Explicit approximation of the sum of the reciprocal of the imaginary parts of the zeta zeros

TL;DR

This paper provides explicit bounds for the sum of the reciprocals of the imaginary parts of the nontrivial zeros of the Riemann zeta function, aiding in understanding their distribution.

Contribution

It introduces explicit upper and lower bounds for the sum over the reciprocals of the zeros' imaginary parts, which was previously not precisely quantified.

Findings

Established explicit bounds for the sum of reciprocals of zeros' imaginary parts.

Enhanced understanding of the distribution of zeta zeros.

Provides tools for further analytical number theory research.

Abstract

In this note, we give some explicit upper and lower bounds for the summation $\sum_{0<\gamma\leq T}\frac{1}{\gamma}$, where $\gamma$ is the imaginary part of nontrivial zeros $\rho=\beta+i\gamma$ of $\zeta(s)$.