Value-Distribution of Logarithmic Derivatives of Quadratic Twists of Automorphic $L$-functions

TL;DR

This paper studies the distribution of logarithmic derivatives of automorphic L-functions twisted by quadratic characters, establishing their limiting behavior and bounds on small values for self-dual cases.

Contribution

It determines the limiting distribution of these derivatives and provides bounds on their small values, advancing understanding of automorphic L-functions in number theory.

Findings

Established the limiting distribution of the family of derivatives.

Provided an upper bound on the discrepancy in convergence.

Derived bounds on small values of derivatives for self-dual representations.

Abstract

Let $d\in\mathbb{N}$, and let $\pi$ be a fixed cuspidal automorphic representation of $\mathrm{GL}_{d}(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. We determine the limiting distribution of the family of values $-\frac{L'}{L}(1+it,\pi\otimes\chi_D)$ as $D$ varies over fundamental discriminants. Here, $t$ is a fixed real number and $\chi_D$ is the real character associated with $D$. We establish an upper bound on the discrepancy in the convergence of this family to its limiting distribution. As an application of this result, we obtain an upper bound on the small values of $\left|\frac{L'}{L}(1,\pi\otimes\chi_D)\right|$ when $\pi$ is self-dual.