Modified Dirichlet character sums over the $k$-free integers

TL;DR

This paper investigates the cancellation in partial sums of modified quadratic Dirichlet characters over $k$-free integers, establishing an $x^{1/(k+1)}$ bound under the Riemann Hypothesis for quadratic Dirichlet $L$-functions.

Contribution

It proves a new conditional bound on the cancellation of sums of modified quadratic Dirichlet characters over $k$-free integers, improving previous results.

Findings

Achieves $x^{1/(k+1)}$ cancellation bound under RH

Extends understanding of character sums over $k$-free integers

Builds upon recent advances in the field

Abstract

The main question of this paper is the following: how much cancellation can the partial sums restricted to the $k$-free integers up to $x$ of a $\pm 1$ multiplicative function $f$ be in terms of $x$? Building upon the recent paper by Q. Liu, Acta Math. Sin. (Engl. Ser.) 39 (2023), no. 12, 2316-2328, we prove that under the Riemann Hypothesis for quadratic Dirichlet $L$-functions, we can get $x^{1/(k+1)}$ cancellation when $f$ is a modified quadratic Dirichlet character, i.e., $f$ is completely multiplicative and for some quadratic Dirichlet character $\chi$, $f(p)=\chi(p)$ for all but a finite subset of prime numbers. This improves the conditional results by Aymone, Medeiros and the author cf. Ramanujan J. 59 (2022), no. 3, 713-728.