The Fyodorov-Hiary-Keating Conjecture. II

Abstract

We prove a lower bound on the maximum of the Riemann zeta function in a typical short interval on the critical line. Together with the upper bound from the previous work of the authors, this implies tightness of $$\underset{|h|\le 1}{\max }|\zeta (\frac{1}{2}+\mathrm{i}\tau +\mathrm{i}h)|\cdot \frac{(\log \log T{)}^{3/4}}{\log T},$$ for large $T$, where $\tau$ is uniformly distributed on $[T,2T]$. The techniques are also applied to bound the right tail of the maximum, proving the distributional decay $\asymp y{e}^{-2y}$ for $y$ positive. This confirms the Fyodorov-Hiary-Keating conjecture, which states that the maximum of $\zeta$ in short intervals lies in the universality class of logarithmically correlated fields.

41 pages