Integral Means Spectrum for the Random Riemann Zeta Function

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.