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 $(a_1\log |L(\rho ,{\chi }_n)|+\cdots +a_1\log |L(\rho ,{\chi }_n)|)$ has an approximate Gaussian distribution with mean $0$ and variance $\frac{1}{2}({a_1}^2+\cdots +{a_n}^2)\log \log T$. Here $a_1,\dots ,a_n\in \mathbb{R}$, each of the ${\chi }_i$ is a primitive Dirichlet character modulo $M_i$ with $M_i\le T$, and $0<\mathrm{Im}\rho \le T$ 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.