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

Abstract

We derive explicit upper bounds for the Riemann zeta-function ζ(σ+it) on the lines σ=1−k/(2k−2) for integer k≥4. This is used to show that the zeta-function has no zeroes in the regionσ>1−log⁡log⁡|t|21.233log⁡|t|,|t|≥3. This is the largest known zero-free region for exp⁡(171)≤t≤exp⁡(5.3⋅105). Our results rely on an explicit version of the van der Corput AnB process for bounding exponential sums.