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

Sami Vihko

TL;DR

The paper proves that Dirichlet L-functions at the critical line, with the Dirichlet character chosen uniformly at random modulo $q$, converge in distribution to a random generalized function $\zeta_{\mathrm{rand}}$ associated with complex Gaussian multiplicative chaos. The authors build a chain of approximations via truncated Dirichlet polynomials and truncated Euler products, transferring randomness from Dirichlet characters to independent unit-modulus variables and finally to the GMC-type limit. Their method yields convergence in the space of Schwartz distributions on $\mathbb{R}$ for $\mathrm{Re}(s)=1/2$ and extends to convergence in the space of analytic functions on the half-plane $\mathrm{Re}(s)>1/2$ after excluding the principal character. This work strengthens the link between arithmetic L-functions and GMC, highlighting a universal probabilistic structure that aligns with prior results for the Riemann zeta function under random shifts.

Abstract

In this paper we prove that the Dirichlet $L$-functions $L(1/2+ix,χ_q)$, where $χ_q$ is uniformly random Dirichlet character modulo $q$ and $x\in \mathbb{R}$, converges to a random Schwartz distribution $ζ_{\mathrm{rand}}$, 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 $ζ(1/2+ix+iωT)$, where $ω\sim \mathrm{Unif} ([0,1])$, converge as $T\to \infty$.

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

TL;DR

The paper proves that Dirichlet L-functions at the critical line, with the Dirichlet character chosen uniformly at random modulo , converge in distribution to a random generalized function associated with complex Gaussian multiplicative chaos. The authors build a chain of approximations via truncated Dirichlet polynomials and truncated Euler products, transferring randomness from Dirichlet characters to independent unit-modulus variables and finally to the GMC-type limit. Their method yields convergence in the space of Schwartz distributions on for and extends to convergence in the space of analytic functions on the half-plane after excluding the principal character. This work strengthens the link between arithmetic L-functions and GMC, highlighting a universal probabilistic structure that aligns with prior results for the Riemann zeta function under random shifts.

Abstract

In this paper we prove that the Dirichlet -functions , where is uniformly random Dirichlet character modulo and , converges to a random Schwartz distribution , 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 , where , converge as .

Paper Structure

This paper contains 13 sections, 13 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Basic definitions and properties of Dirichlet characters and $L$-functions
  3. Short introduction to multiplicative chaos and the randomized zeta function $\zeta_{\mathrm{rand}}$ on the critical line.
  4. The main result
  5. Some preliminary lemmas
  6. Proof of \ref{['thm:main theorem']}
  7. Proof of \ref{['prop:first claim of the proof of the main theorem']}
  8. Proof of \ref{['prop:second claim of the proof of the main theorem']}
  9. Proof of \ref{['prop:third claim of the proof of the main theorem']}
  10. Convergence for $\mathrm{Re} \,(s)>1/2$ in the space of analytic functions
  11. Proof of \ref{['prop:uniform L^2 estimate on arbitrary compact']}
  12. Proof of \ref{['prop:convergence of LcalMq to LcalMomega on fixed compact']}
  13. Proof of \ref{['prop:equivalence of the approximations for fixed compact']}

Key Result

Theorem A

Let $q$ be a positive integer, and denote by $\mathcal{X}_q$ the set of all Dirichlet characters modulo $q$. Assume that $\chi_q$ is a uniformly random Dirichlet character, that is, $\chi_q\sim\mathrm{Unif}(\mathcal{X}_q)$ and $f\in C_c^\infty(\mathbb{R})$. Then as $q\to \infty$.

Theorems & Definitions (23)

  • Definition 1.1: Dirichlet characters
  • Theorem A
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 13 more