Moments of the Riemann zeta function at its local extrema

TL;DR

This paper combines previous results to evaluate the first moment of the Riemann zeta function and its derivatives at local extrema along the critical line, providing a comprehensive asymptotic analysis.

Contribution

It unifies earlier studies by evaluating the first moment of zeta and its derivatives at local extrema, including the functional equation factor, offering a complete asymptotic.

Findings

Derived the full asymptotic for the first moment at local extrema.

Connected moments of zeta and its derivatives at extrema.

Analyzed the functional equation factor at these points.

Abstract

Conrey, Ghosh and Gonek studied the first moment of the derivative of the Riemann zeta function evaluated at the non-trivial zeros of the zeta function, resolving a problem known as Shanks' conjecture. Conrey and Ghosh studied the second moment of the Riemann zeta function evaluated at its local extrema along the critical line to leading order. In this paper we combine the two results, evaluating the first moment of the zeta function and its derivatives at the local extrema of zeta along the critical line, giving a full asymptotic. We also consider the factor from the functional equation for the zeta function at these extrema.