# Lindelof hypothesis

Define

$\displaystyle \mu(\sigma):=\inf\{a\in\mathbf{R}:\zeta(\sigma+it)=O(|t|^a)\}.$

It is trivial to check that $\mu(\sigma)=0$ for $\mu(\sigma)=0$ for $\sigma>1$, and the functional equation of zeta function implies that

$\displaystyle \mu(\sigma)=\mu(1-\sigma)-\sigma+1/2$.

The Phragmen-Lindelof theorem implies that $\mu$ is a convex function. The Lindelof hypothesis states $\mu(1/2)=0$, which together with the above properties of $\mu$ implies that $\mu(\sigma)$ is $0$ for $\sigma\geq 1/2$ and $1/2-\sigma$ for $\sigma\leq 1/2$.