Integral Means Spectrum for the Random Riemann Zeta Function

TL;DR

This paper investigates the integral means spectrum of the randomized Riemann zeta-function and related stochastic objects, confirming conjectured forms and connecting to Gaussian multiplicative chaos and Liouville quantum gravity.

Contribution

It proves the almost sure form of the integral means spectrum for the primitive of the randomized zeta-function, confirming Kraetzer's conjecture and relating it to Gaussian multiplicative chaos.

Findings

The spectrum matches Kraetzer's conjecture for the universal integral means spectrum.

The primitive of the randomized zeta-function is almost surely non-injective.

The spectrum for holomorphic GMC on the unit disc also follows the same Kraetzer form.

Abstract

We study the integral means spectrum associated with the analytic function whose derivative is the so-called randomized Riemann zeta-function, introduced some time ago by Bagchi. The randomized $\zeta$-function, ${\zeta}_{\mathrm{rand}}(\sigma+ih)$, is known to represent the asymptotic statistical behaviour of the random vertical shifts of the actual $\zeta$-function in the critical strip, $1/2 <\sigma\leq 1, h\in \mathbb R$, and appears in a number of recent works on the asymptotic behavior of the moments and maxima of the $\zeta$-function on short intervals along the critical axis $\sigma=1/2$. Using probability and basic analytic number theory, we show that the complex integral means spectrum of the primitive of ${\zeta}_{\mathrm{rand}}$ is almost surely of the form conjectured 30 years ago by Kraetzer, for the so-called universal integral means spectrum of univalent functions in the disc. The Riemann $\zeta$-function and its random version have recently been rigorously related to the so-called Gaussian multiplicative chaos (GMC), initiated by Kahane 40 years ago. In the case of the holomorphic multiplicative chaos on the unit disc -- an important stochastic object closely related to Liouville quantum gravity on the unit circle -- we prove that the integral means spectrum of the primitive is almost surely also of the same Kraetzer form. However, we establish that neither the primitive of the random function ${\zeta}_{\mathrm{rand}}$, nor that of the holomorphic GMC are injective. Building on earlier work by one of the authors and Webb on the convergence of Riemann $\zeta$-function on the critical line to a holomorphic GMC distribution, we finally provide an alternative derivation of the integral means spectrum for the random Riemann $\zeta$-function.