Compute Lindeloef function

Question

How is the Lindeloef $\mu$ function ($ \sigma \longrightarrow \inf \{r/F( \sigma+it)<<t^r\})$ associated to $F=\zeta(s)^2$?

Answer 1

You mean $$\mu(\sigma) = \inf \ \{\ r,\ \zeta(\sigma+it) = \mathcal{O}(t^r) \ (t \to \infty)\}= \inf\ \{ \ r,\ \zeta(\sigma+it)^2 = \mathcal{O}(t^{2r}) \ (t \to \infty)\}$$

It is a continuous and convex function of $\sigma$ by the Phragmén-Lindelöf principle.

$\mu(\sigma) = 0$ for $\sigma > 1$ so that $\mu(1) = 0$. By the functional equation it means $\mu(0) =1/2$ and $\mu(\sigma) = 1/2-\sigma$ for $\sigma < 0$.

The (unproven) Lindelöf hypothesis is that $\mu(1/2) = 0$ (equivalently $\mu(\sigma)=0$ for $\sigma > 1/2$). The convexity gives $\mu(1/2) \le 1/4$.

This has been improved many times, up to $\mu(1/2) < 0.16$.