A further generalisation of sums of higher derivatives of the Riemann Zeta Function

TL;DR

This paper derives an asymptotic formula for sums involving derivatives of the Riemann zeta function evaluated at its zeros, extending previous results and analyzing how the asymptotics vary with the nature of the parameter X.

Contribution

It generalizes existing asymptotic formulas for sums of derivatives of the zeta function at zeros, including the case when X is an integer, revealing new differences in asymptotic behavior.

Findings

Derived asymptotic formulas for sums over zeta zeros involving derivatives.

Identified how asymptotics change when X is an integer versus non-integer.

Extended previous results to higher derivatives of the zeta function.

Abstract

We prove an asymptotic for the sum of $\zeta^{(n)} (\rho)X^{\rho}$ where $\zeta^{(n)} (s)$ denotes the $n$th derivative of the Riemann zeta function, $X$ is a positive real and $\rho$ denotes a non-trivial zero of the Riemann zeta function. The sum is over the zeros with imaginary parts up to a height $T$, as $T \rightarrow \infty$. We also specify what the asymptotic formula becomes when $X$ is a positive integer, highlighting the differences in the asymptotic expansions as $X$ changes its arithmetic nature.