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.

 

 

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