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

TL;DR

This paper establishes explicit bounds for the Riemann zeta-function in the critical strip, leading to the largest known zero-free region for large t, using advanced exponential sum bounds.

Contribution

It provides the first explicit bounds on $ ext{Re}( ho)$ for zeros of $ ext{zeta}(s)$ in a wide region, improving zero-free region estimates.

Findings

Largest known zero-free region for $ ext{zeta}(s)$ for large t

Explicit bounds on $ ext{zeta}(\sigma + it)$ on specific lines

Application of van der Corput exponential sum bounds

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.