Maxima of a random model of the Riemann zeta function on longer intervals (and branching random walks)

We study the maximum of a random model for the Riemann zeta function (on the critical line at height T) on the interval $[-(\log T)^\theta,(\log T)^\theta]$, where $ \theta= = (\log \log T)^{-a}$, with $0<a<1$.  We obtain the leading order as well as the logarithmic correction of the maximum.

As it turns out, a good toy model is a collection of independent BRWs, where the number of independent copies depends on $\theta$. In this talk I will try to motivate our results by mainly focusing on this toy model. The talk is based on joint work in progress with L.-P. Arguin and G. Dubach.