Asymptotic formula of a “hyperbolic” summation related to the Piltz divisor function

Mihoub Bouderbala and Meselem Karras
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 4, Pages 648–655
DOI: 10.7546/nntdm.2022.28.4.648-655
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Authors and affiliations

Mihoub Bouderbala
Department of Mathematics, University of Djilali Bounaama
Khemis Miliana, FIMA Laboratory, Algeria

Meselem Karras
Department of Mathematics, University of Djilali Bounaama
Khemis Miliana, FIMA Laboratory, Algeria

Abstract

In this paper, we obtain asymptotic formula on the “hyperbolic” summation

    \begin{equation*} \underset{mn\leq x}{\sum }D_{k}\left( \gcd \left( m,n\right) \right) \text{ \ \ }\left( k\in \mathbb{Z}_{\geq 2}\right), \end{equation*}

such that D_{k}\left( n\right) = \dfrac{\tau _{k}\left( n\right) }{\tau_{k}^{\ast }\left( n\right) }, where \tau _{k}\left( n\right) =\!\!\sum\limits_{n_{1}n_{2}\ldots n_{k}=n}\!\!1 denotes the Piltz divisor function, and \tau _{k}^{\ast }\left( n\right) is the unitary analogue function of \tau _{k}\left( n\right).

Keywords

  • Number of distinct prime divisors
  • Hyperbolic summation
  • Piltz divisor function

2020 Mathematics Subject Classification

  • 11N37
  • 11A25
  • 11N36

References

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  2. Bordellès, O. (2020). Arithmetic Tales. Advanced Edition, Springer (2nd edition).
  3. Heyman, R., & Tóth, L. (2021). On certain sums of arithmetic functions involving the GCD and LCM of two positive integers. Results in Mathematics, 76, Article 49.
  4. Karras, M., & Derbal, A. (2020). Mean value of an arithmetic function associated with the Piltz divisor function. Asian-European Journal of Mathematics, 13(03), Article 2050062.
  5. Sándor, J. (1989). On the arithmetical functions dk(n). Journal of Numerical Analysis and Approximation Theory. 18(1), 89–94.
  6. Sándor, J. (1996). On the arithmetical functions dk(n) and dk∗(n). Portugaliae Mathematica, 53(1), 107–115.

Manuscript history

  • Received: 29 March 2022
  • Revised: 22 September 2022
  • Accepted: 22 October 2022
  • Online First: 24 October 2022

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Cite this paper

Bouderbala, M., & Karras, M. (2022). Asymptotic formula of a “hyperbolic” summation related to the Piltz divisor function. Notes on Number Theory and Discrete Mathematics, 28(4), 648-655, DOI: 10.7546/nntdm.2022.28.4.648-655.

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