Dirichlet $L$-functions on the critical line and multiplicative chaos

TL;DR

This paper demonstrates that Dirichlet L-functions with random characters on the critical line converge to a distribution linked to Gaussian multiplicative chaos, connecting number theory and probabilistic models.

Contribution

It establishes the convergence of random Dirichlet L-functions to a Gaussian multiplicative chaos-related distribution, extending previous results on the Riemann zeta function.

Findings

Dirichlet L-functions converge to a random Schwartz distribution.

The limiting distribution is related to Gaussian multiplicative chaos.

Connections between number theory and probabilistic models are demonstrated.

Abstract

In this paper we prove that the Dirichlet $L$-functions $L(1/2+ix,\chi_q)$, where $\chi_q$ is uniformly random Dirichlet character modulo $q$ and $x\in \mathbb{R}$, converges to a random Schwartz distribution $\zeta_{\mathrm{rand}}$, which is related to (complex) Gaussian multiplicative chaos. This is the same limiting object that appeared in [34], where the authors proved that the random shifts of the Riemann zeta function on the critical line $\zeta(1/2+ix+i\omega T)$, where $\omega\sim \mathrm{Unif} ([0,1])$, converge as $T\to \infty$.