A Central Limit Theorem for Linear Combinations of Logarithms of Dirichlet $L$-functions Sampled at the Zeros of the Zeta Function
Fatma Çiçek, Steven M. Gonek, Scott J. Kirila
TL;DR
This work proves a central limit theorem for linear combinations $\mathcal L(\rho)=\sum_{j=1}^N a_j\log|L(\rho,\chi_j)|$ sampled at the nontrivial zeros $\rho=\tfrac{1}{2}+i\gamma$ of $\zeta(s)$, under GRH, Hypothesis $\mathscr D$, and a zero-spacing hypothesis. The authors develop Dirichlet-polynomial approximations and a random-model for these polynomials, establish univariate and multivariate Gaussian limit laws with variance $\tfrac12(\sum a_j^2)\log\log T$ and identity covariance, respectively, and prove mutual approximate independence of the log-values. They further apply these results to show that, for any nonzero complex coefficients $c_i$, the equation $c_1L(\rho,\chi_1)+\cdots+c_NL(\rho,\chi_N)=a$ has asymptotic density zero zeros on the zeta zeros. The methods combine Dirichlet polynomial control, probabilistic modeling via random polynomials, and Lévy’s continuity theorem to connect analytic and probabilistic viewpoints, reinforcing the independence of Dirichlet $L$-functions at zeta zeros and providing a robust framework for value distribution at critical zeros.
Abstract
Let $L(s, χ_1), \ldots, L(s, χ_N)$ be primitive Dirichlet $L$-functions different from the Riemann zeta function. Under suitable hypotheses we prove that any linear combination $a_1\log|L(ρ,χ_1)|+\dots+a_N\log|L(ρ,χ_N)|$ 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 $ρ$ runs over the nontrivial zeros of the zeta function with $0< \Imρ\leq T$. From this we deduce that the vectors $\big(\log|L(ρ,χ_1)|/\sqrt{ \frac12 \log\log T}, \ldots, \log|L(ρ,χ_N)|/\sqrt{\frac12 \log\log T}\,\big)$ 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 $ρ$ that are zeros or $a$-values of linear combinations of the form $c_1 L(ρ, χ_1)+ \cdots + c_N L(ρ, χ_N)$ with complex $c_i$ as coefficients.