Accurate estimation of sums over zeros of the Riemann zeta-function

TL;DR

This paper introduces a method to efficiently estimate sums over the zeros of the Riemann zeta-function, enhancing computational techniques for analyzing these sums in various contexts.

Contribution

It presents a simple device to accelerate the numerical estimation of sums over zeta zeros, applicable to both convergent and divergent sums.

Findings

Improved numerical estimation speed for sums over zeta zeros

Applicable to both convergent and divergent sums

Provides practical examples demonstrating the method

Abstract

We consider sums of the form $\sum \phi(\gamma)$, where $\phi$ is a given function, and $\gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We show how the numerical estimation of such sums can be accelerated by a simple device, and give examples involving both convergent and divergent infinite sums.