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On rates of convergence in central limit theorems of Selberg and Bourgade

Po-Han Hsu, Peng-Jie Wong

TL;DR

This work provides explicit rates of convergence in central limit theorems for vectors formed from logarithms of shifted Dirichlet L-functions. Building on Selberg’s CLT and Bourgade’s zeta-function framework, the authors derive a quantitative Dudley-distance bound between the random vector of normalized log-L-values and a limiting N-variate normal with covariance determined by shift interactions, under precise control of shift gaps. The core method propagates through a chain of approximations (from the original X_T to a Gaussian process) and carefully tracks moments and covariance matching, revealing that dependence among components can dramatically affect the rate, and that multidimensional independence cases admit sharper bounds (valid up to N=3). The results extend prior work on Dirichlet L-functions and connect number-theoretic CLTs with multivariate Gaussian-process techniques, with further implications for Dedekind zeta functions and the probabilistic interpretation of log-correlated structures. Overall, the paper provides both a unified framework for high-dimensional CLTs in this setting and concrete rates that quantify how shift structure governs convergence.

Abstract

Based on the recent works of Radziwill-Soundararajan and Roberts, we establish a rate of convergence in Bourgade's central limit theorem for shifted Dirichlet $L$-functions. Our results also indicate that the dependence structure in the components of a random vector could have a dramatic impact on the rate of convergence in such a multivariate central limit theorem.

On rates of convergence in central limit theorems of Selberg and Bourgade

TL;DR

This work provides explicit rates of convergence in central limit theorems for vectors formed from logarithms of shifted Dirichlet L-functions. Building on Selberg’s CLT and Bourgade’s zeta-function framework, the authors derive a quantitative Dudley-distance bound between the random vector of normalized log-L-values and a limiting N-variate normal with covariance determined by shift interactions, under precise control of shift gaps. The core method propagates through a chain of approximations (from the original X_T to a Gaussian process) and carefully tracks moments and covariance matching, revealing that dependence among components can dramatically affect the rate, and that multidimensional independence cases admit sharper bounds (valid up to N=3). The results extend prior work on Dirichlet L-functions and connect number-theoretic CLTs with multivariate Gaussian-process techniques, with further implications for Dedekind zeta functions and the probabilistic interpretation of log-correlated structures. Overall, the paper provides both a unified framework for high-dimensional CLTs in this setting and concrete rates that quantify how shift structure governs convergence.

Abstract

Based on the recent works of Radziwill-Soundararajan and Roberts, we establish a rate of convergence in Bourgade's central limit theorem for shifted Dirichlet -functions. Our results also indicate that the dependence structure in the components of a random vector could have a dramatic impact on the rate of convergence in such a multivariate central limit theorem.
Paper Structure (14 sections, 27 theorems, 315 equations)

This paper contains 14 sections, 27 theorems, 315 equations.

Table of Contents

  1. Introduction
  2. Further discussions - number-theoretic aspect
  3. Further discussions - probabilistic aspect
  4. Structure of the article
  5. Notation and preliminaries
  6. Key propositions and the proof of the main theorem
  7. List of all key propositions
  8. Proof of Theorem \ref{['main-thm']}
  9. Proof of Proposition \ref{['Prop 2 in AR24']}
  10. Proof of Proposition \ref{['Prop 3 in AR24']}
  11. Proofs of Propositions \ref{['Prop 4 in AR24']} and \ref{['Prop 5 in AR24']}
  12. Proof of Proposition \ref{['Prop 6 in AR24']}
  13. Proof of Proposition \ref{['Prop 7']}
  14. Proof of Theorem \ref{['indep-rate']}

Key Result

Theorem 1.1

Let ${\bf X}_{T} = {\bf X}_{T}(U)$ and $\tilde{{\bf X}}$ be defined as in X_T and normal X, respectively. Then for any fixed $0<\varepsilon_{1}<\varepsilon_{2}$, with $\varepsilon_{1} + \varepsilon_{2}<1$, and any sufficiently large $T$, Consequently, as $T\rightarrow\infty$, the vector ${\bf X}_{T}$ converges to $\tilde{{\bf X}}$ in distribution.

Theorems & Definitions (55)

  • Theorem 1.1
  • Remark 1
  • Theorem 1.2
  • Remark 2
  • Theorem 1.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • ...and 45 more