Upper bounds for the moments of zeta prime rho

TL;DR

Under the Riemann Hypothesis, the paper establishes near-optimal upper bounds for the moments of the derivative of the Riemann zeta-function at its zeros, extending methods for continuous moments and L-functions.

Contribution

It provides the first nearly sharp upper bounds for the moments of zeta prime at zeros, using a novel adaptation of Soundararajan's method.

Findings

Derived upper bounds for 2k-th moments of zeta prime at zeros

Bounds are nearly as sharp as conjectured asymptotics

Method extends to moments of L-functions at the central point

Abstract

Assuming the Riemann Hypothesis, we obtain an upper bound for the 2k-th moment of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $\zeta(s)$ for every positive integer k. Our bounds are nearly as sharp as the conjectured asymptotic formulae for these moments. The proof is based upon a recent method of K. Soundararajan that provides analogous bounds for continuous moments of the Riemann zeta-function as well as for moments L-functions at the central point, averaged over families.