The Distribution of Values of $\frac{L'}{L}(1/2+\epsilon,\chi_D)$

TL;DR

This paper analyzes the distribution of the derivatives of L-functions at specific points for quadratic characters, establishing their limiting behavior, convergence rate, and bounds for small values.

Contribution

It determines the limiting distribution of rac{L'}{L}(1/2+ extepsilon, extchi_D) for fundamental discriminants and provides convergence and small value bounds.

Findings

Established the limiting distribution of rac{L'}{L}(1/2+ extepsilon, extchi_D).

Derived an upper bound for the convergence rate to the limiting distribution.

Provided asymptotic bounds for small values of rac{L'}{L}(1/2+ extepsilon, extchi_D).

Abstract

We determine the limiting distribution of the family of values $\frac{L'}{L}(1/2+\epsilon,\chi_D)$ as $D$ varies over fundamental discriminants. Here, $0<\epsilon<\frac12$, 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 $\left|\frac{L'}{L}(1/2+\epsilon,\chi_D)\right|$.