Zero-free regions for the Riemann zeta function

Abstract

We improve existing explicit bounds of Vinogradov-Korobov type for zero-free regions of the Riemann zeta function, both for large height t and for every t. A primary input is an explicit bound of the author (Proc. London Math. Soc. 85 (2002), 565-633) for the growth of $\hat{I}\P (\ddot{I}ƒ+it)$ for $\ddot{I}ƒ$ near 1. Another ingredient is a kind of Jensen formula (Lemma 2.2) relating the growth of a function on vertical lines to a weighted count of its zeros inside a vertical strip. The latter was used recently in the weak subconvexity work of Soundararajan and Thorner (arXiv:1804.03654, Duke Math. J. 168, no. 7 (2019), 1231-1268).

corrected typos in Lemma 2.2, numerics in (8.8) and (8.9), and fixed the argument following Lemma 9.2. Compared to the published version, Theorems 1 and 5 are unchanged, Theorem 4 is very slightly degraded, and Theorems 2 and 3 have different numerics. All corrections to the published version appear in red