On certain sums over ordinates of zeta zeros

TL;DR

This paper evaluates sums over the ordinates of zeta zeros using the function G(s) and explores integrals involving the argument of the zeta function, contributing to understanding the distribution of zeros.

Contribution

It introduces new methods to evaluate sums over zeta zeros and analyzes related integrals, advancing the analytical tools in number theory.

Findings

Explicit formulas for sums over zeta zeros

Connections between sums and the argument function S(T)

Enhanced techniques for studying zero distributions

Abstract

Let $\gamma$ denote imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. Certain sums over the $\gamma$'s are evaluated, by using the function $G(s) = \sum_{\gamma>0}\gamma^{-s}$ and other techniques. Some integrals involving the function $S(T) = (1/\pi)\arg\zeta(1/2+iT)$ are also considered.