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 $\zeta (s)$ of the form $$\sigma >1-\frac{c}{{(log\tau )}^{2/3}{(loglog\tau )}^{1/3}}(\tau ≔|t|+100).$$ For many decades, the general shape of the zero-free region has not changed (although explicit known values for $c$ 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 $\zeta (s)$ 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 $L$-functions associated to nonquadratic Dirichlet characters $\chi$ modulo $q\ge 2$.