Convergence in law of the maximum of the Riemann zeta function on expanding intervals near the critical line

Abstract

Let $T \to \infty$, let $τ$ be uniformly distributed on $[T,2T]$, and let $σ= σ(T)$ be a deterministic real part approaching $1/2$ from the right. We establish the convergence in law of the recentered maximum of $\log|ζ(σ+ \mathrm{i}τ+ \mathrm{i} h)|$ over $I_θ = \smash{[ - (\log T)^θ,(\log T)^θ]}$, for every fixed $θ> 0$. Under explicit conditions on the displacement $σ- 1/2$, the recentered maximum converges to a Gumbel distribution with scale parameter $1/(2\sqrt{1 + θ})$. The recentering is an exact upper quantile of the one-point law of the associated Haar field $F_{h}^{\mathrm{H}} = \Re\sum_{p^{j}\leq X}U_{p}^{j}/(jp^{j(σ+ \mathrm{i} h)})$, where the $U_{p}$ are independent and uniformly distributed on the complex unit circle. The positive constant in the limiting distribution is characterized by a Gaussian process describing the limiting microscopic increments of the Haar field around a high point. The result is established without the Riemann hypothesis.