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

Abstract

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

37 pages