= Richert bound for the Riemann zeta function
{c}
{title2=$|\zeta(\sigma+it)|\ll t^{C(1-\sigma)^{3/2}}\log^{2/3}t$}
For large positive $t$ and $0<\sigma\le1$, 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.
Back to article page
