Zero-free regions inspired by work of Heath-Brown

TL;DR

This paper establishes a new explicit zero-free region for the Riemann zeta-function, improving understanding of its zeros and building on Heath-Brown's influential work on Linnik's constant.

Contribution

It introduces a novel explicit zero-free region for the Riemann zeta-function based on Heath-Brown's methods, extending previous bounds.

Findings

Proves $ ext{ extit{zeta}}(\sigma + it) e 0$ for $t extgreater 3$ and $\sigma extgreater 1 - 1/(4.896 ext{log} t)$.

Provides explicit bounds improving previous zero-free regions.

Enhances techniques for analyzing zeros of the zeta-function.

Abstract

We prove a new explicit zero-free region for the Riemann zeta-function, drawing substantially on Heath-Brown's seminal work on Linnik's constant. Using these ideas we are able to prove that $\zeta(\sigma + it)\ne 0$ whenever $t\geq 3$ and $\sigma \geq 1- 1/(4.896\log t)$.