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A new zerofree region of the Riemann Zeta-function $\zeta$ is identified byusing Tur\'an's localization criterion linking zeros of $\zeta$ with uniformlocal suprema of sets of Dirichlet polynomials expanded over the primes. Theproof is based on a randomization argument. An estimate for local extrema forsome finite families of shifted Dirichlet polynomials, % associated to linearlyindependent sequences of reals is established by preliminary considering theirlocal increment properties, by means of Montgomery-Vaughan's variant ofHilbert's inequality. A covering argument combined with Tur\'an's localizationcriterion allows to conclude.

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