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arXiv · 2506.16115

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

Sami Vihko
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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 $T\to \infty$.

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Sami Vihko. 2025-06-19. Dirichlet $L$-functions on the critical line and multiplicative chaos. https://arxiv.org/abs/2506.16115

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