Modified Dirichlet character sums over the k-free integers

2024/11/13 by Bueno, Caio
#11L40 #11M06 #11N37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2411.08268

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 ± 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 χ, f(p)=χ(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.

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