The mean square of real character sums

TL;DR

This paper evaluates a smoothed sum involving the Kronecker symbol over positive odd square-free integers, providing insights into the mean square of real character sums in number theory.

Contribution

It introduces a new evaluation of a specific smoothed sum of real character sums involving the Kronecker symbol over square-free integers.

Findings

Derived an explicit asymptotic formula for the sum

Established bounds for the mean square of real character sums

Enhanced understanding of character sum behavior in number theory

Abstract

In this paper, we evaluate a smoothed sum of the form $\displaystyle \sideset{}{^*}\sum_{d \leq X}\left(\sum_{n \leq Y} \left(\frac{8d}{n} \right ) \right)^2$, where $\left ( \frac{8d}{\cdot} \right )$ is the Kronecker symbol and $\sideset{}{^*}\sum$ denotes a sum over positive odd square-free integers.