High moments of the Riemann zeta-function. (English) Zbl 1006.11048

Conrey, J. B.; Gonek, S. M.

Duke Math. J. 107, No. 3, 577-604 (2001).

The asymptotic formula for the integral \[ I_k(T) = \int_0^T|\zeta({1\over 2}+it)|^{2k} dt\qquad(k = 1,2,\ldots) \] represents a central problem in the theory of the Riemann zeta-function \(\zeta(s)\) (for an extensive account see the reviewer’s monographs [John Wiley & Sons (1985; Zbl 0556.10026)] and [Springer Verlag (1991; Zbl 0758.11036)]). Except for the classical cases \(k =1,2\) (op. cit.) no asymptotic formula of the form \[ I_k(T) \sim c_kT(\log T)^{k^2}\qquad(T\to\infty)\tag{1} \] is known when \(k > 2\). In fact, it is difficult even to guess what the value of the constant \(c_k\) should be, in case the above formula holds. J. B. Conrey and A. Ghosh [Int. Math. Res. Not. 1998, 775-780 (1998; Zbl 0920.11060)] conjectured that \[ c_3 = {42\over 9!}\prod_p {\left(1 - {1\over p}\right)}^4 \left(1 + {4\over p} + {1\over p^2}\right),\tag{2} \] where \(p\) denotes primes. In the present paper the authors propose that, when \(k=4\), (1) should hold with \[ c_4 = {24024\over 14!}\prod_p {\left(1 - {1\over p}\right)}^9 \left(1 + {9\over p} + {9\over p^2} + {1\over p^3}\right). \tag{3} \] This is achieved by considerations involving Dirichlet polynomials of the form \(D_{k,N}(s) = \sum_{n\leq N}d_k(n)n^{-s}\), where the divisor function \(d_k(n)\) denotes the number of ways \(n\) may be written as a product of \(k\) factors. Such polynomials appear in various approximate functional equations, used to approximate \(\zeta^k(s)\). After squaring and integration, the problem of the asymptotic evaluation of \(I_k(T)\) is reduced to problems involving the so-called general binary additive divisor problem, namely the study of the function \(D_k(x,h) = \sum_{n\leq x}d_k(n)d_k(n+h)\) where the ‘shift parameter’ \(h\) may not be fixed. This condition makes the problem one of extra difficulty, and the authors make a conjecture involving the function \(m_k(x,h)\), the main term approximating \(D_k(x,h)\). This conjecture, too complicated to be reproduced here, is similar in nature to the one previously made by the reviewer [in New Trends Probab. Stat. 4, 69-89 (1997; Zbl 0924.11070)]. By using this conjecture the authors make a detailed analysis which leads to the conjectural value (3).
It is worth remarking that recently J. P. Keating and N. C. Snaith [Commun. Math. Phys. 214, 57-89 (2000; Zbl 1051.11048)] made the general conjecture \[ c_k = {g_k a_k\over \Gamma(1+k^2)}, \] where \[ a_k = \prod_p \left\{{\left(1 - {1\over p}\right)}^{k^2} \sum_{j=0}^\infty d_k^2(p^j)p^{-j}\right\}, \]
\[ g_k = \prod_{j=0}^{k-1}{j!\over(j+k)!}. \] It is remarkable that the values provided by the Keating-Snaith conjecture (an asymptotic formula should hold also when \(k \geq 0\) is not necessarily an integer, in which case they also produce the explicit value of the conjectural \(c_k\)) coincide with the known values when \(k=1,2\) and with the values furnished by (2) and (3), respectively. It is even more remarkable that the Keating-Snaith conjecture follows from considerations totally different from those employed by the authors, namely by modelling the behaviour of \(\zeta({1\over 2}+it)\) by analogies from random matrix theory. Further ramifications in this exciting field of research have been lately obtained by various authors, most notably by J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein and N. C. Snaith [Integral moments of \(L\)-functions, preprint, 58 pp, arXiv:mat.NT/0206018].

Reviewer: Aleksandar Ivić (Beograd)

MSC:

11M06	\(\zeta (s)\) and \(L(s, \chi)\)
11M20	Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\)

Keywords:

mean values; Dirichlet polynomials; Riemann zeta-function; functional equation; generalized additive divisor problem

Citations:

Zbl 0556.10026; Zbl 0758.11036; Zbl 0920.11060; Zbl 0924.11070; Zbl 1051.11048

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