A Central Limit Theorem for Linear Combinations of Logarithms of Dirichlet $L$-functions

TL;DR

This paper extends previous work on the Riemann zeta-function to show that linear combinations of logarithms of Dirichlet L-functions at zeros follow an approximate Gaussian distribution, under certain hypotheses, as T approaches infinity.

Contribution

It generalizes the central limit theorem for the logarithm of the zeta-function to linear combinations of Dirichlet L-functions, establishing their asymptotic Gaussian behavior and independence.

Findings

Sequences of linear combinations are approximately Gaussian distributed.

The distributions of individual logarithms are asymptotically independent.

The variance of the Gaussian distribution depends on the coefficients and log log T.

Abstract

The purpose of this paper is to generalize our earlier work on the logarithm of the Riemann zeta-function to linear combinations of logarithms of primitive Dirichlet $L$-functions with constant real coefficients. Under the assumption of suitable hypotheses, we prove that as $T\to \infty $, a sequence of the form $(a_1\log|L(\rho,\chi_n)|+\dots+a_1\log|L(\rho,\chi_n)|)$ has an approximate Gaussian distribution with mean $0$ and variance $ \tfrac{1}{2}\big({a_1}^2+\dots+{a_n}^2 \big)\log\log T$. Here $a_1, \dots, a_n \in \mathbb{R}$, each of the $\chi_i$ is a primitive Dirichlet character modulo $M_i$ with $M_i\leq T$, and $0< \operatorname{Im}\rho \leq T$ where $\rho$ runs over nontrivial zeros of the zeta-function. From the proof of this result, we also derive the independence of the distributions of sequences $(\log|L(\rho,\chi_1)|), \dots, (\log|L(\rho,\chi_n)|)$ provided that they are suitably normalized.