On criteria related to the reciprocal of the Riemann zeta function

TL;DR

This paper investigates Fourier transforms of the reciprocal of the Riemann zeta function, linking their properties to the Riemann Hypothesis, and addresses a recent open problem by leveraging the non-vanishing of zeta on the line Re(s)=1.

Contribution

It provides new insights into the Fourier transforms related to the reciprocal of the zeta function and offers a partial solution to a recently posed problem using properties of the zeta function.

Findings

Fourier transforms of 1/ζ(s) are connected to the Riemann Hypothesis.

The non-vanishing of ζ(s) on Re(s)=1 is exploited to explore criteria related to the reciprocal.

Partial answers to recent open problems are presented.

Abstract

We explore Fourier transforms of the reciprocal of the Riemann zeta function that have connections to the RH. A partial answer to a recently posed problem is explored by exploiting the fact that $\zeta(s)\neq0$ when $\Re(s)=1.$