On the average order of the gcd-sum function over arbitrary sets of integers

V. Siva Rama Prasad and P. Anantha Reddy
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 3, Pages 16-28
DOI: 10.7546/nntdm.2021.27.3.16-28
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Authors and affiliations

V. Siva Rama Prasad
Professor (Retired), Department of Mathematics, Osmania University
Hyderabad, Telangana-500007, India

P. Anantha Reddy
Government Polytechnic
Kanteshwar, Nizamabad, Telangana-503002, India

Abstract

Let \mathbb{N} denote the set of all positive integers and for j,n\in \mathbb{N}, let (j,n) denote their greatest common divisor. For any S\subseteq \mathbb{N}, we define P_{S}(n) to be the sum of those (j,n)\in S, where j\in \{1,2,3, \ldots, n\}. An asymptotic formula for the summatory function of P_{S}(n) is obtained in this paper which is applicable to a variety of sets S. Also the formula given by Bordellès for the summatory function of P_{\mathbb{N}}(n) can be derived from our result. Further, depending on the structure of S, the asymptotic formulae obtained from our theorem give better error terms than those deducible from a theorem of Bordellès (see Remark 4.4).

Keywords

  • Pillai function
  • gcd-sum function
  • Asymptotic formula
  • Möbius function of S
  • Dirichlet product
  • r-free integer
  • Semi-r-free integer
  • (k, r)-integer
  • Unitary divisor

2020 Mathematics Subject Classification

  • Primary: 11A25
  • Secondary: 11N37

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Cite this paper

Siva Rama Prasad, V., & Anantha Reddy, P. (2021). On the average order of the gcd-sum function over arbitrary sets of integers. Notes on Number Theory and Discrete Mathematics, 27(3), 16-28, DOI: 10.7546/nntdm.2021.27.3.16-28.

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