A Central Limit Theorem for Linear Combinations of Logarithms of Dirichlet $L$-functions Sampled at the Zeros of the Zeta Function

Abstract

Let $L(s, \chi_1), \ldots, L(s, \chi_N)$ be primitive Dirichlet $L$-functions different from the Riemann zeta function. Under suitable hypotheses we prove that any linear combination $a_1\log|L(\rho,\chi_1)|+\dots+a_N\log|L(\rho,\chi_N)|$ has an approximately normal distribution as $T\to \infty $ with mean $0$ and variance $ \tfrac12 \big({a_1}^2+\dots+{a_N}^2\big)\log\log T.$ Here $a_1, a_2, \ldots, a_N \in \mathbb{R}$, and $\rho$ runs over the nontrivial zeros of the zeta function with $0< \Im\rho \leq T$. From this we deduce that the vectors $\big(\log|L(\rho,\chi_1)|/\sqrt{ \frac12 \log\log T}, \ldots, \log|L(\rho,\chi_N)|/\sqrt{\frac12 \log\log T}\,\big)$ have approximately an $N$-variate normal distribution whose components are approximately mutually independent as $T\to \infty$. We apply these results to study the proportion of the $\rho$ that are zeros or $a$-values of linear combinations of the form $c_1 L(\rho, \chi_1)+ \cdots + c_N L(\rho, \chi_N)$ with complex $c_i$ as coefficients.