Maxima of a Random Model of the Riemann Zeta Function over Intervals of Varying Length

We consider a model of the Riemann zeta function on the critical axis and study its maximum over intervals of length $(\log T{)}^{\theta }$, where $\theta$ is either fixed or tends to zero at a suitable rate. It is shown that the deterministic level of the maximum interpolates smoothly between the ones of log-correlated variables and of i.i.d. random variables, exhibiting a smooth transition ‘from $\frac{3}{4}$ to $\frac{1}{4}$’ in the second order. This provides a natural context where extreme value statistics of log-correlated variables with time-dependent variance and rate occur. A key ingredient of the proof is a precise upper tail tightness estimate for the maximum of the model on intervals of size one, that includes a Gaussian correction. This correction is expected to be present for the Riemann zeta function and pertains to the question of the correct order of the maximum of the zeta function in large intervals.