An upper bound for discrete moments of the derivative of the Riemann zeta-function

TL;DR

Under the Riemann hypothesis, the paper derives a sharp upper bound for the discrete moments of the derivative of the Riemann zeta-function at its zeros, aligning with existing conjectures and improving previous results.

Contribution

It provides a new upper bound for discrete moments of the zeta-function's derivative at zeros, confirming conjectures and refining earlier bounds.

Findings

Upper bound matches conjectures of Gonek, Hejhal, Hughes, Keating, and O'Connell.

Sharpens previous results by Milinovich.

Method extends Harper’s approach to discrete moments.

Abstract

Assuming the Riemann hypothesis, we establish an upper bound for the $2k$-th discrete moment of the derivative of the Riemann zeta-function at nontrivial zeros, where $k$ is a positive real number. Our upper bound agrees with conjectures of Gonek and Hejhal and of Hughes, Keating, and O'Connell. This sharpens a result of Milinovich. Our proof builds upon a method of Adam Harper concerning continuous moments of the zeta-function on the critical line.