Fyodorov–Hiary–Keating conjecture on zeta maxima over mesoscopic intervals

Let $\theta\in(-1,0]$, let $T$ tend to infinity, and let $\tau$ be uniformly distributed on $[T,2T]$. Write $\mathcal{N}(0,1)$ for a standard Gaussian random variable, and let $\mathcal{M}$ be a random variable that is $O(1)$ in probability, independent of $\mathcal{N}(0,1)$, with a tail of the form $Cye^{-2y}$. Fyodorov–Hiary–Keating conjecture. The maximum satisfies

$$\max_{h\leq\log^\theta T}\log\left|\zeta\left(\tfrac{1}{2}+i\tau+ih\right)\right|\sim(1+\theta)\log\log T-\frac{3}{4}\log\log\log T+\sqrt{\tfrac{|\theta|}{2}\log\log T}\,\mathcal{N}(0,1)+\mathcal{M}.$$

This conjecture describes the extreme values of the Riemann zeta function on mesoscopic intervals, incorporating both Gaussian fluctuations and an order-one extremal correction. The cited formulation is presented as a prediction from connections with log-correlated processes; its resolution is not supplied here.