A discrete mean-value theorem for the higher derivatives of the Riemann zeta function

TL;DR

This paper proves that the sums of the higher derivatives of the Riemann zeta function over its non-trivial zeros exhibit a mean-value sign pattern depending on the derivative order, supported by a detailed asymptotic expansion.

Contribution

It introduces a discrete mean-value theorem for the derivatives of the Riemann zeta function, revealing sign patterns in the sums over zeros and providing a full asymptotic expansion.

Findings

Sum of odd derivatives over zeros is positive in the mean.

Sum of even derivatives over zeros is negative in the mean.

Provides a comprehensive asymptotic expansion for these sums.

Abstract

We show that the $n$th derivative of the Riemann zeta function, when summed over the non-trivial zeros of zeta, is real and positive/negative in the mean for $n$ odd/even, respectively. We show this by giving a full asymptotic expansion of these sums.