Trigonometric inequalities and the Riemann zeta-function

TL;DR

This paper improves the known zero-free region of the Riemann zeta-function by refining the leading constant through a simple enhancement of a trigonometric polynomial.

Contribution

It introduces a minor but effective modification to the trigonometric polynomial used in zero-free region proofs, leading to a tighter bound.

Findings

Improved the leading constant of the zero-free region

Achieved a marginally larger zero-free region

Demonstrated the effectiveness of simple polynomial modifications

Abstract

We very slightly improve the leading constant of the (currently best) proven asymptotic zero-free region of the Riemann zeta-function, by using an easy improvement to a trigonometric polynomial.