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

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.