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

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