On the moments of the moments of $\zeta(1/2+it)$

TL;DR

This paper introduces the concept of 'moments of moments' of the Riemann zeta function, conjectures their asymptotic behavior for large T, and connects these conjectures to shifted moments and random matrix theory comparisons.

Contribution

It proposes a new conjecture for the asymptotics of moments of moments of the zeta function and relates it to shifted moments and random matrix models.

Findings

Conjectured asymptotics for moments of moments as T→∞.

Established that a function derived from shifted moments approximates these moments.

Motivated similar conjectures for other families of primitive L-functions.

Abstract

Taking $t$ at random, uniformly from $[0,T]$, we consider the $k$th moment, with respect to $t$, of the random variable corresponding to the $2\beta$th moment of $\zeta(1/2+ix)$ over the interval $x\in(t, t+1]$, where $\zeta(s)$ is the Riemann zeta function. We call these the `moments of moments' of the Riemann zeta function, and present a conjecture for their asymptotics, when $T\to\infty$, for integer $k,\beta$. This is motivated by comparisons with results for the moments of moments of the characteristic polynomials of random unitary matrices and is shown to follow from a conjecture for the shifted moments of $\zeta(s)$ due to Conrey, Farmer, Keating, Rubinstein, and Snaith \cite{cfkrs2}. Specifically, we prove that a function which, the shifted-moment conjecture of \cite{cfkrs2} implies, is a close approximation to the moments of moments of the zeta function does satisfy the asymptotic formula that we conjecture. We motivate as well similar conjectures for the moments of moments for other families of primitive $L$-functions.