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

Let $L(s,{\chi }_1),\dots ,L(s,{\chi }_N)$ be primitive Dirichlet $L$-functions different from the Riemann zeta function. Under suitable hypotheses we prove that any linear combination $a_1\log \mathrm{\mid }L(\rho ,{\chi }_1)\mathrm{\mid }+\cdots +a_N\log \mathrm{\mid }L(\rho ,{\chi }_N)\mathrm{\mid }$ has an approximately normal distribution as $T\to \mathrm{\infty }$ with mean $0$ and variance $\frac{1}{2}({a_1}^2+\cdots +{a_N}^2)\log \log T\mathrm{.}$ Here $a_1,a_2,\dots ,a_N\in \mathbb{R}$, and $\rho$ runs over the nontrivial zeros of the zeta function with $0<\mathrm{\Im }\rho \le T$. From this we deduce that the vectors $(\log \mathrm{\mid }L(\rho ,{\chi }_1)\mathrm{\mid }\mathrm{/}\sqrt{\frac{1}{2}\log \log T},\dots ,\log \mathrm{\mid }L(\rho ,{\chi }_N)\mathrm{\mid }\mathrm{/}\sqrt{\frac{1}{2}\log \log T})$ have approximately an $N$-variate normal distribution whose components are approximately mutually independent as $T\to \mathrm{\infty }$. 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.