  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).
