A harmonic sum over nontrivial zeros of the Riemann zeta-function

TL;DR

This paper investigates the sum of the reciprocals of the ordinates of nontrivial zeros of the Riemann zeta-function, revealing its convergence to a specific limit after a smooth approximation, and improves upon previous results.

Contribution

It provides a high-precision calculation of the limiting value of the harmonic sum over nontrivial zeros, refining earlier estimates and employing an improved error analysis.

Findings

The sum tends to a limit H ≈ -0.0171594 as T approaches infinity.

After subtracting a smooth approximation, the sum converges to a well-defined limit.

The method achieves an error of O((log T)/T^2), enhancing previous accuracy.

Abstract

We consider the sum $\sum 1/\gamma$, where $\gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval $(0,T]$, and consider the behaviour of the sum as $T \to\infty$. We show that, after subtracting a smooth approximation $\frac{1}{4\pi} \log^2(T/2\pi),$ the sum tends to a limit $H \approx -0.0171594$ which can be expressed as an integral. We calculate $H$ to high accuracy, using a method which has error $O((\log T)/T^2)$. Our results improve on earlier results by Hassani and other authors.