Negative discrete moments of the derivative of the Riemann zeta-function

TL;DR

This paper establishes near-optimal conditional upper bounds for negative moments of the derivative of the Riemann zeta-function over specific zero subfamilies, advancing understanding of zeta function behavior.

Contribution

It provides new conditional bounds for negative moments of the zeta derivative, extending to larger zero subfamilies under a conjecture about the argument's maximum.

Findings

Bounds are nearly optimal for moments with k ≤ 1/2.

Conditional bounds hold for larger zero subfamilies assuming a conjecture.

Results improve understanding of the zeta function's derivative behavior.

Abstract

We obtain conditional upper bounds for negative discrete moments of the derivative of the Riemann zeta-function averaged over a subfamily of zeros of the zeta function which is expected to have full density inside the set of all zeros. For $k\leq 1/2$, our bounds for the $2k$-th moments are expected to be almost optimal. Assuming a conjecture about the maximum size of the argument of the zeta function on the critical line, we obtain upper bounds for these negative moments of the same strength while summing over a larger subfamily of zeta zeros.