A Simple Proof of Siegel's Theorem Using Mellin Transform

TL;DR

This paper provides a straightforward analytic proof of Siegel's theorem on the lower bounds of $L(1, ext{chi})$ for primitive quadratic characters, utilizing Mellin transforms to compare bounds.

Contribution

It introduces a novel, simplified proof of Siegel's theorem using Mellin transform techniques, offering an alternative to more complex existing proofs.

Findings

Established a new simple proof of Siegel's theorem

Demonstrated the effectiveness of Mellin transform in analytic number theory

Provided clearer bounds for $L(1, ext{chi})$ for quadratic characters

Abstract

In this paper, we present a simple analytic proof of Siegel's theorem that concerns the lower bound of $L(1,\chi)$ for primitive quadratic $\chi$. Our new method compares an elementary lower bound with an analytic upper bound obtained by the inverse Mellin transform of $\Gamma(s)$.