The Distribution of Values of $\frac{L^{\prime }}{L}(1/2+\epsilon ,{\chi }_D)$

Abstract

We determine the limiting distribution of the family of values $\frac{L^{\prime }}{L}(1/2+\epsilon ,{\chi }_D)$ as $D$ varies over fundamental discriminants. Here, $0<\epsilon <\frac{1}{2}$, and ${\chi }_D$ is the real character associated with $D$. Moreover, we also establish an upper bound for the rate of convergence of this family to its limiting distribution. As a consequence of this result, we derive an asymptotic bound for the small values of $\left|\frac{L^{\prime }}{L}(1/2+\epsilon ,{\chi }_D)\right|$.

Journal of Number Theory, in press (2021)