  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 $\sigma \geq 1-1
/\left(54.004(\log |t|)^{2 / 3}(\log \log |t|)^{1 / 3}\right)$ for $|t| \geq 3$
and asymptotically in the region $\sigma \geq 1-1 /\left(48.0718(\log |t|)^{2 /
3}(\log \log |t|)^{1 / 3}\right)$ for $|t|$ sufficiently large.
