The mean value of the function \bm{\frac{d(n)}{d^*(n)}} in arithmetic progressions

Ouarda Bouakkaz and Abdallah Derbal
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 3, Pages 445–453
DOI: 10.7546/nntdm.2023.29.3.445-453
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Authors and affiliations

Ouarda Bouakkaz
Laboratoire d’Equations aux Derivées Partielles Non linéaires et Histoire des Mathématiques,
Ecole Normale Supérieure, Vieux-Kouba Alger, Algérie

Abdallah Derbal
Laboratoire d’Equations aux Derivées Partielles Non linéaires et Histoire des Mathématiques,
Ecole Normale Supérieure, Vieux-Kouba Alger, Algérie

Abstract

Let d(n) and d^*(n) be, respectively, the number of divisors and the number of unitary divisors of an integer n\geq 1. A divisor d of an integer is to be said unitary if it is prime over \frac{n}{d}. In this paper, we study the mean value of the function D(n)=\frac{d(n)}{d^*(n)} in the arithmetic progressions \left\lbrace l+mk \mid m\in\mathbb{N}^* \text{ and } (l, k)=1 \right\rbrace; this leads back to the study of the real function x\mapsto S(x;k,l)=\underset{n\equiv l[k]}{\sum\limits_{ n \leq x}} D(n). We prove that

    \[S(x;k,l)=A(k)x +\mathcal{O}_{k}\left(x\exp \left( -\frac{\theta}{2}\sqrt{(2\ln x)(\ln\ln x)}\right) \right) \left( 0<\theta<1 \right),\]

where \quad A(k)=\dfrac{c}{k}\prod\limits_{p\mid k}\left(1+\dfrac{1}{2}\sum\limits_{n=2}^{+\infty}\dfrac{1}{p^{n}}\right)^{-1}\left( c=\zeta(2)\prod\limits_{p} \left(1-\dfrac{1}{2p^2}+\dfrac{1}{2p^3} \right) \right).

Keywords

  • Divisors
  • Unitary divisors of integer
  • Riemann zeta function
  • Dirichlet function
  • Dirichlet characters modulo k

2020 Mathematics Subject Classification

  • 11M32
  • 11M06
  • 11M38
  • 11N05
  • 11F68

References

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  2. Balazard, M., & Tenenbaum, G. (1998). Sur la Repartition des Valeurs de la Fonction d’Euler. Composito Mathematica, 110, 239–250.
  3. Derbal, A. (2009). Une Forme Effective D’un Theoreme de Batemane Sur la fonction PHI D’Euler. Integers, 9, 735–744.
  4. Derbal, A. (2012). Ordre maximum d’une fonction liée aux diviseurs d’un nombre entier. Integers, 12, Article A44.
  5. Derbal, A., & Karras, M. (2016). Valeurs moyennes d’une fonction liée aux diviseurs d’un nombre entier. Comptes Rendus Mathematique, Acad. Sci. Paris, 354(6), 555–558.
  6. Ford, K. (2002). Vingradov’s integral and bounds for the Riemann zeta function. Proceedings of the London Mathematical Society, 85, 565–633.
  7. Parent, D. P. (1978). Exercices de Théorie des Nombres. Gauthier-Villars, Paris.

Manuscript history

  • Received: 10 October 2022
  • Revised: 13 March 2023
  • Accepted: 16 June 2023
  • Online First: 3 July 2023

Copyright information

Ⓒ 2023 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

Bouakkaz, O., & Derbal, A. (2023). The mean value of the function \frac{d(n)}{d^*(n)} in arithmetic progressions. Notes on Number Theory and Discrete Mathematics, 29(3), 445-453, DOI: 10.7546/nntdm.2023.29.3.445-453.

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