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

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 $(a_1\log|L(ρ,χ_n)|+\dots+a_1\log|L(ρ,χ_n)|)$ 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 $χ_i$ is a primitive Dirichlet character modulo $M_i$ with $M_i\leq T$, and $0< \operatorname{Im}ρ\leq T$ where $ρ$ runs over nontrivial zeros of the zeta-function. From the proof of this result, we also derive the independence of the distributions of sequences $(\log|L(ρ,χ_1)|), \dots, (\log|L(ρ,χ_n)|)$ provided that they are suitably normalized.

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