Mean value of real characters using a double Dirichlet series

TL;DR

This paper improves the understanding of the average value of real characters by analyzing a double Dirichlet series, leading to a better error term and a natural emergence of the main term through complex analysis techniques.

Contribution

It introduces a novel method using inverse Mellin transform twice to analyze double Dirichlet series, enhancing error estimates and clarifying the main term's origin.

Findings

Achieved an improved error term in the asymptotic formula.

Demonstrated the main term arises from residues of the double Dirichlet series.

Provided a new analytical approach using inverse Mellin transforms.

Abstract

We study the double character sum $\sum\limits_{\substack{m\leq X,\\m\odd}}\sum\limits_{\substack{n\leq Y,\\n\odd}}\leg mn$ and its smoothly weighted counterpart. An asymptotic formula with power saving error term was obtained by Conrey, Farmer and Soundararajan by applying the Poisson summation formula. The result is interesting, because the main term involves a non-smooth function. In this paper, we apply the inverse Mellin transform twice and study the resulting double integral that involves a double Dirichlet series. This method has two advantages -- it leads to a better error term, and the surprising main term naturally arises from three residues of the double Dirichlet series.