The moments of the Riemann zeta function on the interval [T,2T] on the critical line play a fundamental role in the distribution of the prime numbers. In this talk, we look at the moments of the Riemann zeta function on typical short intervals, but with length diverging with T. We will show that the moments exhibit a freezing phase transition up to a certain interval length akin to the transition seen in log-correlated processes. As a consequence we prove the leading order of the maximum of the zeta function on such short intervals. The results generalize a conjecture of Fyodorov & Keating and the related results of Arguin et al. and Najnudel on intervals of length one.
Joint work with F. Ouimet and M. Radziwill.