On certain sums over ordinates of zeta-zeros II

TL;DR

This paper investigates the analytic continuation and Laurent expansion of a sum over zeta zeros' ordinates, revisiting second moment estimates and extending previous work on the subject.

Contribution

It extends prior research by analyzing the analytic continuation of a sum over zeta zeros and providing new estimates for its second moment.

Findings

Laurent expansion of G(s) at s=1 obtained

Analytic continuation of G(s) to the left of Re s = -1 achieved

Revised estimates for the second moment on the critical line

Abstract

Let $\gamma$ denote the imaginary parts of complex zeros $\rho = \beta+i\gamma$ of $\zeta(s)$. The problem of analytic continuation of the function $G(s) := \sum\limits_{\gamma > 0}\gamma^{-s}$ to the left of the line $\Re s = -1$ is investigated, and its Laurent expansion at the pole $s=1$ is obtained. Estimates for the second moment on the critical line $\int_1^T|G(1/2+it)|^2\,dt$ are revisited. This paper is a continuation of work begun by the second author in 2001.