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

Abstract

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].