Dense clusters of zeros near the zero-free region of ζ(s)

The methods of Korobov and Vinogradov produce a zero-free region for the Riemann zeta function Formula: see text of the form Formula: see text For many decades, the general shape of the zero-free region has not changed (although explicit known values for Formula: see text have improved over the years). In this paper, we show that if the zero-free region cannot be widened substantially, then there exist infinitely many distinct dense clusters of zeros of Formula: see text lying close to the edge of the zero-free region. Our proof provides specific information about the location of these clusters and the number of zeros contained in them. To prove the result, we introduce and apply a variant of the original method of de la Vallée Poussin combined with ideas of Turán to control the real parts of power sums. We also prove similar results for Formula: see text-functions associated to nonquadratic Dirichlet characters Formula: see text modulo Formula: see text.