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

Abstract

In this article, we prove that the Dirichlet $L$-functions $L(1/2+ix,{\chi }_q)$, where ${\chi }_q$ is a uniformly random Dirichlet character modulo $q$ and $x\in \mathbb{R}$, converge as $q\to \infty$ to a random Schwartz distribution ${\zeta }_{\mathrm{rand}}$, which is related to complex Gaussian multiplicative chaos (GMC). This is the same limiting object that appeared in \cite{SaWe20a}, 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 $T\to \infty$. As an application of our method, we also prove that to the right of the critical line, that is, for $\Re (s)>1/2$, the convergence of the random $L$-functions $L(s,{\chi }_q)$ occurs in the space of analytic functions, if we exclude the zero probability (in the limit $q\to \infty$) event that ${\chi }_q$ is the principal character. This was already proven in \cite{Ba81a}, but our method provides a novel proof of this fact. Our results hold for moduli $q=q_n$ such that the smallest prime divisor of $q_n$ tends to $\infty$ as $n\to \infty$.

arXiv comment28 pages, this version has some minor changes and corrections and added references. Most notably the way the $q\to \infty$ limit needs to be taken is specified