A note on a conjecture of Gonek

TL;DR

This paper establishes a lower bound for the second moment of the reciprocal of the zeta-function's derivative at its zeros, assuming the Riemann Hypothesis and simple zeros, advancing understanding of the zeta-function's behavior.

Contribution

It provides a new conditional lower bound for a key second moment related to the zeta-function's zeros, under standard conjectures.

Findings

Lower bound is half the size of the conjectured value.

Results depend on the Riemann Hypothesis and simplicity of zeros.

Advances understanding of the distribution of zeta zeros.

Abstract

We derive a lower bound for a second moment of the reciprocal of the derivative of the Riemann zeta-function averaged over the zeros of the zeta-function that is half the size of the conjectured value. Our result is conditional upon the assumption of the Riemann Hypothesis and the conjecture that the zeros of the zeta-function are simple.