The second moment of the Riemann zeta function at its local extrema

TL;DR

This paper investigates the second moment of the Riemann zeta function at its local extrema, providing a new method to determine detailed asymptotic expansions including lower order terms.

Contribution

It introduces a novel approach to analyze the second moment at extrema, revealing the full asymptotic expansion with lower order logarithmic terms.

Findings

Leading order behavior matches previous results

New method uncovers lower order logarithmic terms

Asymptotic expansion includes a descending chain of powers of logarithms

Abstract

Conrey and Ghosh studied the second moment of the Riemann zeta function, evaluated at its local extrema along the critical line, finding the leading order behaviour to be $\frac{e^2 - 5}{2 \pi} T (\log T)^2$. This problem is closely related to a mixed moment of the Riemann zeta function and its derivative. We present a new approach which will uncover the lower order terms for the second moment as a descending chain of powers of logarithms in the asymptotic expansion.