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 $$\sigma >1-\frac{\log \log |t|}{21.233\log |t|},|t|\ge 3.$$ 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 $A^{n}B$ process for bounding exponential sums.

arXiv comment51 pages. Updates in response to reviewer feedback