Explicit bounds on $\zeta(s)$ in the critical strip and a zero-free region

Abstract

We derive explicit upper bounds for the Riemann zeta-function $\zeta(\sigma + it)$ on the lines $\sigma = 1 - k/(2^k - 2)$ for integer $k \ge 4$. This is used to show that the zeta-function has no zeroes in the region $$\sigma > 1 - \frac{\log\log|t|}{21.233\log|t|},\qquad |t| \ge 3.$$ This is the largest known zero-free region for $\exp(171) \le t \le \exp(5.3\cdot 10^{5})$. Our results rely on an explicit version of the van der Corput $A^nB$ process for bounding exponential sums.