Zero-free regions for the Riemann zeta function

TL;DR

This paper enhances explicit bounds for zero-free regions of the Riemann zeta function, improving understanding of its zeros near the critical line for all heights t, using advanced growth estimates and Jensen formula techniques.

Contribution

It provides improved explicit bounds for the zero-free regions of the Riemann zeta function, applicable for all heights t, based on refined growth estimates and a Jensen formula approach.

Findings

Enhanced zero-free region bounds for large t

Explicit bounds valid for all t

Application of Jensen formula to zero distribution

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 $\zeta(\sigma+it)$ for $\sigma$ 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).