Refined Fyodorov–Hiary–Keating conjecture for zeta maxima in macroscopic intervals

Fix $\theta>0$, let $\tau$ be uniformly distributed on $[T,2T]$, and let $C_k$ be the moment constants from the moment conjecture. Define

$$\beta=\beta(\theta)=\frac{1}{2\sqrt{1+\theta}}.$$

Let $\mathcal G_\theta$ be a Gumbel random variable with distribution function

$$\mathbf{P}(\mathcal G_\theta\leq x)=\exp\left(-e^{-(x-m)/\beta}\right),$$

where

$$m=m(\theta)=(0.06537\dots)+\beta^2\log C_{\sqrt{1+\theta}}-\frac{\beta^2}{2}\big(\log(1+\theta)-\log(4\pi)\big).$$

Refined Fyodorov–Hiary–Keating conjecture. As a sequence of random variables, the maximum satisfies

$$\max_{|h|\leq\pi(\log T)^\theta}\log|\zeta(1/2+{\rm i}(\tau+h))|=\sqrt{1+\theta}\log\log T-\frac{1}{4\sqrt{1+\theta}}\log\log\log T+\mathcal G_{\theta,T},$$

where $(\mathcal G_{\theta,T},T\geq1)$ converges in distribution to $\mathcal G_\theta$. This refines the conjectural law for maxima of the zeta function on macroscopic intervals by incorporating the moment correction $C_k$ into the order-one centering term. The claim is supported by theoretical and numerical evidence in the paper, but remains open.