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

Caio Bueno

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 $χ$, $f(p)=χ(p)$ 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.

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

Abstract

The main question of this paper is the following: how much cancellation can the partial sums restricted to the -free integers up to of a multiplicative function be in terms of ? 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 -functions, we can get cancellation when is a modified quadratic Dirichlet character, i.e., is completely multiplicative and for some quadratic Dirichlet character , 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.

Paper Structure

This paper contains 14 sections, 7 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Main result and recent progress
  3. Motivation
  4. Other related results
  5. Notation
  6. Main result and discussion about the problem
  7. Auxiliary lemmas and preliminary results
  8. The main propositions
  9. Proposition when k is even
  10. Proposition when k is odd
  11. Proof of
  12. Proof when k is even
  13. Proof when k is odd
  14. Acknowledgments

Key Result

Theorem 1.1

Assume the Generalized Riemann hypothesis. There exists a multiplicative function $f:\mathbb{N}\to\{-1,0,+1\}$ with support on the $k$-free integers such that, at prime numbers, $f$ takes values in $\{-1,+1\}$ and we have

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 1
  • Conjecture 1.1: Aymone Aymone-edp-cubefree
  • Remark 2
  • Remark 3
  • Lemma 4.1: Montgomery-Vaughan, Theorem 5.1 and Corollary 5.3, pp. 138-140
  • Lemma 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • ...and 7 more