Freezing transition and moments of moments of the Riemann zeta function

Abstract

Moments of moments of the Riemann zeta function, defined by \[ \text{MoM}_T (k,β) = \frac{1}{T} \int_T^{2T} \left( \int_{ |h|\leq (\log T)^θ}|ζ(\tfrac{1}{2} + i t + ih)|^{2β} dh \right)^k dt \] where $k,\beta \ge 0$ 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 $T\to \infty$ and obtain sharp upper bounds for ${\text{MoM}}_T(2,\beta )$ for all real $0\le \beta \le 1$ as well as lower bounds of the conjectured order for all $\beta \ge 0$. In particular, we show that the second moment of moments undergoes a freezing phase transition with critical exponent $\beta =\frac{1}{\sqrt{2}}$.

Added references, simplified proofs, and fixed typos