The discrete second moment of mixed derivatives of the Riemann zeta function

TL;DR

This paper derives the complete asymptotic formula for the discrete second moment of mixed derivatives of the Riemann zeta function at its zeros, extending previous work to all derivatives with detailed error analysis.

Contribution

It provides the full asymptotic for the second moment of mixed derivatives of the zeta function at zeros, including all derivatives and error terms, advancing prior leading-order results.

Findings

Unconditional and conditional asymptotic formulas derived

Extension of previous first derivative results to all derivatives

Improved understanding of the distribution of zeta derivatives at zeros

Abstract

We establish the full asymptotic for the discrete second moment of the Riemann zeta function of mixed derivatives evaluated at the zeta zeros, providing both unconditional and conditional error terms. This was first studied by Gonek, where only the leading order asymptotic was given, later extended by Conrey--Snaith and Milinovich to include the lower order terms for the first derivative. We extend the case of the first derivative to all derivatives.