Explicit zero-free regions for the Riemann zeta-function

TL;DR

This paper establishes new explicit zero-free regions for the Riemann zeta-function, improving known bounds and providing bounds applicable to a wide range of large |t| values, which are crucial for understanding the distribution of zeros.

Contribution

The paper introduces improved explicit bounds for zero-free regions of the Riemann zeta-function, refining previous results and covering larger ranges of |t| with sharper constants.

Findings

No zeros for σ ≥ 1 - 1/(55.241(log|t|)^{2/3}(log log |t|)^{1/3}) for |t| ≥ 3

Improved classical zero-free region with σ ≥ 1 - 1/(5.558691 log|t|) for |t| ≥ 2

Enhanced bounds for intermediate |t| values, extending the known zero-free regions

Abstract

We prove that the Riemann zeta-function $\zeta(\sigma + it)$ has no zeros in the region $\sigma \geq 1 - 1/(55.241(\log|t|)^{2/3} (\log\log |t|)^{1/3})$ for $|t|\geq 3$. In addition, we improve the constant in the classical zero-free region, showing that the zeta-function has no zeros in the region $\sigma \geq 1 - 1/(5.558691\log|t|)$ for $|t|\geq 2$. We also provide new bounds that are useful for intermediate values of $|t|$. Combined, our results improve the largest known zero-free region within the critical strip for $3\cdot10^{12} \leq |t|\leq \exp(64.1)$ and $|t| \geq \exp(1000)$.