Divisors and square-free divisors involving the floor function

B. Feng and J. Wu
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 4, Pages 916–923
DOI: 10.7546/nntdm.2025.31.4.916-923
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Authors and affiliations

B. Feng
School of Mathematics and Big Data, Chongqing University of Education
Chongqing 400065, P. R. China

J. Wu
CNRS
Université Paris-Est Créteil
Université Gustave Eiffel
LAMA 8050
F-94010 Créteil, France

Abstract

Let \tau(n) (respectively, \tau_{\sharp}(n)) be the number of divisors (respectively, square free divisors) of natural number n, and let [t] be the integral part of real number t.
In this short note, we prove that for any \varepsilon>0 the asymptotic formula

    \[\sum_{n\le x^{1/c}} g\Big(\Big[\frac{x}{n^c}\Big]\Big) = \Xi_{g, c} x^{1/c} + O_{\varepsilon}(x^{\theta_c^{\rm FW}+\varepsilon})\]

holds for x\to\infty, where g=\tau or \tau_{\sharp} and

    \[\Xi_{g, c} := \sum_{d=1}^{\infty} g(d)\bigg(\frac{1}{d^{1/c}} - \frac{1}{(d+1)^{1/c}}\bigg), \quad \theta_c^{\rm FW} := \begin{cases} \frac{2}{3c+2}, & \text{if $0<c\le \frac{2}{5}$}, \\\noalign{\vskip 1mm} \frac{5}{5c+6}, & \text{if $c\ge \frac{2}{5}$}\cdot \end{cases}\]

This improves and generalises the corresponding results of Feng–Wu for \tau(n) and of Zhang for \tau_{\sharp}(n) with c=1, respectively.

Keywords

  • Exponential sums
  • Square free integers
  • Asymptotic results on arithmetic functions

2020 Mathematics Subject Classification

  • 11L07
  • 11N25
  • 11N37

References

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Manuscript history

  • Received: 21 August 2025
  • Revised: 5 December 2025
  • Accepted: 8 December 2025
  • Online First: 10 December 2025

Copyright information

Ⓒ 2025 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Feng, B., & Wu, J. (2025). Divisors and square-free divisors involving the floor function. Notes on Number Theory and Discrete Mathematics, 31(4), 916-923, DOI: 10.7546/nntdm.2025.31.4.916-923.

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