Explicit bounds for the Riemann zeta function and a new zero-free region

Abstract

We prove that $|ζ(σ+it)|\le 70.7 |t|^{4.438 (1-σ)^{3/2}}\log^{2/3}|t|$ for $1/2\leσ\le 1$ and $|t|\ge 3$. As a consequence, we improve the explicit zero-free region for $ζ(s)$, showing that $ζ(σ+it)$ has no zeros in the region $σ\geq 1-1 /\left(54.004(\log |t|)^{2 / 3}(\log \log |t|)^{1 / 3}\right)$ for $|t| \geq 3$ and asymptotically in the region $σ\geq 1-1 /\left(48.0718(\log |t|)^{2 / 3}(\log \log |t|)^{1 / 3}\right)$ for $|t|$ sufficiently large.