Zero-free regions for the Riemann zeta function

By Kevin Ford. 2019

76 citations recorded by the source corpus.

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).

Subfields: Algebra and Number Theory, Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/29708a7904beebbadcb8665a08c669320563257a

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Citations in the corpus

Une rgion explicite sans zros pour la fonction ? de Riemann (2004, background): “Ford pour m’avoir aimablement transmis une version préliminaire de son article [4].”
Primes in tuples I (2005, background): “We will fix this c for the rest of the paper (we could take, for instance, c D 10 (2), see [8]).”
Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function (2014, background): “Ford [4] established an explicit region of this type, showing that one can take R1 = 57.”
The Riemann hypothesis is true up to 3·1012 (2020, background): “This can be improved slightly with Theorem 1 and will be tackled in a future paper of Mossinghoﬀ and the second author, along with the explicit version of the Vinogradov–Korobov zero-free region, which was proved by Ford [10].”