On certain sums over the nontrivial zeta zeros

TL;DR

This paper investigates coefficients related to the Riemann zeta function's nontrivial zeros, providing new relations, representations, and asymptotic behavior under the Riemann hypothesis.

Contribution

It introduces novel expressions and integral representations for coefficients linked to zeta zeros, advancing understanding of their properties and asymptotics.

Findings

Expressed $b_n$ as sums over zeta zeros

Derived integral representations of $b_n$

Established asymptotic form $b_n o 2^{-n-2} abla n$ under RH

Abstract

We study coefficients $b_n$ that are expressible as sums over the Li/Keiper constants $\lambda_j$. We present a number of relations for and representations of $b_n$. These include the expression of $b_n$ as a sum over nontrivial zeros of the Riemann zeta function, as well as integral representations. Conditional on the Riemann hypothesis, we provide the asymptotic form of $b_n \sim 2^{-n-2}\ln n$.