Fyodorov and Keating's short-interval maximum conjecture for the Riemann zeta function

Let $\zeta(s)$ denote the Riemann zeta function. For $T>0$, consider its maximum on the interval $[T,T+2\pi]$.

Fyodorov and Keating's conjecture. Let $\epsilon>0$. For $T_1$ sufficiently large depending on $\epsilon$, for all $T_1\leq T\leq 2T_1$, outside a set of “bad” values of $T$ of measure at most $\epsilon T_1$, one has

$$\max_{T\leq t\leq T+2\pi}|\zeta(1/2+it)|=\exp(\log\log T-(3/4)\log\log\log T+O_{\epsilon}(1)),$$

where $O_{\epsilon}(1)$ denotes a quantity bounded in terms of $\epsilon$. This refines the expected order of the maximum of $|\zeta(1/2+it)|$ over a short interval and is related to a more precise conjecture for the distribution of its logarithm; its status is not resolved here.