differentialnie Uravnenia, 12, 1(1966). [8] F. W. J. Olver : The asymptotic expansions of Bessel functions of large order, Phil. Trans. Royal Soc. London A 247 (1954), 328368. Computer Centre Far Eastern Department of Academy of Sciences of USSR Ul. Kim Yu Chena 65 680063 Khabarovsk U.S.S.R.

ON RAMANUJAN’S ELLIPTIC INTEGRALS AND MODULAR IDENTITIES By S. Raghavan and S. S. Rangachari 119

Introduction

It is known from Hecke ([5], p. 472) that ‘special integrals of the third kind’ of stufe N (i.e. having logarithmic singularities at most at the cusps of the principal congruence subgroup Γ(N) turn out to be of elementary type, namely, logarithms of functions invariant under Γ(N). Results of this kind have been discovered much earlier by Ramanujan in [11] as may be seen in the sequel, especially in §2. In this connection, Ramanujan was perhaps, one of the first to have considered the problem of ‘evaluating’ elliptic integrals associated with modular curves of small level, although elliptic integrals, in general, have been investigated in depth by various mathematicians such as Jacobi, Cayley and others. In the literature, transformations of orders 2, 3, 5 have been employed with a view to reduce formidable elliptic integrals to simpler (or more explicit) form [4]. In [11], Ramanujan has considered elliptic integrals associated with Γ0 (N) for N = 5, 7, 10, 14, 15 and also a solitary hyperelliptic integral (for Γ0 (35)). Various such elliptic integrals are found in scattered form in [11], with endeavours, via quadratic and higher order transformations and interesting modular relations, to simplify them. The principal objective of this paper is to make a systematic study of all the elliptic integrals and associated formulae recorded by Ramanujan in [11] and provide complete proofs. We shall, in addition, uphold various modular identities involving Eisenstein series stated by Ramanujan in different places in [11] and 138

139

presumably used by him in computing singular values of modular functions. With the help of such identities, we exhibit a nonlinear differential equation for Eisenstein series denoted in §3 by E p , for p = 5, 7; for p = 5, this differential equation is essentially equivalent to the nonlinear differential equation written down by ramanujan [11] for a function F(λ5 ), where λ5 is a ‘Hauptmodul’ for Γ0 (5). One has to compare these with nonlinear differential equations obtained by Eichler-Zagier [2] for divisor values of Weierstrass’ elliptic function. 120

1 Notation and preliminary results 1.1 Let H denote the complex upper half-plane and for z ∈ H, let x = e 2πiz , so that |x| < 1. By Γ(1) = Γ, we mean the modular group { ac db |a, b, c, d ∈ Z, ad − bc = 1}, acting on H via the analytic home  omorphisms z → (az + b)(cz + d)−1 . As usual, Γ(N) := { ac db ∈ Γ|c ≡ 0(mod N)}. By Q, R! we mean the normalized Eisenstein ! series ∞ ∞ P P 3 2πinz P P 5 E4 (z) = 1 + 240 d e , E6 (z) = 1 − 504 d × e2πinz n=1 0 −ǫ −5 > −ǫ. To evaluate the (real-valued) integral (18), it is enough to consider the range ǫ 5 < τ < ∞, since for other values of τ, say ǫ −1 < τ < ǫ 5 , the expression inside the radical sign viz. (τ + ǫ −5 )(τ − ǫ 5 )(τ − ǫ −1 ) × (τ + ǫ) becomes negative and hence the integrand will be purely imaginary. Using formula (78) in §66 of Greenhill [4] (with α = ǫ 5 , β = ǫ −1 , γ = −ǫ −5 , δ = −ǫ) we have, for ǫ 5 < τ < ∞.

Zτ

√

dτ

τ4 − 10τ3 − 13τ2 + 10τ + 1 s (ǫ −1 + ǫ)(τ − ǫ 5 ) 2 sn−1 = p (ǫ 5 + ǫ)(τ − ǫ −1 ) (ǫ 5 + ǫ −5 )(ǫ −1 + ǫ) s √ 2 5 (τ − ǫ 5 ) = sn−1 √ 5 6 + 3 5 (τ − ǫ −1 )

ǫ5

152

3

ELLIPTIC INTEGRALS ARISING FROM CUSP FORMS

2 = 5

Zφ

dφ q

0

1−

9 25

(19) sin2 φ

9 (ǫ −1 + ǫ −5 )(ǫ 5 + ǫ) . The upper limit φ in (19) is given = 5 −5 −1 (ǫ +√ǫ )(ǫ + ǫ) 25 √ 5 (τ − ǫ 5 ) 6 + 2 5 (τ + ǫ) 2 2 by sin φ = or cos φ = so that √ √ 6 + 3 5 !(τ − ǫ −1 ) 6 + 3 5 (τ − ǫ −1 ) 51/4 τ − ǫ 5 φ = tan−1 . From Ramanujan ([10], p. 208), we know that, 2ǫ τ + ǫ q

since κ2 =

1 − κ2 sin21 .

for φ1 , φ2 with cot φ1 · tan( 21 φ2 ) = 2

Zφ1 0

dφ q

1−

=

κ2 sin2 φ

However, from above, we have

Zφ2 0

dφ q

1−

κ2 sin2 φ

r  s √ 1/4 5  5 τ − ǫ 9 5 (τ − ǫ 5 )    2 2  (tan φ) 1 − κ sin φ =   1− 2ǫ τ+ǫ  25 3ǫ 3 (τ − ǫ −1 ) s r 51/4 τ − ǫ 5 4ǫ 2 (τ + ǫ −5 ) = 2ǫ τ+ǫ 53/2 (τ − ǫ −1 ) s 1 (τ − ǫ 5 )(τ + ǫ −5 ) = √ 5 (τ + ǫ)(τ − ǫ −1 ) s 1 (τ2 − 11τ − 1) = √ (τ2 + τ − 1) 5 q

131

Thus from (19) and (20), we obtain Zτ ǫ5

√

dτ τ4 − 10τ3 − 13τ2 + 10τ + 1

(20)

3.1 r

√1 5

2 tan−1

153

(τ2 −11τ−1) (τ2 +τ−1)

Z

1 = 5

!

0

2 tan−1

=

1 5



√1 5

Z

q

1−11v−v2 1+v−v2

Z∞ τ

√



1−

sin2 φ

9 25

dφ q

0

where v =

dφ q

1−

9 25

sin2 φ

3 3 1 η η15 = 3 3 . Hence τ η3 η5

dτ τ4 − · · · + 1

=

Z∞

√

ǫ5

=

1 5

dτ τ4 − · · · + 1

√ −1 (1/ 5) 2 tanZ 2 tan−1 0

1 5 2 tan−1

=

1 5

1 5

√

ǫ5

√

1−

(1−11v−v2 ) Z 1+v−v2

√ 2 tan−1 (1/ 5)

dτ τ4 − · · · + 1

dφ q

9 25 ! √ 1 1−11v−v2 √ 2 5 1+v−v Z

0

−

−

Zτ

sin2 φ

!

dφ q

1−

9 25

sin2 φ

dφ q

1−

9 25

sin2 φ

which together with (18) gives us Ramanujan’s formula (??). The right hand side of (??) admits of interesting reduction when one resorts to well-known transformations available for elliptic integrals 132 such as Landen’s transformation or Gauss’ transformation. On page 67

154

3

ELLIPTIC INTEGRALS ARISING FROM CUSP FORMS

+ 1 of [11], Ramanujan also notes that √ −1 (1/ 5) 2 tanZ

1 5

=

 

1 = 9

 2 tan−1  √1 5

2 tan−1

r

dφ 

 (1−11v−v2 )   1+v−v2

(1−v/ǫ 3 ) (1+vǫ 3 )

q

1−

Zπ/2 r

sin2 φ

9 25

dψ

(1+vǫ)(1−vǫ 5 ) (1−v/ǫ)(1+v/ǫ 5 )

!

q

1−

1 81

(21) sin2 ψ

also equal to 1 4

√ tan−1Z(3− 5)

√ tan−1 (3− 5)

r

dψ

(1−v/ǫ)(1−vǫ 5 ) (1−vǫ)(1+v/ǫ 5 )

!

q

1−

15 16

(22) sin2 ψ

with v and ǫ as above. To prove (21), let us use Landen’s transformation ([12], p. 496) : Z Z 1 dψ dφ = q r √  2 √ 2 2 1 − κ2 sin2 φ 1 + 1 − κ 1 − 1− √1−κ sin2 ψ 1+ 1−κ2

p

for tan(ψ − φ) − ( 1 − κ2 ) tan φ

(23)

with κ = 3/5 and determine the limits of integration on the right hand side corresponding to those on the left hand side of (23). We have only to verify that the relations r 1 1 − 11v − v2 4 tan(ψ − φ) = tan φ, tan(φ/2) = √ 5 1 + v − v2 5 together imply that 1 − v/ǫ 3 tan(ψ/2) = 1 + vǫ 3

s

(1 + vǫ)(1 − vǫ 5 ) , (1 − v/ǫ)(1 + v/ǫ 5 )

3.1

155

√ so that the upper limit π/2 for ψ in (21) will correspond to 2 tan−1 (1/ 5) arising for v = 0. Setting t1 = tan(φ/2), t2 = tan((ψ − φ)/2), we have then 4 2t1 2t2 = tan(ψ − φ) = 2 5 1 − t12 1 − t2 and so 5(1 − t12 ) 1 ± t2 = 4t1 2

v t

133

25 (1 − t12 )2 + 4. 16 t12

Since 1−t12 = 1−(1−11v−v2 )/[5(1+v−v2 )] = (4/5)(1+4v−v2 )/(1+v−v2 ) and (25/16)(1 − t12 )2 /t12 + 4 = 9(1 + v2 )2 /{(1 + v − v2 ) × (1 − 11v − v2 )}, we have √ − 5(1 + 4v − v2 ) + 3(1 + v2 ) t2 = p 2 (1 + v − v2 )(1 − 11v − v2 ) noting that only the positive root has to be taken for t2 , in view of t2 having to be positive for large v. Thus ǫ 2 (ǫ − v)(ǫ −5 − v) t2 = p (1 − v/ǫ)(1 + vǫ)(1 − vǫ 5 )(1 + v/ǫ 5 ) s (1 − v/ǫ)(1 − vǫ 5 ) = ǫ −2 . (1 + vǫ)(1 + v/ǫ 5 ) Finally t1 + t2 tan(ψ/2) = = 1 − t1 t2 q

q

1−11v−v2 1 1+v−v2

(1−vǫ 5 ) (1−v/ǫ)(1+vǫ)(1+v/ǫ 5 )

=

=

√1 5

1/(1 + vǫ)

s

+ v − v2 + ǫ −2 1−

ǫ√−2 (1−vǫ 5 ) 5 (1+vǫ)

(1−v/ǫ)(1−vǫ 5 ) (1+vǫ)(1+v/ǫ 5 )

√ (1 + v/ǫ 5 ) + 5ǫ −2 (1 − v/ǫ) · √ 5(1 + vǫ) − ǫ −2 (1 − vǫ 5 )

(1 − vǫ 5 )(1 + vǫ) (1 − vǫ −3 ) · (1 − v/ǫ)(1 + v/ǫ 5 ) (1 + vǫ 3 )

establishing the validity of (21).

q

156

3

ELLIPTIC INTEGRALS ARISING FROM CUSP FORMS

In order to prove (22), apply Gauss’ transformation [12] : Z Z 2 dφ dψ = q q 1+κ 1 − κ2 sin2 φ 1 − [4κ/(1 + κ)2 ] sin2 ψ for sin(2ψ − φ) = κ sin φ

134

with κ = 3/5 again and determine the limits of integration which correspond to each other on either side. From sin(2ψ − φ) = (3/5) sin φ, we get a quadratic equation for t := tan ψ, viz. √ p 5t 5 (1 − 11v − v2 )(1 + v − v2 ) 2 tan φ/2 = = tan φ = 2 4 − t2 (1 + 4v − v2 ) 1 − tan2 (φ/2) proceeding as in the earlier case. Since by the same calculations, 36(1 + v2 )2 20(1 + 4v − v2 )2 + 16 = , (1 − 11v − v2 )(1 + v − v2 ) (1 − 11v − v2 )(1 + v − v2 )

we get

√ − 5(1 + 4v − v2 ) + 3(1 + v2 ) tan ψ = t = p (1 − 11v − v2 )(1 + v − v2 ) taking the positive square root as before. Thus s 2 (1 − v/ǫ)(1 − vǫ 5 ) t= 2 ǫ (1 + vǫ)(1 + v/ǫ 5 ) √ −1 (3− 5) clearly giving the lower limit for ψ in (22); the upper limit tan √ corresponds to the upper limit 2 tan−1 (1/ 5) arising when v = 0. Consequently Ramanujan’s formula (22) is proved. η2 (3z)η2 (15z) Using a different ‘uniformiser’ v1 := 2 related to Γ0 (15), η (z)η2 (5z) Ramanujan ([11], p. 70/78) also records the formula : Zx

√ −1 (1/ 5) 2 tanZ

dx 1 = ηη3 η5 η15 x 5

0

2 tan−1

(1 − 3v1 ) √ 5(1 + 3v1 )

p

dφ

r

1−

9 sin2 φ 25

(24)

3.1

157

In view of formula (??), it suffices to show that r 1 − 11v − v2 1 − 3v1 = 1 + 3v1 1 + v − v2

(25)

in order to prove (24). Substituting for v and v1 on both sides of (25) and simplifying further, (25) will follow from (η3 η5 )6 −(ηη5 )5 η3 η15 −5(ηη3 η5 η15 )3 −9(ηη5 )(η3 η15 )5 −(ηη15 )6 = 0. (26) Formula (26), however, is a consequence of the following Proposition 2. η53 η55 ηη15

135

− η4 η45 − 5(ηη3 η5 η15 )2 − 9(η3 η15 )4 −

η5 η515 η3 η5

= 0.

(27)

Proof. Each term on the left hand side of (27) can be shown to be a modular form of weight 4 for Γ0 (15) and it is not hard to derive the following Fourier expansions (writing x for e2πiz ) : η53 η55 ηη15

= x + x2 + 2x3 − 2x4 − 0 · x5 − 8x6 − 4x7 − 15x8 + 7x9 − 0 · x10 + · · ·

− η4 η45 = −x + 4x2 − 2x3 − 8x4 + 5x5 + 8x6 − 6x7 + 0 · x8 + 23x9

− 20 · x10 + · · ·

− 5(ηη3 η5 η15 )2 = −5x2 + 10x3 + 5x4 + 0 · x5 − 25x6 − 10x7 + 15x8 − 10x9 + 25x10 + · · ·

− 9(η3 η15 )4 = −9x3 + 0 · x4 + 0 · x5 + 36x6 + 0 · x7 + 0 · x8 − 18x9 + 0 · x10 + · · ·

−

(ηη15 )5 = −x3 + 5 · x4 − 5 · x5 − 11x6 + 20x7 + 0 · x8 − 2x9 η3 η5 − 5x10 + · · ·

From these expansions, the left hand side of (27) is a modular form of weight 4 for Γ0 (15) all of whose Fourier coefficients corresponding to

158

3

ELLIPTIC INTEGRALS ARISING FROM CUSP FORMS

e2πinz for 0 ≤ n ≤ 8 = 4[Γ(1) : Γ0 (15)]/12 vanish. By Hecke’s theorem ([5], p. 811-see also p. 953) again, this modular form has to vanish identically.  Remark. We may rewrite (27) in terms of the above modular functions v, v1 for Γ0 (15) as 1 1 − v − 5 − 9v1 − =0 (28) v v1 If we show that the left hand side (which is already regular on all of H is also regular at all the four cusps of Γ0 (15), we can obtain an alternative proof for (27), via (28). 3.2 Ramanujan has considered on the same page 67 + 1 of [11] an elliptic integral coming now from the group Γ0 (14) of genus 1 and has noted the formula Zx Z dx dφ ηη2 η7 η14 (29) = ... r √ x   16 2−13 2 √ √ √ 0 1+v 1− sin φ cos−1

136

1 2 7 1−v2

13+16 2

32 2

where v2 := (ηη14 /η2 η7 )4 .

2πi

dτ are connected dz by the relation σ2 = τ4 − 14τ3 + 19τ2 − 14τ + 1, from Fricke ([3], pp. 451-453) and in fact, they generate the field of modular functions for Γ0 (14). Proceeding as in §3.1, we have We know that τ := 1/v2 and σ :=

Zx

dx ηη2 η7 η14 = −πi x

0

=

Z∞ τ

Zi∞

4π2 ηη2 η7 η14

ηη2 η7 η14 dz (with z = iy, y > 0)

z

√

dτ τ4 − 14τ3 + 19τ2 − 14τ + 1

(30)

2 + n2 ) where Now X 4 − 14X 3 + 19X 2q− 14X + 1 = (X − α)(X − β)[(X − m)q √ √ √ √ √ √ α = 12 (7 + 4 2 + 7 11 + 8 2), β = 21 (7 + 4 2 − 7 11 + 8 2),

3.2

159

√ √ m = 12 (7−4 2), n2 = 47 (8 2−11). From Greenhill ([4], p. 61 as quoted from page 23 of “Elliptische Funktionen” by Enneper), we obtain Zx

dX p

(X − α)(X − β)[(X − m)2 + n2 ) ) ( 1 H(X − β) − K(X − α) ,κ = √ cn−1 H(X − β) + K(X − α) HK Zφ dφ 1 = √ q HK 1 − κ2 sin2 φ 0 ! √ H(X − β) − K(X − α) with φ = cos−1 , H 2 = (α − m)2 + n2 = 4 2(7 + H(X − β) + K(X − α) q q √ √ √ √ √ √ √ 4 2+ 7 11 + 8 2), K 2 = (β−m)2 +n2 = 4 2(7+4 2− 7 11 + 8 2), √ and κ2 = 21 − 41 {(α − β)2 − H 2 − K 2 }/HK. Also, HK = 8 2 and √ √ κ2 = (16 2 − 13)/(32 2). Hence, for the (real-valued) integral (30) 137 wherein τ > α necessarily, we have the value  x  Z Zτ  dτ    −  √ 4 τ − ···+ 1 α

α

α

1 = q √ 8 2

cos−1



H−K cos−1 Z( H+K )

H−K−v2 (Hβ−Kα) H−K−v2 (Hβ+Kα)

cos−1

1 = q √ 8 2

cos−1

√

√

Z



s

dφ √ 16 2 − 13 2 1− sin φ √ 32 2

√ ! 13+16 2 7

! √ 13+16 2 1+v2 7 1−v2

dφ r 1−

√ 16 2−13 √ 32 2

sin2 φ

since H 2 β = K 2 α = HK, (Hβ − Kα)/(K − H) = 1 = (Hβ + Kα)/(H + K)

160

3

ELLIPTIC INTEGRALS ARISING FROM CUSP FORMS

(H 2 − K 2 )/(H 2

and (H − K)/(H + K) = This proves formula (29).

+ K 2 + 2HK)

=

1 7

q √ 13 + 16 2.

3.3 An integral of the same form as above but not of elliptic type has been mentioned by Ramanujan on page 70 /78 of [11] : Zx

dx 1 ηη5 η7 η35 = x 2

0

Zv3 0

where v3 = (ηη35 η5 η7 )2 . √ If τ := 1/ v3 and σ =

dt q

(1 − t +

√

t)(1 − t3 −

√

t(5 + 9t + 5t2 )) (31)

4πi

dτ , then by Fricke ([3], pp. 444dz 445), σ and τ are modular functions for Γ0 (35) connected by the relation σ2 = τ8 − 4τ7 − 6τ6 − 4τ5 − 9τ4 + 4τ3 − 6τ2 + 4τ + 1. [The right hand side factorizes as (τ2 + τ − 1)(τ6 − 5τ5 − 9τ3 − 5τ − 1)]. As in earlier examples, Zx 0

dx = ηη5 η7 η35 x

Z∞ τ

1 = 2

2π2 η2 η235

τdτ p

(τ2 + τ − 1)(τ6 − 5τ5 − 9τ3 − 5τ − 1)

Zv3 √ q √ √ dt (1 − t + t)(1 − t3 − t(5 + 9t + 5t2 )) 0

√ (setting τ = 1/ t)

138

Remarks. The further reduction of this hyperelliptic integral can be carried out by known methods in the theory of elliptic functions ([4], pp. 159-160). It is interesting to note the following relation between P = η/η7 and Q = η5 /η35 on page 303 of [10] : (PQ)2 − 5 + 49/(PQ)2 = (Q/P)3 − 5(Q/P)2 − 5(P/Q)2 − (P/Q)3

3.4

161

which is the same as equation (29) on page 446 of Fricke (3); the latter is itself a consequence of the above relation between σ and τ. 3.4 Ramanujan has also considered elliptic integrals wherein the integrand involves (higher) powers of η. On page 45 /54 of [11], he has written down the following formulae : 53/4

Zx

η2 η25

dx = x

0

√ 2 tan−1Z(53/4 λk ) 0

Zπ/2

=2

cos−1 [(ǫu)5/2 ]

=

√

5

dφ q

(32)

1 − ǫ −5 5−3/2 sin2 φ dφ

q

1−

(33)

ǫ −5 5−3/2

√ 2 tan−1 [51/4Z xψ(x5 )/ψ(x)] 0

q

2

sin φ

dφ √ 1 − ǫ/ 5) sin2 φ

(34)

x1/5 x x2 . . ., ψ(x) = x−1/8 × η22 (z)/η(z) recalling that λ5 = n65 /η6 , u = 1+ 1+ 1+ √ and ǫ = ( 5 + 1)/2. Before proving (32)–(34), we state Proposition 3. q (i) E2 (z) − 5E2 (5z) = −4(η5 /η5 ) 1 + 22λ5 + 125λ25

/η2 (ii) E4 (z) = η10 /η25 + 250η4 η45 + 3125η10 5 /η2 . (iii) E4 (5z) = η10 /η25 + 10η4 η45 + 5η10 5

2 Proof. We know that η10 /η25 , η4 η45 , η10 5 /η form a basis for the space of modular forms of Haupttypus (−4, 5, 1) and their Fourier expansions are given by

η10 /η25 = 1 − 10e2πiz + 35e4πiz + · · ·

162

3

ELLIPTIC INTEGRALS ARISING FROM CUSP FORMS

η4 η45 = e2πiz − 4e4πiz + · · ·

2 4πiz + ··· η10 5 /η = e

139

2 Writing E4 = αη10 /η25 + βη4 η45 + λη10 5 /η and comparing the first three Fourier coefficients, we have α = 1, −10α + β = 240, 35α − 4β + λ = 2160 i.e. α = 1, β = 250, λ = 3125, proving (ii). The proof of (iii) is identical. For proving (i), we have only to argue instead with [E2 (z) − 5E2 (5z)]2 of Haupttypus (−4, 5, 1) and identify it likewise with /η2 ). Identity (i) is stated by Ramanujan on 16(η10 /η25 + 22η4 η45 + 125η10 5

page 73 /81 of [11].



d λ5 ) = η2 η25 Corollary. x( dx

q

λ5 + 22λ25 + 125λ35 .

Proof is immediate from (??), (10) and (i) of Proposition 3. We now proceed to prove (32). In fact, from the Corollary, we have 3/4

5

Zx

dx η2 η25 x

=

1 53/4

Zλ5 0

0

=

1 53/4

Zλ 0

dλ5 q q

λ35 +

22 2 125 λ5

(since λ5 (i∞) = 0) +

1 125 λ5

dλ λ[(λ +

11 2 125 )

(35)

, 2 2 + ( 125 ) ]

dropping the suffix 5 from λ5 . Now, using formula (24) on page 40 of 3 , n = 2/53 , H 2 := (α−m)2 + Greenhill ([4], §46) with α = 0, m = −11/5 √ 1 n2 = 1/53 , κ2 := 2 [1 − (α − m)/H] = (5 5 − 11)/(2.53/2 ) = ǫ −5 /53/2 , we see that (35) is the same as cos−1

1 √

53/4

H

cn

−1

(

) H − λ ǫ −5 = , H + λ 53/2



5−3/2 −λ 5−3/2 +λ

Z 0



dφ q

1 − ǫ −5 5−3/2 sin2 φ

√ But cos φ = (5−3/2 − λ)/(5−3/2 + λ) implies that tan(φ/2) = 53/4 λ and so (32) is proved.

3.4

163

To prove that the right hand side of (32) is the same as (33), let us first invoke the following transformation formulae from Ramanujan ([10], Chapter XVII, 7(vi) and 7(ii), pp. 207-208) : Zβ 0

dφ q

=2

1 − κ2 sin2 φ

Zα

dφ q

1 − κ2 sin2 φ p (where tan(β/2) = (tan α)| 1 − κ2 sin2 α) 0

=2

Zλ 0

p (where tan λ = (tan α) 1 − κ2 )

which together imply that Zβ 0

Now taking

dφ q

=2

1 − κ2 sin2 φ

κ2 = ǫ −5 /53/2

dφ p 1 − κ2 cos2 φ

and

Zπ/2

π/2−λ

dφ q

1 − κ2 sin2 φ

λ = (π/2) − cos−1 ((ǫu)5/2 ),

we have 5/2

q

/ 1 − (ǫu)5 , q p 3/4 5/2 2 tan α = (tan λ)/ 1 − κ = 5 u / 1 − (ǫu)5 ,

tan λ = (ǫu)

1 − κ2 sin2 α = 1/(1 + ǫ −5 u5 ) and

p tan(β/2) = (tan α) 1 − κ2 sin2 α q 3/4 5/2 = 5 u / (1 − ǫ 5 u5 )(1 + ǫ −5 u5 )

140

(36)

164

3

ELLIPTIC INTEGRALS ARISING FROM CUSP FORMS

p p = 53/4 / u−5 − u5 − 11 = 53/4 λ5 ,

by (3). Our assertion above concerning (32) and (33) is immediate. Finally, we show that the right hand side of (32) coincides with (34). For this, we need to use Ramanujan’s transformation formula ([10], p. 231 – see also Smith [12], p. 469) : ZA 0

where

κ2

dφ q

1−

κ2 sin2 φ

3 = 1 + 2α

ZB 0

α3 (2 + α) 2 2+α = ,µ =α 1 + 2α 1 + 2a

!3

dφ q

1−

(37)

µ2 sin2 φ

and

tan((A − B)/2) = (1 − α)/(2α + 1) tan B. (38) √ √ √ Taking α = √ 1/(ǫ 2 5), we have 1 − α = 3/(ǫ 5), 2α + 1 = 3/ 5, √ 2 + α = 3ǫ/ 5, κ2 = √ǫ −5 /53/2 and µ2 = ǫ/ 5. Further, in (37), if we √ take A = 2 tan−1 (53/4 λ5 )B = 2 tan−1 [51/4 xψ(x5 )/ψ(x)], we have to verify that (38) holds. But the latter is the same as η35 η3

=

ηη210 η22 η5

×

1 − (ηη210 /η22 η5 )2

1 − 5(ηη210 /η22 η5 )2

which, on the other hand, follows at once from Proposition 1. It is now clear that (37) implies our assertion concerning (32) and (34). Remark. On the same page in [11] wherein Ramanujan has noted down formulae (32)-(34) as well as the integrals in (39)-(41) considered in §3.5, one finds also values of κ2 , κ2 (1 − κ2 ) corresponding to two succes141 sive “cubic” transformations. We should mention here that classically the object of applying such transformations to elliptic integrals repeatedly was the realisation of the complete integral in the limit ([4], p. 322). 3.5 Formulae (32)-(34) were connected with the modular relation (3) for u. The following formulae due to Ramanujan ([11], p. 45 /54) are

3.5

165

analogous and connected with the modular relation (2) instead : −3/4

5

Zx 0

η5 dx =2 √ η1/5 η5 x

cos−1 ( 1/4

=

2 tan−1 (5Z

1 = √ 5

Zπ/2

√

η5 /η1/5

√

su)

)

q

dφ √ 1 − (ǫ −1 / 5) sin2 φ

dφ √ 1 − (ǫ −1 / 5) sin2 φ 0 √ 2 tan−1 (53/4 ((η1/5 +η5 )/(η1/5 +5η5 )) η5 /η1/5 ) Z

(40)

q

0

(39)

dφ q

1 − (ǫ 5 /53/2 ) sin2 φ (41)

1 du 1 5 Before proving (39), we note that, by virtue of (9), = η /η5 u dx 5x    1 R1 η5 dx  √ . But u(1) = 1 1 . . . = ( 5 − and so u = u(1) exp − 5 x η5 x 1+ 1+ −1 −1 −2πy 1)/2 = ǫ . Moreover, u < ǫ since x = e < 1 (for y > 0) and 1 5 R 1 η dx < 0. Using (2) it is now easily seen that the left hand side − 5 x η5 x of (39) is the same as Zu Zu du du 1/4 1/4 5 =5 √ p u − u2 − u3 −u(u + ǫ)(u − ǫ −1 ) 0

0

(noting 0 < u < ǫ −1 )  ǫ −1 ǫ −1 Z Z  du   = 51/4  −  p   −u(u + ǫ)(u − ǫ −1 ) u

(42)

0

But from Greenhill ([4], formula (14), p. 36), we know that 51/4

Zǫ −1 u

du p

−u(u + ǫ)(u − ǫ −1 )

=2

−1 ( √ǫ) cosZ

0

q

dφ √ 1 − (ǫ −1 / 5) sin2 φ

142

166

3

ELLIPTIC INTEGRALS ARISING FROM CUSP FORMS

√ taking M = 21 (ǫ −1 + ǫ)1/2 and (κ′ )2 = ǫ −1 /(ǫ + ǫ −1 ) = ǫ −1 / 5 therein. Letting u tend to 0 in (??) and substituting into (42), we see that formula (39) is true. In order to verify that (40) is the same as the right hand side√of (39), we have only to use Ramanujan’s formula (36) with κ2 = ǫ −1 / 5 √ and λ = π2 − cos−1 ( ǫu); indeed, tan λ = (ǫu/(1 − ǫu))1/2 , tan2 α = √ √ 5 u/(1 − ǫu), 1 − (ǫ −1 / 5) sin2 α = 1/(1 + ǫ −1 u) and consequently tan β/2 = (ǫu/(1 − ǫu))1/2 51/4 /(ǫ + u)1/2 = 51/4 (η5 η/η1/5 ) in view of (2). For proving that (40) and (41) are the same, we appeal to the Legendre transformation ([4], p. 323) : Z∞ 0

dφ q

1 − κ2 sin2 φ

1 = 2α + 1

Zψ

dψ q

1 − µ2 sin2 ψ φ+ψ = (α + 1) tan φ) (with tan 2 0

where κ2 = (α4 + 2α3 )/(2α + 1) and µ2 = α[(α + 2)/(2α + 1)]3 ; we need √ only to take α = ǫ −1 , φ = 2 tan−1 (51/4 η1/5 η5 ) and ψ = 2 tan−1 (53/4 · p φ+ψ = (α + η5 /η1/5 · [(η1/5 + η5 )/(η1/5 + 5η5 )] and verify that tan 2 1) tan φ. 3.6 Finally, we take up an interesting formula stated by Ramanujan ([11], p. 45 /54) concerning u, namely    Zx 8  Z1 8    η5 dx    η dx   u−5 + u5 = 3  + 125 C +   4 4   η x η5 x 2η5     x x η3

where

  Zπ/2 q    3/4  1 − ǫ −5 5−3/2 sin2 φ dφ− C=5  −π + 4     0

(44)

3.6

Zπ/2 p

dφ

0

q

167

     2  −5 −3/2 1−ǫ 5 sin φ     √

(u5

143

+ u−5 )

To is the same √ prove (44), we first remark that G := 2 λ5 as 2 125λ5 + 22 + 1/λ5 , in view of the modular relation (3). Hence 125 − 1/λ25 dG dλk = √ dx 125λ5 + 22 + 1/λ5 dx   η85 η8  1  = 125 4 − 4  , by Corollary to Proposition 3. x η η5

Now, by the fundamental theorem of the integral calculus, −2π/θ

G(x) − G(e

)=

Zx

125

η85 dx η4 x

Zx

−

e−2π/θ

e−2π/θ

η8 dx (for any θ > 0) η45 x

Consequently, we have η3 u−5 + u5 = 3 2η5 where ′

−2π/θ

C := G(e

  Zx 8  Z1 8  η  dx dx η  ′ 5  + 125 C + 4 4 η x  η5 x x

) − 125

(45)

0

−2π/θ eZ η85 η4 0

dx − x

Z1

e−2π/θ

η8 dx . η45 x

It is easy to see that C ′ is independent of θ and moreover C ′ < G(e−2π/θ ) for all θ > 0. From the transformation formula for the η-function, we find that 125 · 125 ·

η85 η4

η6 (− 1z )

=

η6 (z) η65 (z)

1 η6 (− 5z ) ! η8 (z) 1 = 5 · (z/i)2 · 4 − 5z η5 (z)

(46)

168

3

ELLIPTIC INTEGRALS ARISING FROM CUSP FORMS

and therefore 125

−2π/θ eZ η85 η4 0

dx = x

Z1

e−2πθ/5

! 1 η8 dx · using z → − . 5z η45 x

As a result, C ′ = G(e−2π/θ ) − 250 144

−2π/θ eZ η85 η4 0

dx = G(e− 2π/θ ) − 2 x

Z1

e−2πθ/5

η8 dx . η45 x

Formula (45) may now be rewritten as   −2π/θ eZ Zx 8   3 8 η η  η dx 5 dx   , + 125 u−5 + u5 = 3 C +  4 4 x x η η5 2η5  x

e−2π/θ

denoting ′

C +

Z1

e−2π/θ

η8 dx + 125 η45 x

−2π/θ eZ η85 η4 0

dx (= G(e−2π/θ ) clearly) by C. x

This last stated formula for u−5 + u5 may be deemed to be obtained from (45), merely by replacing the limits of integration 0, 1 therein by e−2π/θ and further the constant of integration C ′√there by C(= G(e−2π/θ√)). Ramanujan evaluates the constant C for θ√= 1, 5 and 5. Taking θ√= 5 first, we√obtain from √ (46) with z = i/ 5, that 125λ5 (−1/(5i/ 5)) = 1/λ5 (i/ 5), i.e. λ5 (i/ 5) = 5−3/2 . Hence √ √ C = G(e−2π/ 5 ) = 2(125 · 5−3/2 + 22 + 53/2 )1/2 = 4(11 + 5 5)/2)1/2 , in agreement with Ramanujan’s value for C for this case. Again, using the functional equation for λ5 implied by (46) with z = i, we have 125λ5 (i) = 1/λ5 (i/5) and therefore G(e−2π ) = G(e−2π/5 ). From Ramanujan’s identity (iii) in §4, viz. E62 = η24 (η3 /η35 − 500η35 /η3 − 56 · η95 /η9 )2 (1 + 22λ5 + 125λ25 ),

169

we infer λ5 (i) is a root of the polynomial 1 − 500X − 15625X 2 since E6 (i) = 0 and the polynomial 1 + 22X + 125X 2 has no real root. Thus √ λ5 (i) = 1/(5ǫ)3 and consequently, G(e−2π/5 ) = G(e−2π ) = 6·51/4 (3+ 5); however, this does not agree with the value for C given by Ramanujan. Remark . It is possible to express C ′ in terms of complete elliptic integrals of the first and the second kind as Ramanujan has noted in connection with C in (44).

4 Identities for Eisenstein series

We discuss in this section several useful identities for E4 (z), E4 (pz), E6 (z) and E6 (pz) for p = 5, 7 written down by Ramanujan ([11], p. 70 /78, also p. 67 /75) : 2 3 (i) E43 (z) = (η10 /η25 + 250η4 η45 + 55 · η10 5 /η )

(ii) E43 (5z) = (η10 /η25 + 10η4 η45 + 5η10 /η2 )3 5 12 (iii) E62 (z) = η24 (η3 /η35 −500η35 /η3 −56 η95 /η9 )2 (1+22η65 /η6 +125η12 5 /η ) 12 12 6 6 9 9 3 3 2 15 15 (iv) E62 (5z) = η24 5 (η /η5 +4η /η5 −η /η5 ) (1+22η5 /η +125η5 /η )

(v) E43 (z) = (η7 /η7 + 5 · 72 η3 η37 + 74 η77 /η)3 (η7 /η7 + 13η3 η37 + 49η77 /η) (vi) E43 (7z) = (η7 /η7 + 5η3 η37 + η77 /η)3 (η7 /η7 + 13η3 η37 + 49η77 /η) √ √ (vii) E62 (z) = (η7 /η7 − 72 (5 + 2 7)η3 η37 − 73 (21 + 8 7)η77 /η)2 × √ √ (η7 /η7 − 72 (5 − 2 7)η3 η37 − 73 (21 − 8 7)η7 /η)2 √ √ (viii) E62 (7z) = (η7 /η7 − (7 + 2 7)η3 η37 + (21 + 8 7η77 /η)2 × √ √ (η7 /η7 − (7 − 2 7)η3 η37 + (21 − 8 7)η77 /η)2 Identities (i) and (ii) have already been proved as (ii) and (iii) in Proposition 2 in §3.4. One can derive the other identities in a similar fashion. We can also deduce these identities from the results of Klein −1 ([7], p. 46). If τ1 := −λ−1 5 and τ2 := λ7 , then for elliptic modular function j(z), we have from Klein (see also [4], p. 329) the following relations :

145

170 5 DIFFERENTIAL EQUATIONS SATISFIED BY ‘EISENSTEIN SERIES’

(i) 1728 j(z) = (τ21 − 250τ1 + 55 )3 /(−τ51 ) (ii) 1728 j(5z) = (τ21 − 10τ1 + 5)3 /(−τ1 ) (iii) 1728( j(z) − 1) = (τ21 + 500τ1 − 56 )2 (τ21 − 22τ1 + 125)/(−(τ51 )) (iv) 1728( j(5z) − 1) = (τ21 − 4τ1 − 1)2 (τ21 − 22τ1 + 125)/(−τ1 ) (v) 1728 j(z) = (τ22 + 5 · 72 τ2 + 74 )3 (τ2 + 13τ2 + 49)/τ72 (vi) 1728 j(7z) = (τ22 + 5τ2 + 1)3 (τ22 + 13τ2 + 49)/τ2 (vii) 1728[ j(z) − 1) = (τ42 = 10 · 72 τ32 − 9 · 74 τ22 − 2 · 76 τ2 − 77 )2 /τ72 (viii) 1728( j(7z) − 1) = (τ42 + 14 · τ32 + 9 · 7τ22 + 10 · 7τ2 − 7)2 /τ2 Substituting for j(z) = E43 /(E43 − E62 ) = E43 /(1728 η24 ), we deduce Ramanujan’s identities (i)-(viii) immediately from the above identities (i)(viii). One finds a few more striking identities involving the Eisenstein series E4 and E6 , stated by Ramanujan ([11], p. 67 + 1) : (E42 (z) + 94E4 (z)E4 (5z) + 625E42 (5z))1/2 √ 2 = 12 5(η10 /η25 + 26η4 η45 + 125η10 5 /η )

(47)

(5(E6 (z) + 125E6 (5z))2 − (126)2 E6 (z)E6 (5z))1/2

(48)

10 2 1/2 4 4 2 10 2 = 252(η10 /η25 + 62η4 η45 + 125η10 5 /η )(η /η5 + 22η η5 + 125η4 /η )

Identity (47) may be verified by squaring both sides, substituting the expressions for E4 (z) and E4 (5z) from the identities (i)-(ii) above and checking that the coefficients corresponding to the monomials η24 , 24 6 18 η18 η65 , η12 η12 5 , η η5 and η5 are the same on both sides. A similar remark applies to identity (48), using now identities (iii)-(iv) for E6 above.

171

5 Differential equations satisfied by ‘Eisenstein series’ We discuss, in this section, a differential equation mentioned 146

by Ramanujan ([11], p. 73 /81) for certain ‘Eisenstein series’. First, let us recall ‘Hecke summation’ ([5], p. 469) for Eisenstem series of weight 2 : X (cz + d)−2 |cz + d|−s Lt s→0

(c,d)=1 c≥0

  ∞ X  X   −6i  d e2πinz = + 1 − 24 π(z − z) n=1 0 0, i = −1. We discusses them in many places in the Notebooks and more importantly in the ‘Lost’ Notebook. In partic√ ular, he evaluated R(τ) and S (τ) for τ = i n for many rational values of n > 0. Some of these evaluations were sent by him to Hardy in his early letters from India. A number of evaluations of R(τ) and S (τ) contained in the ‘Lost’ Notebook were discussed and upheld by us [4] using the Kronecker limit formula which seems to be well adapted for these problems. We do not, of course, know Ramanujan’s methods. They could not be the method using the limit formula. There are two evaluations [7, p. 46] which are particularly intriguing. The are √

(−3 +

√

−3 +

S (i 3) =

S (i/ 3) =

√ √

5) +

5+

4 q

4 176

q

6(5 +

6(5 −

√

√

5) (3)R

5) (4)R

177

As far as we know, these two results have not been proved until now. In attempting to prove these, we encountered another of Ramanujan’s evaluations. If λn for integers n ≥ 1 is defined by √

√ √ √ e(π/2) n/3 {(1 + e−π n/3 )(1 − e−2π n/3 )(1 − e−4π n/3 ) . . .}6 λn = √ 3 3

the Ramanujan states √ √ λ1 = 1, λ9 = 3, λ17 = 4 + 17, λ25 = (2 + 5)2 , √ √ λ33 = 18 + 3 33, λ41 = 32 + 5 41, √ √ λ49 = 55 + 12 21, λ89 = 500 + 53 89, . . .

(5)R

The function λn seems to have been introduced earlier in the Notebooks 152 (Vol 2, p. 393) where Ramanujan gives a formula for evaluating λn for n 3 = 11, 19, 43, 67, 163 and others. It is to be noticed that these values of − 3n are precisely the discriminates ≡ 5(mod 8) of imaginary quadratic fields of class number one. If we use Dedekind’s modular form η(τ) = eπiτ/12

∞ Y (1 − e2nπiτ ). n=1

then

 1 6 √     η[ 2 (1 + i n/3)]   λn = √     1 3 3  η[ 2 (1 + i 3n)]  1 √

(6)

As was shown by us in [5], if −3n √ is a fundamental discriminant of an imaginary quadratic field K = Q( −3n) which has only one class in each genus of ideal classes, then λn can be evaluated fairly easily using L-series. For example, for n = 17, 41, 89 this property is true. However, for n = 25, 49 the numbers −3.25 and −3.49 are√not fundamental discriminant of an imaginary quadratic field K = Q( −3n) which has only one class in each genus of ideal classes, then λn can be evaluated fairly easily using L-series. For example, for n = 17, 41, 89 this property is true. However, for n = 25, 49 the numbers −3.25 and −3.49 are not fundamental discriminants but nevertheless they are discriminants in

178

2

√ the orders in Q( −3) with conductors 5 and 7, with similar properties with regard to genera of ring ideal classes. One has then an analogue of the Kronecker limit formula for the L-series of such ideal classes which leads to the evaluation of λ25 and λ49 and consequently to the proof √ of √ (3) and (4). For n = 9 and 33, the subrings of Q( −3) and Q( −11) with conductors 3 have similar properties but the evaluation of λ9 and λ33 depends on different ideas. Ramanujan, in every case, seems to consider only dicriminants, fundamental or not, which have only two classes. We shall do the same in this note and restrict further to odd discriminants with class number 2 since we are dealing only with S (τ). √ Let K = Q( d), d < 0 be an imaginary quadratic field with discriminant d and class number h(d). Let R be the maximal order in K and for any rational integer f ≥ 1, R f the ring with conductor f . Clearly R = R1 . The ring R f has discriminant d f 2 and a minimal basis (1, θ) where √   −1+i D f 2   , if d f 2 ≡ 1(mod 4)   2 (7) θ= √    2 ≡ 0(mod 4) i Df2 , if d f 2

2

153

where D = |d|. We consider in R f only ideals which are prime to f . As is well known, there is a (1, 1) correspondence between ideals in R prime to f and those in R f prime to f . If a and b are two ideals prime to f , in R f , then they are said to be in the same ideal class in R f if there exist λ and µ in R f both prime to f such that λa = µb This leads to a class division of ideals of R f into ideal classes. The number h(d f 2 ) of these ideal classes is given by h(d f 2 ) =

h(d) · ϕ([ f ]) e · ϕ( f )

(8)

179

where ϕ([ f ]) denotes the Euler function of the ideal [ f ] in R so that Y 1 ) (9) ϕ([ f ]) = f 2 (1 − Nκ κ/ f where κ runs through all prime ideals in R dividing f and ϕ( f ) in the denominator is the ordinary totient function. The number e is the index of the group of units in R f in the group of units of R. It is to be noted that formula (8) is still true if d > 0. Let C be any ideal class in R f . The zeta function ζ ∗f (s, C) of the class C is defined by X ζ ∗f (S , C) = (Na)−s , Re s > 1 (10) a∈C (a, f )=1

where a runs through all integral ideals in C which are prime to f . If ℓ is an ideal in the class C −1 which is prime to f , then ζ ∗f (s, C) =

(Nℓ) s X |Nα|−s w 0,α∈ℓ

(11)

(α, f )=1

w being the number of roots of unity in R f . If f > 1, then w = 2. If f > 1, because of the restrictive summation in (11), it is not possible to apply at once the Kronecker Limit formula to ζ ∗f (s, C). We shall see that for our purposes, the zeta function of the class C in the extended sense defined below would be sufficient. Put (Nℓ) s X |Nα|−s (12) ζ f (s, C) = w α,0 α∈ℓ

with α running through all elements of ℓ not equal to zero. The sum in 154 (12) is then an Epstein zeta function. We shall choose the ideal class C in a particular way using the (1, 1) correspondence between ring ideal classes and binary, positive, primitive integral quadratic forms with discriminant d f 2 , d being, of course, a negative fundamental discriminant.

180

2

Let p be a prime number dividing d but not f (we assume that such primes exist). We shall construct a binary, primitive, positive form which represents p primitively. Let px2 + bxy + cy2 be the quadratic form with discriminant d f 2 so that b2 − 4pc = d f 2 .

(13)

Clearly p|b and so if b = pb1 , d = pd1 , then pb21 − 4c = d1 f 2 . Let p be odd. If d f 2 is odd, then p − d1 f 2 ≡ 0(mod 4) and so we choose b1 = 1 and c = (p − d1 f 2 )/4. The quadratic form px2 + pxy +

p − d1 f 2 2 y 4

is primitive since p ∤ d1 and is odd. It has discriminant d f 2 . We choose the ideals class C to be the inverse of the ideal class C −1 represented by the ideal ℓ with basis (1, z) where p −1 + d f 2 /p (14) z= 2 The ideal clearly has norm equal to 1/p. Note that (p, f ) = 1. If p is odd and d f 2 is even, then, by (13), 2p|b1 . One easily sees that we can again take the quadratic form to be px2 + 2pxy + (p − d1 f 2 /4)y2

181

which means that the ideal class C −1 is represented by the ideal (1, z) with q (15) z = d f 2 /2p

In a similar way, we obtain for C −1 the ideal class represented by the ideal (1, z) with  p 1  2    2 (1 + d f /2) , p = 2, (d/4) odd z= (16) p    d f 2 /4 , p = 2, (d/4) even

If we go back to formula (12) and take C = C0 as the principal class and 155 apply the Kronecker limit formula ([5, formula 6]), we have − lim [ζ f (s, C0 ) − ζ f (s, C) = ps→1 = (4π/w D f 2 ) log(N[1, z])1/2 |η(z)/η(θ)|2 )

(17)

where w = w f is the number of roots of unity in R f , z is given by (14), (15) and (16) and θ by (7). The two functions ζ f (s, C0 ) and ζ f (s, C) are the zeta functions of the classes C0 and C respectively in the extended sense.

3

In order to proceed further, it is necessary to obtain another expression for the left side of (17). Let χ be any character of the ring ideal class group of R f . We define the L-function X χ(a) , Re s > 1. (18) L f (s, χ) = (Na) s a∈R f

(a, f1 )=1

since χ is a multiplicative function on the ideal of R f prime to f . Y L f (s, χ) = (1 − χ(κ)Nκ−s )−1

(19)

κ| f

Furthermore L f (s, χ) =

X C

χ(C)ζ ∗f (s, C)

(20)

182

3

where C runs through all ring ideal classes of R f . If χ is a non-principal character, it is shown by Meyer that we have even the relation X L f (s, χ) = χ(C)ζ f (s, C) (21) C

in terms of the zeta functions of classes in the extended sense. Formula (21) has the advantage that one can apply the Kronecker limit formula. If now we assume that every genus of ring ideal classes of R f has only one class in it, then one has     X r−2  (22) L(s, χ) − 2 ζ f (s, C0 ) − ζ f (s, C) = χ(c)=−1

where the sum runs through all characters which take the value −1 on C, 2r−1 being the number of genera. We now define the genus characters. Let d f 2 have the decomposition d f 2 = d0 d0∗ 156

where d0 is a fundamental discriminant and d0∗ a discriminant. For such a decomposition, we have a character of the class group of R f ! d0 χd0 (κ) = Nκ ! d0 2 for all prime ideals κ which do not divide d f ; being the Kronecker symbol ([9, p. 380 et seq]). For prime ideals not dividing d0 , (??) also makes sense. If κ divides d0 , then take ! d0∗ χd0 (κ) = Nκ It is to be noted that d0 and d0∗ have only divisors of f as common divisor. We shall now confine ourselves to the case d f 2 odd, h(d f 2 ) = 2, (d, f ) = 1.

(23)

183

Since d is a fundamental discriminant, d f 2 = −p f 2 , p ≡ −1(mod 4) Further d f 2 has only one non-trivial decomposition     −p f · f , if f ≡ 1(mod 4) df2 =    − f · p f , if f ≡ −1(mod 4)

(24)

(25)

Following Siegel [8], we see that there is only one L-series and     L−p f (s) · L f (s), if f ≡ 1(mod 4) L f (s, χ) =    L− f (s) · L p f (s), if f ≡ −1(mod 4) where L∗ (s) is the ordinary Dirichlet L series. From (22), we get

− lim (ζ f (s, C0 ) − ζ f (s, C)) = L f (1, χ) s→1

and therefore, from (17) using the fact that w = 2 for f > 1, we get      η   1   √   p     η

√

! 2        h(−p f )·h( f ) , f ≡ 1(mod 4) 2     (ǫ( f )  ! =   √    (ǫ(p f )h(p f )·h(− f )·2/w0 , f ≡ −1(mod 4)  −1+i f 2 p      2

−1+i

f 2 /p

where h(−p f ), . . . are class numbers,pǫ( f ) and ǫ(ppf ) are the fundamental units in the real quadratic fields Q( pf ) and Q( p f ) respectively and w0 the number of roots of unity in Q( − f ). From the definition of λn and formula (??), we see that we can evaluate λn if p = 3 and the conditions (23) are satisfied. They are indeed satisfied in cases d f 2 = −3 · 52 , −3 · 72 as seen from the tables in [1]. In case p = 3, f = 5, we have

157

184

3

√ ǫ( f ) = ( 5 + 1)/2 and h(−15) = 2. We therefore have

λ25

   6 √  −1+i 25/3    √ 6   η   √ 2   5 + 1  1     = (2 + 5)2  =  = √ , √     2 −1+i 75    3 3      η 2

In a similar way, if p = 3, f = 7, h(21) = 1, ǫ(21) = (5 + gives since w0 = 2, λ49

√

(5)R

21)/2. This

√ 3  √  5 + 21   = 55 + 12 21. =  2

(5)R

√ √ η(i 3/5) f (i 3/5) + 1 − S (i 3) = √ · √ η(i5 3) f (i5 3)

(26)

The value of λ25 enables us to prove √ Ramanujan’s statements (??) and (??). It is known, by taking τ = i 3 in ([4, p. 700]) that √

−1

[S (i 3)]

√

where f (τ) is Schlefli’s modular function f (τ) = e−πi/24 ·

η((1 + η)/2) = f (−1/τ) η(η)

(27)

If we use the formula η(−1/τ) = (−iτ)1/2 η(τ) Then

√ √ √ !1/2 η(i 3/5) f (i 3/5) 5 η[(1 + i 25/3)/2] = √ √ √ √ η(i5 3) f (i5 3) 3 η(1 + i 75/2)

so that, by definition of λ25 , √ √ √ √ √ = 5( 5 + 1)/2 [S (i 3)]−1 + 1 − S (i 3) = 5 − λ1/6 25

(28)

185

√ Solving √ the above quadratic equation for S (i 3) and using the fact that S (i 3) > 0, we get q √ √ −(3 + 5) + 6(5 + 5) √ S (i 3) = (3)R 4 √ The value of S (i/ 3) can be obtained by again using (27) and (28) or by using the formula √ √  √     5 − 1   5 − 1  √  5 − 1  (29) S (τ) +  S (−1/τ) +  = 5   2 2 2

which was stated by Ramanujan in his Notebooks. It was first proved 158 by Watson. (See also [4]). If we use the formulae (29) and  5  √  √ 5 !     5 − 1  5−1 5 (S (τ))5 +    + (S (−1/5τ) + 2  2 (30) √ 5 √  5 − 1   = 5 5  2 √ proved by us, one can obtain the values of S [i5k ( 3)1 ] where k is any rational integer and 1 = ±1.

4

We shall prove now the other statements of Ramanujan in (??). In the first place, √ √ 6  1  η(i/ 3) f (i/ 3)  λ1 = √  √ · √  . 3 3 η(i 3) f (i 3)

If we now use the formulae (27) and (28) we get λ1 = 1

Consider now λ9 . By definition, √  6 1  η((1 + i 3)/2)   λ9 = √  √ 3 3 η((1 + i3 3)/2)

186

4

If we use the product expansion of the η-function, then −i η(ω) λ9 = √ 3 3 η(3ω)

!6

√ −1 + i 3 . , ω= 2

(31)

On the other hand, η(3ω) α = 27 η(ω)

!6

is a root of the equation x4 + 18x2 + λ3 (ω)χ − 27 = 0 where λ3 (ω) =

p

j(ω) − 1728

and j(ω) is the well-known Klein’s invariant ([9, p. 504]). Weber has shown that √ √ λ3 ((−1 + i 3)/2) = i.24 3 159

and therefore λ = λ9 is a root, positive, of x4 − 8x3 + 18x2 − 27 = 0. This however equals (x + 1)(x − 3)3 which shows that λ9 = 3

(5)R

√

Consider now ω = (1 + i 11)/2. Then λ33 where

−i η(ω) = √ 3 3 η(3ω)

!6

√ 3 3i = α

√ 6   η(3(−1 + i 11)/2)   = 27 ·  √ η((−1 + i 11)/2)

REFERENCES

187

From Weber [9, p. 504], √   √  −1 + i 11   = 56i 11 λ3  2

and hence λ is the positive root of √ 9x4 − 56 33x3 + 18 · 32 · x2 − 35 = 0

(32)

This quartic equation can be solved by the classical methods of the theory of algebraic equations. One obtains √ λ = λ33 = 3(6 + 33). √ In fact, Weber (loc. cit) has given the values of λ3 ((−1 + i n)/2) for n = 19, 43, 67 and 163 and thus λ3n is a root of a quadratic equation like (32) from which λ3n can be evaluated.

References [1] Z. I. Borevich and I. R. Shafarevich : Number Theory, Academic Press, New York (1966). [2] R. Fricke : Die elliptische Funktionen und ihre Anwendungen, Bd II, B.G. Teubner, Berlin (1922). [3] C. Meyer : Die Berechnung der Klassenzahl abelscher K¨orper 160 u¨ ber quadratischen Zahlk¨orpern, Akademie Verlag, Berlin (1957). [4] K. G. Ramanathan : Ramanujan’s continued fraction, Indian Jour. Pure Appl. Math., 16(1985), 695-724. [5] K. G. Ramanathan : Some applications of Kronecker’s limit formula, Jour. Ind. Math. Soc., 52(1987), 71-89. [6] S. Ramanujan : Notebooks, Vol. 2, Tata Institute of Fundamental Research, Bombay (1957).

188

REFERENCES

[7] S. Ramanujan : The Lost Notebook and other unpublished papers, Narosa Publishing House, New Delhi (1987). [8] C. L. Siegel : Analytische Zahlentheorie II, G¨ottingen (1963). [9] H. Weber : Lehrbuch der Algebra, Bd. III, Braunschweig (1908). A1 Sri Krishna Dham 70, L. B. S. Marg Mulund (West) Bombay 400 080

THE ADJOINT HECKE OPERATOR II By R. A. Rankin 161

1 Introduction

Progress on the theory of modular forms and associated Euler products can be divided roughly into three stages. At the first fundamental stage there is the work of Hecke [3], who introduced the linear operators T n now associated with his name. The second stage comprises the work of Petersson [8], who observed that the space M of cusp forms of given level, weight and character is a finite-dimensional Hilbert space, and showed that the adjoint Hecke operator T n∗ is a scalar multiple of T n , provided that n is a prime to the level N of M. The foundations of the third stage were laid by Atkin and Lehner [1], who separated off from M the subspace M − consisting essentially of forms of lower level, and concentrated their attention on its orthogonal complement M + , showing by delicate methods that M + has an orthogonal basis of forms that are eigenforms for all the operators T n and not only for those with n prime to N. The present paper arose from an effort to simplify the arguments of the third stage, by investigating the properties of the adjoint operator T n∗ for all n, and showing, if possible, that it commutes with T n on the subspace M + . We recall that, for any forms f and g in M, T n∗ is defined by ( f |T n , g) = ( f, g|T n∗ ). (1.1) Petersson proved that T n∗ = χ(n)T n if (N, n) = 1, where χ is the associated Dirichlet character. For this he provided two proofs. Of these one [8] was fairly direct, but had a combinatorial part in which a common left and right transversal of a certain group was shown to exist (Hilfssatz 2), and did not seem applicable to other values of n. On the other hand, his other earlier proof [7] (p. 68), although only valid when the 189

190

2 GROUPS, MATRICES AND CHARACTERS

weight k of the space exceeds 2, seemed more promising, although technically somewhat complicated, as it involved the evaluation of G|T n for an arbitrary Poincar´e series G in M. This is the method developed in my first paper under the same title [10], where is yielded an apparently previously unknown explicit definition of T n∗ for (n, N) , 1. The case when N is a prime number was then investigated in detail using properties of Poincar´e series. However, for composite N this method becomes decidedly more complicated, because of increased number of incongruent cusps of verying cusp widths 162 and parameters. In the present paper the general case is considered in a relatively simple way without the use of Poincar´e series, and the explicit definition of the adjoint operator, found in [10], is proved by a different method.

2 Groups, matrices and characters

As is customary, we

write Γ(1) := SL(2, Z)

(2.1)

for the modular group and, for any positive real number m, we denote by Ωm the set of all matrices " # a b T := (2.2) c d belonging to GL(2, R) and having determinant m. We shall be particularly concerned with the group Γ0 (N) = {T ∈ Γ(1) : c ≡ 0(mod N)},

(2.3)

where N is a positive integer, and require the following special matrices in Γ(1) : " # " # " # " # 1 0 1 1 0 −1 1 0 I= , U= , V= , W= . (2.4) 0 1 0 1 1 0 1 1 Write also Γ(N) = {T ∈ Γ(1) : T ≡ I(mod N)}.

(2.5)

191

For various positive rational values of m we write " # 1 0 Jm = . 0 m

(2.6)

Throughout k will be a positive integer and, for typographical reasons we write 1 (2.7) K = k − 1. 2 Let H = {z ∈ C : Imz > 0}, (2.8) and put, as customary, e(z) = exp(2πiz)

(z ∈ C).

(2.9)

For T ∈ Ωm , we define T z :=

az + b , T : z = cz + d. cz + d

(2.10)

For any function f : H → C and T ∈ Ωm (m > 0) we define f (z)|T := (T : z)−k (det T )k/2 f (T z).

(2.11)

This depends of course on k, which is fixed. Note that f (z)|Jm = m−k/2 f (z/m), f (z)|Jm−1 = mk/2 f (mz).

(2.12)

The letters p and q will always denote prime numbers, and we write P = {0, 1, . . . , p − 1}, P∗ = {1, 2, . . . , q − 1},

(2.13)

Q = {0, 1, . . . , q − 1}, Q∗ = {1, 2, . . . , q − 1}.

(2.14)

Throughout, χ denotes a character modulo N such that χ(−1) = (−1)k ;

163

(2.15)

it follows that, when k is odd, N ≥ 3. We denote by N(χ) the conductor of χ and put n(χ) = N/N(χ). (2.16)

192

3

M(N, K, χ) AND ITS SUBSPACES

Note that, for any positive integer r, r|n(χ) ⇔ N(χ)|(N/r).

(2.17)

The principal character modulo m is denoted by ǫm . Accordingly, χ may be written as χ = χ∗ ǫN ,

(2.18)

where χ∗ is a primitive character modulo N(χ). When (2.17) holds χr := χ∗ ǫN/r

(2.19)

is a character modulo N/r with conductor χ∗ .

3 M(N, k, χ) and its subspaces

A standard notation for the vector space of cusp forms belonging to a group Γ and having weight k and multiplier system v is {Γ, k, v}0 (3.1) and we shall write M = M(N, k, χ) = {Γ0 (N), k, χ}0 .

(3.2)

Thus M is a space of level N, weight k and character χ. For any positive integers r and s satisfying r|n(χ), s|r

(3.3)

C(r, s, χr ) := M(N/r, k, χr )|Js−1 ,

(3.4)

we define where χr is defined by (2.19). Note that C(1, 1, χ1 ) = M.

(3.5)

C(r, s, χr ) ⊆ M(N s/r, k, χr/s ) ⊆ M.

(3.6)

It is easy to see that

193

Whenever r > 1 and s < r the level of C(r, s, χr ) is less than N. When r = s > 1 the level is N, but the space is isomorphic to M(N/r, k, χr ) and so can be regarded as a space of essentially lower level. For this reason we define M M− = C(r, s, χr )(r > 1, r|n(x), s|r). (3.7) r,s

Then M − is a subspace of M of essentially lower level and any member of M − is called an oldform. Now M is a finite-dimensional Hilbert space, and we define M + to be the orthogonal complement of M − in M, so that M = M− ⊕ M+. (3.8) The definition of M − can be simplified, as the following Theorem 164 shows. Theorem 3.1. We have M− =

M

C(p, χ p ),

(3.9)

p|n(χ)

where C(p, χ) = C(p, 1, χ p ) ⊕ C(p, p, χ p ).

(3.10)

Proof. We observe that C(rs, t, χrs ) ⊆ C(r, t, χr )(t|r, rs|n(χ))

(3.11)

C(rs, rt, χrs ) ⊆ C(s, t, χs )(t|s, rs|n(χ))

(3.12)

and It is clear that (3.11) holds. To prove (3.12) take any F ∈ C(rs, rt, χrs ), so that we can put F = g|Jt−1 = f |Jrt−1 ,

194

4

THE FRICKE INVOLUTION HR

where f ∈ M(N/rs, k, χrs ). Now take any T ∈ Γ0 (N/s), so that T 1 := Jr−1 T Jr ∈ Γ0 (N/(rs)). Then g|T = f |Jr−1 T = f |T 1 Jr−1 = χrs (T 1 ) f |Jr−1 = χrs (T 1 )g = χs (T )g, so that g ∈ M(N/s, k, χs ), and this proves (3.12). By successive applications of (3.11) and (3.12) we complete the proof of the theorem. 

4 The Fricke involution Hr

For any r ∈ N define " # 0 −1 Hr = Jr V = r 0

(4.1)

so that Hr2 = −rI and Hr−1 = −r−1 Hr . It is easily verified that Hr−1 Γ0 (r)Hr = Hr Γ0 (r)Hr−1 = Γ0 (r).

(4.2)

Lemma 4.1. Let N = rs where r|n(χ). Then M(N/r, k, χr )|H s = M(N/r, k, χr ).

(4.3)

M(N, k, χ)|HN = M(N, k, χ).

(4.4)

In particular Further, if N = pt, where p|n(χ), then C(p, p, χ p )|HN = C(p, 1, χ p )

(4.5)

C(p, 1, χ p )|HN = C(p, p, χ p ).

(4.6)

and

195 165

Proof. Let f ∈ M(N/r, k, χr )|H s so that f = g|H s , where g ∈ M(N/r, k, χr ). Take any T ∈ Γ0 (N/r) so that H s−1 T H s ∈ Γ0 (s). Then f |T = g|T H s = g|H s H s−1 T H s = χr (H s−1 T H s )g|H s = χr (H s−1 T H s ) f = χr (T ) f. Moreover C(p, p, χ p )|HN = C(p, 1, χ p )|J −1 p H N = C(p, 1, χ p )|Ht = C(p, 1, χ p ), by (4.4) with N replaced by t. This gives (4.5) and (4.6) follows by  replacing χ by χ and operating again on the right with HN . Lemma 4.2. If N = pt, then HN J p U u HN−1 = pW −ut J −1 p . Proof. Straightforward.

(4.7) 

5 The Hecke operators T n

For any n ∈ N and any f ∈ M(N, k, χ) define the operator T n (N, χ) = T n by f |T n = nk

d XX

χ(a) f |Jd Uu Ja−1 ,

(5.1)

ad=n u=1

where J is given by (2.7), and observe that, for any prime p we have, in particular,    X u −1 K (5.2) f |J p U + χ(p) f |J p  ; f |T p = p  u∈P

196

5 THE HECKE OPERATORS T N

see (2.12). It is clear that, in (5.2), u can run through any complete set of residues modulo p. If N = pt, p ∤ t and f ∈ C(p, 1, χ p ), (5.3) if follows from (5.2) that f |T p (t, χ p ) = f |T p (N, χ) + pK χ p (p) f |J −1 p

(5.4)

and we note that χ p (p) , 0 in this case. We now summarize some of the known properties of the operators. For any f ∈ M, let ∞ X f (z) = a(r)e(rz). (5.5) r=1

166

Then f (z)|T n =

∞ X

an (r)e(rz),

(5.6)

dk−1 χ(d)a(nr/d2 ).

(5.7)

(n ∈ N)

(5.8)

r=1

where an (r) =

X

d|(n,r)

Moreover, we have M|T n ⊂ M and ( f |T m )|T n =

X

dk−1 χ(d) f |T mn/d2 (m ∈ N, n ∈ N).

(5.9)

d|(m,n)

It follows that the operators commute and that T n is completely determined when T p is known for each prime p|n. Moreover, as shown in Petersson [8], if f and g belong to M, then ( f |T n , g) = χ(n)( f, g|T n ) for (n, N) = 1.

(5.10)

Here the inner product is defined for cusp forms f and g of weight k on a subgroup Γ of finite index h in Γ(1) by " 1 f (z)g(z)yk−2 dxdy. (5.11) ( f, g) = ( f, g; Γ) = h F

197

where x = Re z, y = Im z and F is any fundamental region in H for Γ. In §6 we shall require this definition for various subgroups Γ contained in Γ(1). It follows from (5.10) that T n∗ , the adjoint operator, is given by T n∗ = χ(n)T n for (n, N) = 1.

6 The adjoint operator T p∗ for p|N

(5.12)

For any prime p|N and

f ∈ M define the operator T p∗ = T p∗ (N, χ) by

f |T p∗ : = f |HN T p HN−1 = f |HN−1 T p HN X = pK f |HN J p U u HN−1 =p

K

u∈P X

f |W −ut J −1 p

(6.1)

(6.2)

u∈P

by Lemma 4.2. Since T p (N, χ) = T p (N, χ) it follows from (4.4) that M|T p∗ ⊆ M.

(6.3)

Theorem 6.1. For any prime p|N, T p∗ is the adjoint operator to T p ; i.e. ( f |T p∗ , g) = ( f, g|T p ) for f and g in M.

(6.4)

For the proof we require the following Lemma, which we quote from Theorem 5.2.1 of [9]. Lemma 6.2. (i) If Γ1 and Γ2 are subgroups of Γ(1) of finite index in Γ(1) 167 and Γ1 ⊆ Γ2 , then ( f, g : Γ1 ) = ( f, g : Γ2 ) whenever f and g both belong to {Γ2 , k, v}0 .

198

6

THE ADJOINT OPERATOR T P∗ FOR P|N

(ii) Let Γ be a congruence subgroup of Γ(1) of finite index and let, for any prime p, Γ p = Γ ∩ Γ(p). (6.5) Suppose that f and g belong to {Γ, k, v}0 and let L ∈ Ω p . Then ( f, g; Γ) = ( f |L, g|L; L−1 Γ p L).

(6.6)

Proof of Theorem. Take any f and g in M and write F = f |T p∗ , so that f ∈ M by (6.3). Note that, if S ∈ Γ(pN), then ′ −ut −1 f |W −ut J −1 Jp , p S = f |S W

where S ′ ∈ Γ(N), so that χ(S ′ ) = 1. Hence, for any u ∈ Z, f |W −ut J −1 p ∈ {Γ(pN), k, 1}0 , and so, by Lemma 6.2(i) and (6.2), (F, g : Γ0 (N)) = (F, g; Γ(pn)) = pK

X ( f |W −ut JP−1 , g; Γ(pN)) u∈P

X = pK ( f |U −u W −ut JP−1 , g; Γ(pN)). u∈P

Write Au = J p W ut U u = W uN J p U u ∈ Ω p and note that, when Γ = Γ(pN), Γ p = Γ(pN), by (6.5), so that −1 2 A−1 u Γ p Au = J p Γ(pN)J p ⊇ Γ(pN ).

199

Taking L = Au in (6.6), we get (F, g; Γ0 (N)) = pK

X ( f, g|Au ; Γ(pN 2 )) u∈p

    X uN u 2 K g|W J p U ; Γ(pN ) = p  f, u∈P

   X  K u 2 = p  f, g|J p U ; Γ(pN ) u∈P

= ( f, g|T p ; Γ(pN 2 )) = ( f, g|T p ; Γ0 (N)).

This completes the proof of the theorem. Theorem 6.3. Let m and n be positive integers. Then the following pairs 168 of operators on M commute : (i) T m , T n ; (ii) T m∗ , T n∗ ; (iii) T m , T n∗ provided that (m, n, N) = 1. Proof. (i) follows from (5.9) and this yields (ii), since ( f |T m∗ T n∗ , g) = ( f, g|T n T m ). By (5.12) and (i) we need only prove that T p∗ and T q commute when p and q are different primes dividing N. Write N = pqs and define S u,w by 2

−wq s −1 S u,w Jq U up W −wqs J −1 J p Jq U u p =W

for (u, w) ∈ Q × P. Then it is easy to see that S u,w ∈ Γ0 (N) and that χ(S u,w ) = 1. Now, if f ∈ M, since up and wq run through complete sets of residues modulo p and modulo q respectively, we have XX 2 u f |T p∗ T q = (pq)K f |W −wq s J −1 p Jq U u∈Q w∈P

= (pq)K

XX

f |Jq U up W −wqs J −1 p

w∈P u∈Q

= f |T q T p∗ . 

200

7

THE ACTION OF THE OPERATORS ON M

7 The action of the operators on M Theorem 7.1. For all n ∈ N M − |T n ⊆ M − · M − |T n∗ ⊂ M − .

(7.1)

In particular, if p is any prime dividing N and N = pt, we have : (i) For (n, N) = 1 and p|n(χ) C(p, 1, χ p )|T n ⊆ C(p, 1, χ p ), C(p, p, χ p )|T n ⊆ C(p, p, χ p ). (7.2) (ii) If p|n(χ), C(p, p, χ p )|T p ⊆ C(p, 1, χ p ), C(p, 1, χ p )|T p∗ ⊆ C(p, p, χ p ). (7.3) (iii) If p and q are different primes dividing n(χ), C(q, 1, χq )|T p ⊆ C(q, 1, χq ), C(q, 1, χq )|T p∗ ⊆ C(q, 1, χq ), (7.4) C(q, q, χq )|T p ⊆ C(q, q, χq ), C(q, q, χq )|T p∗ ⊆ C(q, q, χq ). (7.5) (iv) If p|n(χ) and p2 |N, C(p, 1, χ p )|T p ⊆ C(p, 1, χ p ), C(p, p, χ p )|T p∗ ⊆ C(p, p, χ p ) (7.6) (v) If p|n(χ p ) and p2 ∤ N. C(p, 1, χ p )|T p ⊆ C(p, χ p ), C(p, p, χ p )|T p∗ ⊂ C(p, χ p ).

(7.7)

Proof. In view of Theorem 3.1, (7.1) will follow if we prove parts (i)(v) of the theorem. For the proof of (i) see pp. 321-322 of [9]. By (6.1), (4.5) and (4.6) it is only necessary to prove those parts of (7.3)-(7.7) that involve the operator T p . 169 For (7.3) we note that C(p, p, χ p )|J p U u = C(p, 1, χ p )|U u = C(p, 1, χ p ).

201

For (7.4) note that the operator T p (N, χ) is the same as the operator T p (N/q, χq ) since p divides N/q and the latter operator maps the space M(N/q, k, χq ) into itself. Also, if N = pqs and f = g|Jq−1 ∈ C(q, q, χq ) then g ∈ C(q, 1, χq ) and X X f |T p = pK g|Jq−1 J p U u = pK g|J p U uq Jq−1 u∈P

u=P

= g|T p Jq−1 ⊆ C(p, 1, χq )|Jq−1 = C(q, q, χq ). which proves (7.5). (7.6) follows for the same reason as (7.4), since p divides N/p and the operators T p (N, χ) and T p (N/p, χ p ) are identical. Finally, assume that p|n(χ) but p2 ∤ N. We assume (5.3) and deduce that f |T p (N, χ) ∈ C(p, 1, χ p ) ⊕ C(p, p, χ p ) = C(p, χ p ), from (5.4).



Theorem 7.2. For all n ∈ N M + |T n ⊆ M + and M + |T n∗ ⊆ M + . Proof. Take any f ∈ M + and g ∈ M − . Then ( f |T n , g) = ( f, g|T n∗ ) = 0 by Theorem 7.1. Hence f |T n ∈ M + . The proof of the second part is similar.  Theorem 7.3. Let MH = M|HN . Then (MH )− = M − |HN . Proof. By (4.4) MH = M(N, k, χ), so that (MH )− is a vector sum of the spaces C(p, 1, χ p ) and C(p, p, χ p ); the result follows. 

202

8

THE OPERATORS T P∗ T P AND T P T P∗ (P|N)

8 The operators T p∗ T p and T p T p∗ (p|N) Lemma 8.1. Let R be a right transversal of Γ0 (N) in Γ0 (t), where N = tp, and put [ W −wt . (8.1) R0 = w∈P

Then we may take (i) R = R0 , when p ∤ t, and (ii) R = R0 ∪ R∗ , when p ∤ t, where ! (1 + st)/p s R = p JpU W Jp = t p ∗

−1

s

t

(8.2)

and s is chosen so that s ∈ P and st ≡ −1(mod p). 170

This is straightforward : note that R∗ ∈ Γ(1). Lemma 8.2. Let f ∈ M(N, k, χ), where p|n(χ). Then X χ(R) f |R ∈ M(t, k, χ p ), F := R∈R

so that F ∈ M − . Proof. Let the members of R be Rr (r = 1, 2, . . . , h) where h = [Γ0 (t) : Γ0 (N)], and take any S ∈ Γ0 (t). Then RS is also a right transversal of Γ0 (N) in Γ0 (t) and so Rr S = S r R′r , where R′r ∈ R and S r ∈ Γ0 (N). Note that χ(Rr )χ(S ) = χ(S r )χ(R′r ). Then F|S =

h X r=1

χ(Rr ) f |Rr S =

h X r=1

χ(Rr ) f |S r R′r

203

=

h X

X(Rr )χ(S r ) f |R′r

=

h X

χ(S )χ(R′r ) f |R′r

r=1

r=1

= χ(S )F. It follows that F ∈ M(t, k, χ p ) and this proves the lemma.



Lemma 8.3. Suppose that N = pt, where p|t, and that χ is a character modulo N. Then, for some integer m ∈ P, χ(1 + rt) = e(mr/p)(r ∈ Z).

(8.3)

m = 0 if and only if p|n(χ).

(8.4)

Moreover Proof. Since (1 + t) p ≡ 1(mod N), χ(1 + t) = e(m/p) for some m ∈ P and (8.3) follows since 1 + rt ≡ (1 + t)r (mod p). If p|n(χ), then N(χ)|t and so m = 0, Conversely, if N(χ) ∤ t, then χ(n) , χ(n + t) for some n prime to N and so, taking rn ≡ 1(mod N). 1 , χ(1 + rt), from which it follows that m , 0, by (8.3). We now define δ(χ) = 0 if p ∤ n(χ); δ(χ) = 1 if p|n(χ). Further, for any prime p dividing define    0 if     k−2 α(p) =  p if      pk−1 if

Then we have

(8.5)

N we put N = pt, as usual, and p|t, p|n(χ), p ∤ t, p|n(χ), p ∤ n(χ).

(8.6)  171

204

8

THE OPERATORS T P∗ T P AND T P T P∗ (P|N)

Theorem 8.4. Let f ∈ M(N, k, χ) and suppose that p|N. Then f |T p∗ T p − α(p) f ∈ M − ,

(8.7)

f |T p T p∗ − α(p) f ∈ M − .

(8.8)

and Accordingly, if f ∈ M + , then f |T p T p∗ = f |T p∗ T p = α(p) f.

(8.9)

Proof. We write N = pt and, in the first instance, assume that p ∤ t. Then we can write χ = ψ p ψt , where ψ p and ψt are characters modulo p and modulo t, respectively. For any integers u, v, w we write S (u, v, w) := W −wt U u W −vt " # 1 − uvt u = t(−v − w + uvwt) 1 − uwt and, for any n ∈ P∗ we define n′ = P∗ by nn′ ≡ 1(mod p). The finite set of ordered pairs P × P can be written as [ Av P × P = A∗ ∪

(8.10) (8.11)

(8.12)

v∈P

where A∗ = {(w, u) ∈ P2 : w , 0, u = (wt)′ }

(8.13)

A0 = {(w, u) ∈ P2 : w = 0}

(8.14)

and, for v ∈ P∗ Av = {(w, u) ∈ P2 : u , (vt)′ , w = (ut − v′ )′ }.

(8.15)

It is easily checked that these p + 1 sets are disjoint and that their union is P2 . Note also that, for (w, u) ∈ Av (v , 0), we have S (u, v, w) ∈ Γ0 (N).

205

For any f ∈ M(N, k, χ) we write X X s∗ = f |W −wt U u , sv = f |W −wt U u (v ∈ P) (w,u)∈A∗

(8.16)

(w,u)∈Av

so that f |T p∗ T p = pk−2 {s∗ +

X

sv }.

v∈P

For (w, u) ∈ A∗ , we put S w = W −wt U u (R∗ )−1 ,

(8.17)

where R∗ is defined by (8.2). Then S w ∈ Γ0 (N) and χ(S w ) = χ{wst + (1 − wut)(1 + st)/p} = ψ p (−w)ψt (p) and so, by (8.16),

172

s∗ =

X

f |S w R∗ = ψt (p)

w∈P∗

X

ψ p (−w) f |R∗

w∈P∗

= (p − 1)δ(χ)ψ t (p) f |R∗ ,

(8.18)

by (8.5), since p|n(χ) if and only if ψ p is the principal character. Clearly X s0 = f |U u = p f (8.19) w∈P∗

and, for v , 0, sv =

X

f |S (u, v, w)W vt

X

χ(1 − uvt) f |W vt

(w,u)∈Av

=

(w,u)∈Av

=

X u∈P

χ(1 − uvt) f |W vt =

X u∈P

ψ p (1 − uvt) f |W vt

206

8

THE OPERATORS T P∗ T P AND T P T P∗ (P|N)

= δ(χ)(p − 1) f |W vt

(8.20)

Accordingly, by (8.18, 19, 20) we have      X      vt ∗ k−2  ∗ f |W + f T p T p = p (p f + δ(χ)(p − 1)  ψ (p) f |R  . (8.21)  t    ∗  v∈P

This gives

f |T p∗ T p = α(p) f for p ∤ t, p ∤ n(χ).

(8.22)

If, however, p|n(χ), then ψt = χt and f |T p∗ T p = α(p) f + pk−2 (p − 1)F,

(8.23)

where F=

X

f |W vt + ψt (p) f |R∗

=

X

χt (R) f |R.

v∈P

(8.24)

R∈R

It follows from Lemma 8.2 that F ∈ M − . It remains to consider the case when p|t. In this case S (w, w, u) ∈ Γ0 (N) and we write [ P×P= Bw , (8.25) w∈P

where Bw = {(w, u) : u ∈ P}. 173

Then f |T p∗ T p = pk−2

X

tw ,

w∈P

where tw =

X

f |W −wt U u

(w,u)∈Bw

=

X u∈P

f |S (u, w, w)W wt

(8.26) (8.27)

207

X

=

χ(1 − wut) f |W wt

(8.28)

t0 = p f.

(8.29)

u∈P

Hence When w ∈ P∗ , we have by Lemma 8.3 that χ(1 − wut) = e(−mwu/p)(m ∈ P), where m = 0 if and only if p|n(χ). Hence, by (8.5), tw = pδ(χ) f |W wt and so f |T p∗ T p = pk−1 { f + δ(χ)

X

f |W wt }.

(8.30)

w∈P∗

Accordingly, if p|n(χ), then f |T p∗ T p = pk−1

X

f |R ∈ M −

(8.31)

R∈R0

while, when p ∤ n(χ), f |T p∗ T p = pk−1 f.

(8.32)

Accordingly, (8.7) holds in either case. To prove (8.8) we write f = g|HN , where g ∈ M(N, k, χ). Then f |T p T p∗ = g|HN T p T p∗ = g|HN T p HN−1 · HN T P∗ = g|T p∗ HN T p∗ = g|T p∗ T p HN = {α(p)g + h}|HN = α(p) f + h|HN = α(p) f + h′ , say, where h ∈ M − (χ) and therefore, by Theorem 7.3. h′ ∈ M − . It remains to prove (8.9). If f ∈ M + , then f |T p∗ T p ∈ M + and so ∗ f |T p T p − α(p) f ∈ M + . It follows from (8.7) that f |T p∗ T p = α(p) f. The proof that f |T p T p∗ = α(p) f is similar. Theorem 8.4 is proved.



9 THE ACTION OF THE OPERATORS ON M +

208

Corollary 8.5. Suppose that f ∈ M + and that, for some p|N we have 174 p|n(χ). Then, in the notation of Lemma 8.2, X χ(T ) f |T = 0. T ∈R

This follows from (8.24) and (8.31).

9 The action of the operators on M +

In this section we are concerned solely with the space M + . We recall that a linear operator on a Hilbert space is said to be normal if it commutes with its adjoint. Let F be the family of all the operators T n and T n∗ (n ∈ N). Then it follows from Theorems 6.3 and 8.4 that F is a family of normal operators acting on M + and that any two members of F commute. Now M + is a finite-dimensional Hilbert space and so, from a standard theorem on operators on such spaces (see pp. 267 and 291 of [2]), we deduce Theorem 9.1. M + has an orthogonal basis of forms, each of which is an eigenvector for all the operators in F . Moreover, if f is such a basis element, with Fourier expansion (5.5), we may assume that f is primitive, i.e. that a(1) = 1, and then f |T n = a(n) f, f |T n∗ = a(n) f (n ∈ N).

(9.1)

a(n) = χ(n)a(n) for (n, N) = 1,

(9.2)

Further, and, for any prime p dividing N, |a(p)|2 = α(p).

(9.3)

Proof. If f is an eigenvector of all the operators T n (n ∈ N) with eigenvalue λ(n), then we have, taking r = 1 in (5.7) λ(n)a(1) = an (1) = a(n)(n ∈ N), which shows that a(1) , 0; by division, we may assume that a(1) = 1 and then we have λ(n) = a(n). Since T n∗ = χ(n)T n for (n, N) = 1, (9.2) follows and (8.9) gives (9.3).

REFERENCES

209

Each basis element is called a newform and M + is the newform space. It may be noted that, from (8.6) and (9.3), the absolute value of the eigenvalue a(p), where p divides N, emerges naturally from the proof of Theorem 8.4. In certain cases one can determine a(p) rather than |a(p)|; see [1], [4], [6], [9]. In this connexion I take the opportunity to correct an error in the statement of Theorem 9.4.8 (iii) of [9], where condition (d) should be replaced by p ∤ t1 , p ∤ (N/Nχ ). In conclusion, it may be noted that, although the paper [5] is not 175 concerned with the determination of adjoint Hecke operators, the linear operator Cq there introduced has points of similarity with the operator T q + T q∗ , which is clearly normal on each of the subspaces M, M − and M+. 

References [1] A.O.L. Atkin and J. Lehner : Hecke operators on Γ0 (m), Math, Ann. 185 (1969), 134-160. [2] F.R. Gantmacher : Matrix theory, vol. 1. Chelsea, 1960. [3] E. Hecke : Ueber Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung I, Math. Ann. 144(1937), 1-28 Math. Ann. 114(1937), 316-351. [4] W.C.W. Li : Newforms and functional equations, Math. Ann. 212 (1975), 285-315. [5] W.C.W. Li : Diagonalizing modular forms, J. Algebra 99(1986), 210-231. [6] A.P. Ogg : On the eigenvalues of Hecke operators, Math. Ann. 179 (1969). 101-108.

210

REFERENCES

[7] H. Petersson : Ueber eine Metrisierung der ganzen Modulformen, Jber, Deutsch. Math. Verein. 49(1939), 49-75. [8] H. Petersson : Konstruktion der samtlichen L¨osungen einer Riemannschen Funktionalgleichung durch Dirichlet-Reihen mit Eulerscher Produktentwicklung II, Math. Ann. 166 (1939), 39-64. [9] R.A. Rankin : Modular forms and functions, Cambridge University Press 1977. [10] R. A. Rankin : The adjoint Hecke operator I, J. Madras University (to appear). University of Glasgow Glasgow G12 8QW Scotland

ON ZETA FUNCTIONS ASSOCIATED WITH SELF-DUAL HOMOGENEOUS CONES By Ichiro Satake Let C be and (irreducible) self-dual homogeneous cone is a vec- 177 tor space V with a Q-structure such that the automorphism group G = Aut(V, C )◦ is defined over O. Let M be a lattice in V and let Γ = {g ∈ G|gM = M}. Then by definition the zeta function associated with C is given by X |Γ x |−1 N(x)−s (s ∈ C), (1) ZC (M; s) = x:Γ\C ∩M

where Γ x = {λ ∈ Γ|λx = x} and N(x) is the “norm” of x (see 1). The purpose of this note is to supplement our previous report [SO] in the following points. First, in 2, we will show that, except for the case C = Pr (R) and G is O-split (treated in [?]), the fundamental assumption (2.6) in [SS] is satisfied, so that we can apply the general results of Sato-Shintani on the zeta functions of prehomogeneous vector spaces to our case. In §5, we will determine that poles and the residues of the zeta functions, and, in 6, the functional equations (cf. [SO], Th. 2.3.1, 2.3.3). These will be done including the case d is odd, which was excluded in [SF] and [SO]. In particular, we will show that the matrix U (r) (x) giving the functional equations is always diagonalizable.

1

Let V be a real vector space of dimension n endowed with a positive definite inner product h i. Let C be a self-dual homogeneous cone in V, i.e. an open convex cone with vertex at 0 satisfying the following two conditions : 211

212

1

(i) C is “self-dual”, i.e. one has C = C ∗ = {x ∈ V|hx, yi i0 for all y ∈ C − {0}}. (ii) The automorphism group of C , G = Aut(V, C )◦ = {g ∈ GL(V)|gC = C }◦ , is transitive on C (◦ denotes the connected component of the identity.) In what follows, we assume for simplicity that C is irreducible and exclude the trivial case C = R+ (the half-line of positive numbers). Then G is (the identity connected component of) a reductive algebraic group defined over R with R-rank r ≥ 2 and one has G = G s × R+ , where G s is R-simple. (For the treatment of the reducible case, see [SO].) For any 178 c0 ∈ C , the stabilizer K = G c0 is a maximal compact subgroup of G and one has G/K  C . We can (hence will) assume that the base point c0 and the inner product h i are so chosen that for g ∈ G one has gc0 = c0 if and only if t g−1 = g. We further normalize h i by hc0 , c0 i = r. We set g = Lie G, k = Lie K, k = Lie K and let g = k + p be the corresponding Cartan decomposition. As is well-known (see e.g. [S1]), there exists a unique structure of Jordan algebra on V with the unit element c0 such that, denoting by T x (x ∈ V) the Jordan multiplication y 7→ xy(y ∈ V), one has p = {T x (x ∈ V)}. We denote by N(x) the reduced norm of this Jordan algebra. Then N : V → R is a polynomial function of degree r defined over R satisfying the following conditions : N(c0 ) = 1, N(gx) = det(g)r/n N(x)

(g ∈ G, x ∈ V).

(2)

It is then clear that χ(g) = det(g)r/n is a rational character of the algebraic group G. One can find a system of mutually orthogonal primitive idempotents {ei (1 ≤ i ≤ r)} such that c0 =

r X i=1

ei ,

ei e j = δi j ei ,

213

which we call a “primitive decomposition” of c0 . Then a = {T ei (1 ≤ i ≤ r)}R is a maximal (abelian) subalgebra in p. It is known that the system of R-roots (relative to a) is of type (Ar ) and all the R-roots have the same multiplicity d. One has a direct sum decomposition M V= Vkl , k≤l

where

   {x ∈ V|ek x = x} Vkl =   {x ∈ V|ek x = el x = 1 x} 2

(k = l), (k < l),

and one has dim Vkk = 1 and dim Vkl = d(k < l). Hence one has the relation n d = 1 + (r − 1). (3) r 2 We assume that there is given a O-structure on V (i.e. a O-vector space VO in V with V = VO ⊗O R) such that G is defined over O and c0 ∈ VO . Then, clearly, K, h i, N, χ are all defined over O. We denote by r0 the O-rank of G. Then it can be shown that r0 is a divisor of r. So we set δ = r/r0 . The possible values of δ are as listed below. C Pr (R) Pr (C) Pr (H) P3 (O) P(1, n − 1)

2

r ≥2 ≥2 ≥3 3 2

d 1 2 4 8 ≥3

δ 1 or 2 (r even) δ|r 1 1 1

To define the zeta function, we fix a lattice M in V compatible with the given O-structure and let Γ be the stabilizer of M in G. Then Γ is an arithmetic subgroup of G acting properly discontinuously on C . For x ∈ V, we denote by G x and Γ x the stabilizers of x in G and Γ. Let S denote the singular set {x ∈ V|N(x) = 0} and put V × = V − S . Let Vi× denote the set of all x ∈ V × with “signature” (r − i, i) (see §3).

179

214

2

Then one has an (open G-orbit decomposition V× =

r a

Vi× .

(4)

i=0

× , V× = C . Clearly one has Vi× = −Vr−i 0 × For x ∈ VO , G x is a reductive subgroup defined over O and Γ x is an arithmetic subgroup. We denote by µ(x) the volume of Γ x \G x with respect to a suitably normalized Haar measure on G x . In particular, if x ∈ C , then G x is compact, Γ x is finite, and one has µ(x) = |Γ x |−1 < ∞. × (1 ≤ i ≤ r − 1), one has µ(x) < ∞ except for the case For all x ∈ ViQ r = 2, d = δ = 1. In what follows, we exclude this case, which is treated in [Si] and [?]. For 0 ≤ i ≤ r we define a zeta function associated with the G-orbit Vi× by X µ(x)|N(x)|−s , (5) ξi (M; s) = x∈Γ\M∩Vi×

where the summation is taken over a complete set of representatives of the Γ-orbits in M ∩ Vi× . Clearly one has ξi = ξr−i and ξ0 (M; s) is the zeta function ZC (M; s) associated with the self-dual homogeneous cone C . To discuss the convergence of these zeta functions, we need Lemma 1. Let G1 = {g ∈ G| det(g) = 1} and for f ∈ S (V) (the Schwartz space) set   Z   X  f (gx) d1 g. I( f, M) =  G 1 /Γ∩G 1

180

x∈M

where d1 g is a (suitably normalized) Haar measure on G1 . Then, if dδ ≥ 2, the integral on the right hand side is absolutely convergent and the map f 7→ I( f, M) is a tempered distribution on V. This is proved by applying Weil’s criterion ([W], p. 90, Lem. 5). For c > 0 put Ac = {diag(t1 , . . . , tr0 )|ti ∈ R+ ,

r0 Y i=1

ti = 1, ti /ti+1 ≥ c

215

(1 ≤ i ≤ r0 − 1)}. Then, since every O-root has the multiplicity dδ2 , it is enough to show that Z  r0   Y Sup(1, ti−2 )δ(1+(d/2)(δ−1)) ×    Ac

i=1

×

Y

2

dδ −1 −dδ Sup(1, ti−1 t−1 j ) (ti t j )

1≤i< j≤r0

2

1/2  r0 −1    Y −1 ti dti < ∞.     i=1

−1 )1/r0 , one has for some (See [SS], p. 166, Lem. 4.3.) Putting τi = (ti ti+1 c1 > 0

Sup(1, ti−2 ) ≤ c1 Sup(1, tt−1 t−1 j ) ≤ c1

i−1 Y

τ2k k

k=1

i−1 Y

τ2k k ,

k=1

Y

0 (i < j), τ2k−r k

i≤k< j k≥r0 /2

rY 0 −1 Y −1 (ti t−1 ) = τi−i(r0 −i)r0 . j i< j

i=1

In view of these estimates, one sees that the above integral is 1/2 −1 Z rY 0  0 −1 i(r0 −i)δvi  rY   ti−1 dti τ ≤ c2   i Ac

i=1

for some c2 > 0, where    2 − d(δi + 1) vi =   2 − d(δ(r0 − i) + 1)

i=1

for 1 ≤ i ≤ [r0 /2], for [r0 /2] + 1 ≤ i ≤ r0 − 1.

If dδ ≥ 2, one has vi < 0 for all 1 ≤ i ≤ r0 − 1, which proves our assertion.

216

3

In what follows, we assume that dδ ≥ 2. Then Lemma 1 assures that the fundamental assumption (2.6) in [SS] is satisfies, so that we can 181 apply the general results obtained there. (As we shall see in §3, the condition (2.13) in [SS] is also satisfied). In particular, by [SS], Theorem 2, (i), the Dirichlet series on the right hand side of (5) converges absolutely for Re s > n/r and the function ξi (M; s) thus defined can be continued to a meromorphic function on the whole complex plane. It is known that, even in the case d = δ = 1, the Dirichlet series defining ZC (M; s) has the same property ([?]).

3

We now consider the G-orbit decomposition of the singular set S = V − V × . Every element x in V can be expressed in the form   r  X  (6) x = k  αv ev  with k ∈ K, αv ∈ R, v=1

where (α1 , . . . , αr ) is uniquely determined up to the order (independently of the choice of the primitive decomposition {ev }) ([S3], Prop. 3). We say that x is of rank ρ and of signature (ρ − i, i) if, in a suitable order of (αv ), one has α1 , . . . , αi < 0, αi+1 , . . . , α p > 0, αρ+1 = . . . = αr = 0. For 0 ≤ ρ ≤ r − 1 and 0 ≤ i ≤ ρ, we set S (ρ) = {x ∈ V| rank x = ρ}, (ρ)

S i = {x ∈ V| sign x = (ρ − i, i)}. Then it is easy to see that the G-orbit decomposition of S is given by a (ρ) Si . (7) S = 0≤ρ≤r−1 0≤i≤ρ

(ρ)

Since G1 is transitive on each S i , (7) is also the G1 -orbit decomposition of S . Thus the condition (2.13) in [SS] is certainly satisfied. By [SS], Lemmas 2.7 and 2.8, (i), there exists a G1 -invariant measure dv(ρ) (v) on S (ρ) satisfying the relation dv(ρ) (gv) = χ(g)sρ ,i dv(ρ) (v)

(ρ)

(g ∈ G, v ∈ S i )

(8)

217

for some sρ,i ∈ R. To describe the measure dv(ρ) explicitly, we use the following parametrization of S (ρ) . Set r−ρ r X X e= ev , e′ = c0 − e = ev v=1

v=r−ρ+1

and Vλ = Vλ (e) = {x ∈ V|ex = λx}. Writing v ∈ V in the form v = v1 + v1/2 + v0 , vλ ∈ Vλ (e), we set

182

S (ρ) (e′ ) = {v ∈ S (ρ) |N0 (v0 ) , 0},

(9)

where N0 denotes the norm of the Jordan subalgebra V0 (e). Then S (ρ) (e′ ) is a Zariski open set in S (ρ) and by [S3], Lemma 1 every element v in S (ρ) (e′ ) can be written uniquely in the form v = exp(ey)v0

with y ∈ V1/2 , v0 ∈ V0 ,

(10)

where in general xy = T xy + [T x , T y ] (Koecher’s notation). By a well-known identity in the Jordan algebra one has for any y, y′ in V1/2 (e) [[T y , T y′ ], T e ] = T y(y′ e)−y′ (ye) = 0. Hence one has [ey, ey′ ] = 0 (by [S3], (4)) and exp(ey) · exp(ey′ ) = exp(e(y + y′ )). Therefore S (ρ) (e′ ) can be viewed as a principal bundle of the additive group V1/2 (e) with base space V0 = V0 (e) by the action v 7→ exp(ey)v(y ∈ V1/2 (e), v ∈ S (ρ) (e′ )). It follows, that, if one puts dµ(v) = dy · dv0

for

v = exp(ey)v0 , y ∈ V1/2 , v0 ∈ V0 ,

then there exists a continuous function ϕ = ϕρ,i : V0 (e) → R such that dv(ρ) (v) = ϕ(v0 )−1 dµ(v). Putting g = λ1 in (8), one sees that ϕ is homogeneous of degree ρ(1 + d2 (ρ − 1)) − rsρ,i .

218

3

Not let G0 be the subgroup of G generated by exp T x (x ∈ V0 (e)). Then the Vλ (e)′ s are G0 -invariant. For g0 ∈ G0 , one has g0 |V1 (e) = id. and N0 (g0 v0 ) = χ0 (g0 )N0 (v0 ) for v0 ∈ V0 (e), where χ0 is a rational character of G0 satisfying the relation det(g0 |V0 (e)) = χ0 (g0 )1+(d/2)(ρ−1) .

(11)

Lemma 2. For g0 ∈ G0 , one has det(g0 |V1/2 ) = χ0 (g0 )(d/2)(r−ρ) , χ(g0 ) = χ0 (g0 ).

(12) (13)

Proof. Since g0 e = e, one has for y ∈ V1/2 g0 (ey)v0 = (et g−1 0 y)g0 v0 . Hence g0 (yv0 ) = (t g−1 0 y) · (g0 v0 ), or g0 T v0 t g0 = T g0 v0

on V1/2 (e).

(*)

By [SF], Lemma 2, (i), one has det(2T v0 |V1/2 ) = N0 (v0 )d(r−ρ) . Taking the determinant of both sides of (*), one has det(g0 |V1/2 )2 N0 (v0 )d(r−ρ) = N0 (g0 v0 )d(r−ρ) , 183

whence follows (12). Since χ(g0 )1+(d/2)(r−1) = det(g0 ) = det(g0 |V1/2 ) · det(g0 |V0 ), (13) follows from (11) and (12). By (11) and (12) one has dµ(g0 v) = d(t g−1 0 y) · d(g0 v0 ) = χ0 (g0 )−(d/2)(r−ρ)+1+(d/2)(ρ−1) dµ(v). Hence by (8) and (13) one has ϕ(g0 v0 ) = χ0 (g0 )1−(d/2)(r−2ρ+1)−sρ,i ϕ(v0 ).



219

In particular, ϕ is homogeneous of degree ρ(1− d2 (r−2ρ+1)− sρ,i ). Combining this with what we mentioned above, we see that sρ,i = d2 ρ, which is independent of 0 ≤ i ≤ ρ, and that ϕ(v0 ) is given by cN(v0 )1−(d/2)(r−ρ+1) for some c > 0. We normalize dv(ρ) (v) by putting c = 2−dρ(r−ρ) . Summing up, we have Lemma 3. In the expression (10), aG1 -invariant measure on S (ρ) is given by dv(ρ) (v) = 2−dρ(r−ρ) N0 (v0 )(d/2)(r−ρ+1)−1 dy dv0 . (14) and one has dv(ρ) (gv) = χ(g)(d/2)ρ dv(ρ) (v)

(g ∈ G, v ∈ S (ρ) ).

(15)

4

We set Vk = ⊕ j 0 and r − ρ + 1 ≤ k ≤ r, put √ −1 (k−1) Qk = Qk (λ, ǫk , u ) = λ1Vk − ǫk T u(k−1) |Vk . 2 ` Then, for u ∈ Vη× , one has hu, vi =

√

r X √ −1 lim tk (λ − 2 −1ǫk ξk + 2 λ→0 k=r−ρ+1 √    ′  1 −1 −1 + Qk  xk − ǫk Qk uk  + Q−1 [uk ]) 2 4 k

√ 1 [uk ], one has Hence, putting qk (λ) = λ − 2 −1ǫk ξk + Q−1 4 k Iǫ =

Z(ρ)

Sǫ

b f (v)dv(ρ) (v)

222

4

  Z Z    f (u)e(hu, vi)du dv(ρ) (v) =   (ρ)

Sǫ

V

−(d/2)ρ(2r−ρ−1)

=2

X Z

η∈E (r)

Vη×

f (u)du × √

 −1  + lim tk qk (λ) dtk × λ→0 2 k=r−ρ+1 0 √ √   Z  −1  ′ −1  −1    × e tk Qk  xk − ǫk Qk uk  dx′k . 2 2 Z∞ r X

 (d/2)(2k−r+ρ−1)−1  tk e 

???????????

186

Vη× ,

For u ∈ one has, in the notation of [SF], sign χk (u) = ηk (1 ≤ k ≤ r). Hence √   Z √  −1   ′ −1 −1 e  tk Qk  xk − ǫk Qk uk  dx′k = tk−(d/2)(k−1) det(Qk )−1/2 2 2 Vk

→

 

 d 2d(k−1) tk−(d/2)(k−1) e  ǫk 8

 k−1  X  ηl  |N (k−1) (u(k−1) )|−d/2 l=1

(λ → 0), √   −1  t(d/2)(k−r+ρ)−1 e  tk qk (λ) dtk 2 0 ! d = Γ (k − r + ρ) (πqk (λ))−(d/2)(k−r+ρ) 2 ! ! d d → Γ (k − r + ρ) e ǫk ηk (k − r − ρ) (2|χk (u)|)−(d/2)(k−r+ρ) 2 8

Z∞

(λ → 0). Moreover, one has r Y

k=r−ρ+1

N (k−1) (u(k−1) )|χk (u)|k−r+ρ = N(u)ρ .

223

Therefore, putting k′ = k − r + ρ(r − ρ + 1 ≤ k ≤ r), one has X Z (d/2)ρ(2r−ρ−1) f (u)du × Iǫ = 2 η∈E (r) V × η

  ! r  X  d ′ det(Qk )−1/2 Γ k′ (πqk (λ))−(d/2)k  × lim  λ→0 2 k=r−ρ+1

ρ  Y i ′ = 2π)−(d/2)k Γ((d/2)k′ ) k′ =1

  Z   r X  d X  X  f (u)|N(u)|−(d/2)ρ du, ǫk  ηl  − ηk (k − r + ρ) × e  8 η l 0, then no such simple result holds. However, in the case F = Q( −a), then using the representation ζF (s) = ζ(s)L(s) and the formula ζ(2) = π2 /6 and writing the periodic function (d/n) as a finite linear combination of terms e2πin d , we obtain ! |d|−1 π2 X d D(e2πin d ) (F imaginary quadratic), ζF (2) = √ n 6 |d| n=1 e.g.,

 π2  √ D(e2πi/7 ) + D(e4πi/7 ) − D(e6πi/7 ) 3 7 Thus the values of ζF (2) for imaginary quadratic fields can be expressed in closed form in terms of values of the Bloch-Wigner function D(z) at algebraic arguments z. ζQ( √−7) (2) =

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By using the ideas of the last section we can prove a much stronger statement. Let O denote √ the ring of√integers of F (this is the Z-lattice in C spanned by 1 and −a or (1 + −a)/2, depending whether d = −4a or d = −a). Then the group Γ = S L2 (O) is a discrete subgroup of S L2 (C) and therefore acts on hyperbolic space H3 by isometries. A classical result of Humbert gives the volume of the quotient space H3 /Γ 246 as |d|3/2 × ζF (2)/4π2 . On the other hand, H3 /Γ (or, more precisely, a certain covering of it of low degree) can be triangulated into ideal tetrahedra with vertices belonging to P1 (F) ⊂ P1 (C), and this leads to a representation π2 X ζF (2) = nv D(zv ) 3|d|3/2 v with nv in Z and zv in F itself rather than in the much larger √field Q(e2πi d )([8], Theorem 3). For instance, in our example F = Q( −7) we find √  √      −1 + −7  4π2   1 + −7   + D   . ζF (2) = √ 2D  2 4 21 7

This equation together with the fact that ζF (2) = 1.89484144897 . . . , 0 implies that the element (13) has infinite order in BC . In [8], it was pointed out that the same kind of argument works for all number fields, not just imaginary quadratic ones. If r2 = 1 but N > 2 then one can again associate to F (in many different ways) a discrete subgroup Γ ⊂ S L2 (C) such that Vol(H3 /Γ) is a rational multiple of d|1/2 ζF (2) × π2(1−N) . This manifold H3 /Γ is now compact, so the decomposition into ideal tetrahedra is a little less obvious than in the case of imaginary quadratic F, but by decomposing into non-ideal tetrahedra (tetrahedra with vertices in the interior of H3 ) and writing these as differences of ideal ones, it was shown that the volume is an integral linear combination of values of D(z) with z of degree at most 4 over F. For F completely arbitrary there is still a similar statement, except that now one gets discrete groups Γ acting on Hr32 ; the final result ([8], Theorem 1) is that |d|1/2 × ζF (2)/π2(r1 +r2 ) is a rational linear combination of r2 -fold products D(z(1) ) . . . D(z(r2 ) ) with each z(i) of degree ≤ 4 over F (more

291

precisely, over the ith complex embedding F (i) of F, i.e. over the subfield Q(α(i) ) of C where α(i) is one of the two roots of the ith quadratic factor of f (x) over R). But in fact much more is true : the z(i) can be chosen in F (i) itself (rather than of degree 4 over this field), and the phrase “rational linear combination of r2 -fold products” can be replaced by “rational multiple of an r2 × r2 determinant.” We will not attempt to give more than a very sketchy account of why this is true, lumping together work of Wigner, Bloch, Dupont, Sah, Levine, Merkuriev, Suslin, . . . for the purpose (references are [1], [3], and the survey paper [7]). This work connects the Bloch group defined in the last section with the algebraic K-theory of 247 the underlying field; specifically, the group1 BF is equal, at least after tensoring it with O, to a certain quotient K3ind (F) of K3 (F). The exact definition of K3ind (F) is not relevant here. What is relevant is that this group has been studied by Borel [2], who showed that it is isomorphic (modulo torsion) to Zr2 and that there is a canonical homomorphism, the “regulator mapping,” from it into Rr2 such that the co-volume of the image in a non-zero rational multiple of |d|a/2 ζF (2)/π2r1 +2r2 ; moreover, it is known that under the identification of K3ind (F) with BF this D

mapping corresponds to the composition BF → (BC )r2 −→ Rr2 , where the first arrow comes from using the r2 embeddings F ⊂ C(α → α(i) ). Putting all this together gives the following beautiful picture : The group BF /{torsion}, is isomorphic to Zr2 . Let ζ1 , . . . , ζr2 by any r2 linearly independent elements of it, and form the matrix with entires D(ζ (i) j ), (i, j = 1, . . . , r2 ). Then the determinant of this matrix is a non-zero rational multiple of |d|1/2 ζF (2)/π2r1 +2r2 . If instead of taking any r2 linearly independent elements we choose the ζ j to be a basis of BF /{torsion}, then this rational multiple (chosen positively) is an invariant of F, independent of the choice of ζ j . This rational multiple is then conjecturally related to the quotient of the order of K3 (F)torsion by the order of the 1 It should be mentioned that the definition of BF which we gave for F = C or Q must be modified slightly when F is a number field because F × is no longer divisible; however, this is a minor point, affecting only the torsion in the Bloch group, and will be ignored here.

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finite group K2 (OF ) where OF denotes the ring of integers of F (Lichtenbaum conjectures). This all sounds very abstract, but it is fact not. There is a reasonably efficient algorithm to produce many elements of BF for any number field F. If we do this, for instance, for F an imaginary quadratic field, and compute D(ζ) for each element ζ ∈ BF which we find, then after a while we are at least morally certain of having identified the lattice D(BF ) ⊂ R exactly (after finding k elements at random, we have only about one chance in 2k of having landed in the same non-trivial sublattice each time). By the results just quoted, this lattice is generated by a number of the form κ|d|3/2 ζF (2)/π2 with κ rational, and the conjecture 3 where T is the orreferred to above says that κ should have the form 2T der of the finite group K2 (OF ), at least for d < −4 (in this case the order of K3 (F)torsion is always 24). Calculations done by H. Gangl in Bonn for several hundred imaginary quadratic fields support this; the κ he found 3 248 all have the form 2T for some integer T and this integer agrees with the order of K2 (OF ) in the few cases where the latter is known. Here is a small excerpt from his tables : |d| 7 8 11 15 19 20 23 24 31 35 39 40 . . . 303 472 479 491 555 583 T 2 1 1 2 1 1 2 1 2 2 6 1 . . . 22 5 14 13 28 34 (the omitted values contain only the primes 2 and 3; 3 occurs whenever d ≡ 3(mod 9) and there is also some regularity in the powers of 2 occurring). Thus one of the many virtues of the mysterious dilogarithm is that it gives, at least conjecturally, an effective way of calculating the orders of certain groups in algebraic K-theory! To conclude, we mention that Borel’s work connects not only ind (F) and ζ (m) for any inteind K3 (F) and ζF (2) but more generally K2m−1 F ger m > 1. No elementary description of the higher K-groups analogous to the description of K3 in terms of B is known, but one can at least speculate that these groups and their regulator mappings may be related to the higher polylogarithms and that, more specifically, the value of ζF (m) is always a simple multiple of a determinant (r2 ×r2 or (r1 +r2 )×(r1 +r2 ) depending whether m is even or odd) whose entries are linear combinations of values of the Bloch-Wigner-Ramakrishnan function Dm (z) with

REFERENCES

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arguments z ∈ F. As the simplest case, one can guess that for a real quadratic field F the value of ζF (3)/ζ(3) = L(3), where L(s) is a Dirichlet L-Function of a real quadratic character of period d) is equal to d−5/2 times a simple rational linear combination of differences D3 (x) − D3 (x′ ) with x ∈ F, where x′ denotes the conjugate of x over Q. Here is one (numerical) example of this : −5 5/2

2 5

ζQ( √5) (3)/ζ(3)

√  √     1 − 5   1 + 5     −  = D3   − D3  2 2 √ √ 1 − [D3 (2 + 5) − D3 (2 − 5)] 3

(both sides are equal approximately to 1.493317411778544726). I have found many other examples, but the general picture is not yet clear.

References [1] S. Bloch : Applications of the dilogarithm function in algebraic 249 K-theory and algebraic geometry, in : Proc. of the International Symp. on Alg . Geometry, Kinokuniya, Tokyo, 1978. [2] A. Borel : Commensurability classes and volumes of hyperbolic 3-manifolds, Ann. Sc. Norm. Sup. Pisa, 8 (1981) pp. 1-33. [3] J. L. Dupont and C. H. Sah : Scissors congruences II, J. Pure and Applied Algebra, 25 (1982), 159-195. [4] L. Lewin : Polylogarithms and associated functions (title of original 1958 edition : Dilogarithms and associated functions). North Holland, New York, 1981. [5] W. Neumann and D. Zagier : Volumes of hyperbolic 3-manifolds, Topology, 24 (1985), 307-332. [6] D. Ramakrishnan : Analogs of the Bloch-Wigner function for higher polylogarithms, Contemp. Math., 55 (1986), 371-376.

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[7] A. A. Suslin : Algebraic K-theory of fileds, in : Proceedings of the ICM Berkeley 1986 A.M.S. (1987), 222-244. [8] D. Zagier : Hyperbolic manifolds and special values of Dedekind zeta-functions, Invent. Math., 83 (1986), 285-301. [9] D. Zagier : On an approximate identity of Ramanujan, Proc. Ind. Acad. Sci. (Ramanujan Centenary Volume) 97 (1987), 313-324. [10] D. Zagier : Green’s functions of quotients of the upper half-plane (in preparation.) Max-Planck-Institut f¨ur Mathematik, Gottfried-Claren-Straße 26, D-5300 Bonn, FRG. and Department of Mathematics, University of Maryland, College Park, Maryland 20742, USA.

This book contains original papers presented at the Srinivasa Ramanujan Birth Centenary International Colloquium on Number Theory and Related Topics held at the Tata Institute of Fundamental Reserach in January 1988.