Simple Zeros Of The Zeta Function

TL;DR

This paper investigates the Laurent series expansion of the inverse zeta function at nontrivial zeros and explores its implications for the zeros' simplicity, contributing to understanding the Riemann zeta function's zeros.

Contribution

It provides a novel analysis of the Laurent series of 1/ζ(s) at zeros to study the simplicity of these zeros, linking series properties to zero multiplicity.

Findings

Laurent series of 1/ζ(s) at zeros reveals zero simplicity

Connection established between series behavior and zero multiplicity

Insights potentially inform the Riemann Hypothesis

Abstract

This note studies the Laurent series of the inverse zeta function $1/\zeta(s)$ at any fixed nontrivial zero $\rho$ of the zeta function $\zeta(s)$, and its connection to the simplicity of the nontrivial zeros.