Richert bound for the Riemann zeta function ($|\zeta (\sigma +it)|\ll t^{C(1-\sigma {)}^{3/2}}{\log }^{2/3}t$)

For large positive $t$ and $0<\sigma \le 1$, a fixed sufficiently large $C$ gives the displayed upper bound; to the right of one use $|\zeta (\sigma +it)|\ll {\log }^{2/3}t$. Its nonlinear dependence on $1-\sigma$ permits a wider zero-free region of the Riemann zeta function through the Landau zero-free-region theorem. Its proof uses estimates for exponential sums; its use as an upper-bound input is separate from a proof of a zero-free region.