High Moments of the Riemann Zeta{function

In 1918 G. Hardy and J. Littlewood proved an asymptotic estimate for the Second moment of the modulus of the Riemann zeta-function on the segment [1/2,1/2+iT] in the complex plane, as T tends to infinity. In 1926 Ingham proved an asymptotic estimate for the fourth moment. However, since Ingham’s result, nobody has proved an asymptotic formula for any higher moment. Recently J. Conrey and A. Ghosh conjectured a formula for the sixth moment. We develop a new heuristic method to conjecture the asymptotic size of both the sixth and eighth moments. Our estimate for the sixth moment agrees with and strongly supports, in a sense made clear in the paper, the one conjectured by Conrey and Ghosh. Moreover, both our sixth and eighth moment estimates agree with those conjectured recently by J. Keating and N. Snaith based on modeling the zeta-function by characteristic polynomials of random matrices from the Gaussian unitary ensemble. Our method uses a conjectural form of the approximate functional equation for the zeta-function, a conjecture on the behavior of additive divisor sums, and D. Goldston and S. Gonek’s mean value theorem for long Dirichlet polynomials. We also co:nsider the question of the maximal order of the zeta-function on the critical line. 0. Introduction One of the most important goals of number theorists in the past century has been to determine the moments of the Riemann zeta-function on the critical line. These are important because they can be used to estimate the maximal order of the zeta-function on the critical line and because of their applicability to the study of the distribution of prime numbers, often through zero-density estimates, and to divisor problems. The most significant early results were obtained by Hardy and Littlewood [HL] in 1918 and by A. Ingham [I] in 1926. Hardy and Littlewood proved that DUKE MATHEMATICAL JOURNAL Vol. 107, No. 3, c © 2001 Received 7 July 1999. Revision received 8 August 2000. 2000 Mathematics Subject Classification. Primary 11M06; Secondary 11M26. Authors’ research supported in part by the American Institute of Mathematics and by grants from the National Science Foundation.