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

In this article, we prove that the Dirichlet $L$-functions $L(1/2+ix,χ_q)$, where $χ_q$ is a uniformly random Dirichlet character modulo $q$ and $x\in \mathbb{R}$, converge as $q\to \infty$ to a random Schwartz distribution $ζ_{\mathrm{rand}}$, which is related to complex Gaussian multiplicative cha