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

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.