Explicit zero-free regions for the Riemann zeta-function

We prove that the Riemann zeta-function \(\zeta (\sigma + it)\) $\zeta (\sigma +it)$ has no zeros in the region \(\sigma \ge 1 - 1/(55.241(\log |t|)^{2/3} (\log \log |t|)^{1/3})\) $\sigma \ge 1-1/{(55.241(log|t|)}^{2/3}{(loglog|t|)}^{1/3})$ for \(|t|\ge 3\) $|t|\ge 3$ . In addition, we improve the constant in the classical zero-free region, showing that the zeta-function has no zeros in the region \(\sigma \ge 1 - 1/(5.558691\log |t|)\) $\sigma \ge 1-1/(5.558691log|t|)$ for \(|t|\ge 2\) $|t|\ge 2$ . We also provide new bounds that are useful for intermediate values of \(|t|\) $|t|$ . Combined, our results improve the largest known zero-free region within the critical strip for \(3\cdot 10^{12} \le |t|\le \exp (64.1)\) $3·{10}^{12}\le |t|\le exp(64.1)$ and \(|t| \ge \exp (1000)\) $|t|\ge exp(1000)$ .