Effective Lower Bounds for L(1,{\chi}) via Eisenstein Series

TL;DR

This paper establishes effective lower bounds for the L(1,χ) values using Eisenstein series, combining analytic and sieve methods, extending techniques previously applied to the Riemann zeta function.

Contribution

It introduces a novel approach to bounding L(1,χ) by integrating Eisenstein series analysis with sieve theory, providing explicit bounds.

Findings

Derived explicit lower bounds for L(1,χ)

Applied Maass-Selberg relation to Eisenstein series

Extended Sarnak's methodology to L-functions

Abstract

We give effective lower bounds for $L(1,\chi)$ via Eisenstein series on $\Gamma_0(q) \backslash \mathbb{H}$. The proof uses the Maass-Selberg relation for truncated Eisenstein series and sieve theory in the form of the Brun-Titchmarsh inequality. The method follows closely the work of Sarnak in using Eisenstein series to find effective lower bounds for $\zeta(1+it)$.