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Distribution of sums involving Dirichlet characters over the $k$-free integers

Caio Bueno

Abstract

Assuming the generalized Riemann hypothesis and a bound for the negative discrete moments of the Riemann zeta function (resp. Dirichlet $L$-functions), we prove the existence of a logarithmic limiting distribution for the normalized partial sums $x^{-\frac{1}{2k}}\sum_{n\leq x}f(n)$, where $f$ is either a quadratic Dirichlet character or a modified Dirichlet character, restricted to the $k$-free integers. Moreover, we strengthen a conjecture made by Aymone, Medeiros and the author (cf. Ramanujan J. 59(3):713-728, 2022) concerning the precise order of magnitude for these partial sums.

Distribution of sums involving Dirichlet characters over the $k$-free integers

Abstract

Assuming the generalized Riemann hypothesis and a bound for the negative discrete moments of the Riemann zeta function (resp. Dirichlet -functions), we prove the existence of a logarithmic limiting distribution for the normalized partial sums , where is either a quadratic Dirichlet character or a modified Dirichlet character, restricted to the -free integers. Moreover, we strengthen a conjecture made by Aymone, Medeiros and the author (cf. Ramanujan J. 59(3):713-728, 2022) concerning the precise order of magnitude for these partial sums.
Paper Structure (11 sections, 18 theorems, 113 equations)

This paper contains 11 sections, 18 theorems, 113 equations.

Table of Contents

  1. Introduction and main results
  2. Notation
  3. Discussion of results concerning limiting distributions
  4. Preliminaries
  5. Ng's result and the existence of a limiting distribution
  6. Main lemmas
  7. Proof of
  8. Large deviations
  9. Additional results
  10. Final remarks
  11. Acknowledgements

Key Result

Theorem 1.1

Assume the generalized Riemann hypothesis. Let $\chi$ modulo $q$ be a primitive real non-principal Dirichlet character and $k\geq 2$ a fixed integer. Additionally, suppose that one of the following holds: where $\rho=\frac{1}{2}+i\gamma$ are the non-trivial zeros of $\zeta(s)$ in the first case or $L(s,\chi)$ in the second case. Then $e^{-\frac{y}{2k}}\sum_{n\leq e^y}f(n)$ has a limiting distribu

Theorems & Definitions (42)

  • Definition 1.1
  • Remark 1
  • Remark 2
  • Definition 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.1: Gonek1989Hejhal1989
  • Conjecture 1.2: Hughes-Keating-OConnell2000
  • Definition 1.3
  • Conjecture 1.3
  • ...and 32 more