Generalisations of the Landau--Gonek Theorem and applications to mean values of zeta

TL;DR

This paper generalizes the Landau--Gonek Theorem to include multiplicative characters and explores their impact on sums over zeta zeros, providing new insights into the behavior of the zeta function and applications to mean value calculations.

Contribution

It extends the Landau--Gonek Theorem to sums involving the functional equation's character, revealing deep dependence on whether X is an integer and its size relative to T.

Findings

Sum over zeros depends on whether X is an integer

Results split into three cases based on X's size relative to T

Application to alternative proof of Shanks' conjecture

Abstract

The Landau--Gonek Theorem evaluates $X^\rho$ summed over the non-trivial zeros of the Riemann zeta function. Their result shows great sensitivity to the arithmetic nature of $X$. We prove a related result concerning the sum of $\chi(\rho) X^\rho$ over the zeros of zeta, where $\chi(s)$ is the term arising in the functional equation for the zeta function. Again, this result depends deeply on whether $X$ is an integer or not. We show the result splits into three cases, depending on whether $X$ is smaller than $T$, about the same size as $T$, or bigger than $T$. The reason this result is useful is that it easily permits the calculation of discrete moments of the Riemann zeta function via the approximate functional equation. As an application of this result, we provide an alternative proof of Shanks' conjecture.