Freezing transition and moments of moments of the Riemann zeta function

TL;DR

This paper investigates the moments of moments of the Riemann zeta function, revealing a phase transition at a critical exponent, and provides bounds for these moments as the parameter T grows large.

Contribution

It offers sharp bounds and demonstrates a freezing phase transition in the second moment of moments of the zeta function for large T.

Findings

Sharp upper bounds for MoM_T(2,β) for 0≤β≤1

Lower bounds matching conjectured order for all β≥0

Identification of a freezing phase transition at β=1/√2

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 $\beta = \tfrac{1}{\sqrt{2}}$.