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

TL;DR

This paper establishes a rate of convergence in a central limit theorem for shifted Dirichlet L-functions, highlighting how dependence structures affect convergence rates in multivariate CLTs.

Contribution

It provides new quantitative convergence rates in Bourgade's CLT for Dirichlet L-functions and explores the impact of dependence structures on multivariate CLT convergence.

Findings

Established a convergence rate in Bourgade's CLT for shifted Dirichlet L-functions.

Showed that dependence structures significantly influence multivariate CLT convergence rates.

Connected recent work of Radziwill-Soundararajan and Roberts to these convergence results.

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.