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 for 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).
Keywords
Cite
@article{arxiv.1910.08205,
title = {Zero-free regions for the Riemann zeta function},
author = {Kevin Ford},
journal= {arXiv preprint arXiv:1910.08205},
year = {2025}
}
Comments
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