On the joint second moment of zeta and its logarithmic derivative

TL;DR

This paper investigates the weighted second moment of the Riemann zeta function and its logarithmic derivative, providing bounds under the assumption of the Riemann Hypothesis, and explores its connection to pair correlation conjecture.

Contribution

It introduces a weighted version of the second moment problem for zeta and its derivative, offering new bounds and insights into their joint behavior.

Findings

Provided upper and lower bounds for the weighted second moment.

Connected the second moment to the pair correlation conjecture.

Extended previous unweighted analyses to a weighted context.

Abstract

Assuming the Riemann Hypothesis, Goldston, Gonek and Montgomery \cite{GGM} studied the second moment of the log-derivative of $\zeta$, shifted away from the half line by $a/\log T$, and its connection with the pair correlation conjecture. In this paper, we consider a weighted version of this problem, where the average is tilted by $|\zeta(\frac{1}{2}+it)|^2$. More precisely, we provide an upper and a lower bound for the second moment of zeta times its logarithmic derivative, $a/\log T$ away from the critical line.