Uniform effective estimates for $\vert L(1,\chi)\vert$

TL;DR

Under the assumption of the Generalised Riemann Hypothesis, the paper establishes effective bounds for the absolute value of Dirichlet L-functions at 1, with implications for class numbers of imaginary quadratic fields.

Contribution

It proves uniform effective estimates for |L(1,χ)| under GRH, extending prior results and applying them to class numbers of imaginary quadratic fields.

Findings

Validates estimates of |L(1,χ)| under GRH

Extends results to class numbers of imaginary quadratic fields

Provides uniform bounds for these quantities

Abstract

Let $L(s,\chi)$ be the Dirichlet $L$-function associated to a non-principal primitive Dirichlet character $\chi$ defined modulo $q$, where $q\ge 3$. We prove, under the assumption of the Generalised Riemann Hypothesis, the validity of estimates given by Lamzouri, Li, and Soundararajan on $\vert L(1,\chi) \vert$. As a corollary, we have that similar estimates hold for the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$, $q\ge 5$.