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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 $ζ(σ+it)$ for $σ$ 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).
arXiv 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
Journal reference
In the book Number Theory for the Millenium (Urbana, IL, 2000) (M. A. Bennett, B. C. Berndt, N. Boston, H. G. Diamond, A. J. Hildebrand and W. Phillip, eds), vol. II (2002), 25-56
"It has been proved all the zeroes are within a narrow strip centred on the line and you can make the strip as arbitrarily narrow as you like." Nothing close to this is known. The nontrivial zeros of zeta lie within the critical strip, i.e., 0 <= Re(s) <= 1 (in analytic number…
"It has been proved all the zeroes are within a narrow strip centred on the line and you can make the strip as arbitrarily narrow as you like." Nothing close to this is known. The nontrivial zeros of zeta lie within the critical strip, i.e., 0 <= Re(s) <= 1 (in analytic number…
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