Freezing transition and moments of moments of the Riemann zeta function (2301.10634v2)

Abstract: Moments of moments of the Riemann zeta function, defined by [ \text{MoM}T (k,\beta) = \frac{1}{T} \int_T^{2T} \left( \int{ |h|\leq (\log T)^\theta}|\zeta(\tfrac{1}{2} + i t + ih)|^{2\beta} dh \right)^k dt ] where $k,\beta \geq 0$ and $\theta > -1$, were introduced by Fyodorov and Keating when comparing extreme values of zeta in short intervals to those of characteristic polynomials of random unitary matrices. We study the $k = 2$ case as $T \rightarrow \infty$ and obtain sharp upper bounds for $\text{MoM}_T(2,\beta)$ for all real $0\leq \beta \leq 1$ as well as lower bounds of the conjectured order for all $\beta \geq 0$. In particular, we show that the second moment of moments undergoes a freezing phase transition with critical exponent $\beta = \tfrac{1}{\sqrt{2}}$.