High moments of the Riemann zeta-function

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. Gho...

Moments of the Riemann Zeta-function

0 |ζ( 1 2 + it)| dt. For positive real numbers k, it is believed that Mk(T ) ∼ CkT (logT ) 2 for a positive constant Ck. A precise value for Ck was conjectured by Keating and Snaith [9] based on considerations from random matrix theory. Subsequently, an alternative approach, based on multiple Dirichlet series and producing the same conjecture, was given by Diaconu, Goldfeld and Hoffstein [4]. R...

Hybrid Moments of the Riemann Zeta-function

The “hybrid” moments Z 2T T |ζ( 1 2 + it)| „ Z t+G t−G |ζ( 1 2 + ix)| dx m dt of the Riemann zeta-function ζ(s) on the critical line Re s = 1 2 are studied. The expected upper bound for the above expression is Oε(T G). This is shown to be true for certain specific values of k, l,m ∈ N, and the explicitly determined range of G = G(T ; k, l,m). The application to a mean square bound for the Melli...

Riemann Zeta Moments

The behavior of the Riemann zeta function () on the critical line Re() = 12 has been studied intensively for nearly 150 years. We start with a well-known

Moments of the Riemann Zeta Function and Eisenstein Series I