Local Suprema of Dirichlet Polynomials and Zerofree Regions of the Riemann Zeta-Function

TL;DR

This paper introduces a new zero-free region for the Riemann Zeta-function by linking it to local suprema of Dirichlet polynomials, using a probabilistic approach and advanced inequalities.

Contribution

It establishes a novel zero-free region for , utilizing Ture1n's localization criterion and a randomization method to analyze Dirichlet polynomial extrema.

Findings

Identifies a new zero-free region for .

Provides estimates for local extrema of Dirichlet polynomials.

Employs a probabilistic approach with Montgomery-Vaughan's inequality.

Abstract

A new zerofree region of the Riemann Zeta-function $\zeta$ is identified by using Tur\'an's localization criterion linking zeros of $\zeta$ with uniform local suprema of sets of Dirichlet polynomials expanded over the primes. The proof is based on a randomization argument. An estimate for local extrema for some finite families of shifted Dirichlet polynomials, % associated to linearly independent sequences of reals is established by preliminary considering their local increment properties, by means of Montgomery-Vaughan's variant of Hilbert's inequality. A covering argument combined with Tur\'an's localization criterion allows to conclude.