High moments of the Riemann zeta-function

In 1918 G. Hardy and J. Littlewood proved an asymptotic estimate for the Se cond 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 prove d an asymptotic estimate for the fourth moment. However, since Ingham's res ult, nobody has proved an asymptotic formula for any higher moment. Recentl y J. Conrey and A. Ghosh conjectured a formula for the sixth moment. We dev elop a new heuristic method to conjecture the asymptotic size of both the s ixth 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 a gree with those conjectured recently by J. Keating and N. Smith based on mo deling the zeta-function by characteristic polynomials of random matrices f rom the Gaussian unitary ensemble. Our method uses a conjecture 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 val ue theorem for long Dirichlet polynomials. We also consider the question of the maximal order of the zeta-function on the critical line.