Freezing transition and moments of moments of the Riemann zeta function

Moments of moments of the Riemann zeta function, defined by $${\text{MoM}}_T(k,\beta )=\frac{1}{T}{\int }_T^{2T}{({\int }_{\mathrm{\mid }h\mathrm{\mid }\le (\log T{)}^{\theta }}\mathrm{\mid }\zeta (\frac{1}{2}+it+ih){\mathrm{\mid }}^{2\beta }dh)}^{k}dt$$ 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 $T\to \mathrm{\infty }$ 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 $\beta =\frac{1}{\sqrt{2}}$.