  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.
