Zero-free strips for the Riemann zeta-function derived from the Prime Number Theorem

TL;DR

This paper uses the Prime Number Theorem to establish the existence of zero-free regions for the Riemann zeta function near the line Re(s)=1, providing new insights into its zeros.

Contribution

It introduces a method to derive zero-free strips for the Riemann zeta function directly from the Prime Number Theorem.

Findings

Existence of a positive delta such that zeta(s) has no zeros for Re(s)>1-r when 0≤r<delta

Zero-free regions are established close to the line Re(s)=1

The approach links prime number distribution to zero-free regions of zeta(s)

Abstract

We use the Prime Number Theorem to prove the existence of zero-free strips for the Riemann-zeta function. Precisely, we prove that there exists $\delta>0$ for which if $0\leq r<\delta $ then $\zeta(s)\neq 0$ for Re$(s)>1-r$.