Explicit zero-free regions for the Riemann zeta-function

Abstract

We prove that the Riemann zeta-function $\zeta (\sigma +it)$ has no zeros in the region $\sigma \ge 1-1/(55.241(\log |t|{)}^{2/3}(\log \log |t|{)}^{1/3})$ for $|t|\ge 3$. In addition, we improve the constant in the classical zero-free region, showing that the zeta-function has no zeros in the region $\sigma \ge 1-1/(5.558691\log |t|)$ for $|t|\ge 2$. 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 $3\cdot 10^{12}\le |t|\le \exp (64.1)$ and $|t|\ge \exp (1000)$.

27 pages