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

**Full paper (PDF, 244 Kb)**

## Details

### 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 denote the set of all positive integers and for , let denote their greatest common divisor. For any , we define to be the sum of those , where . An asymptotic formula for the summatory function of is obtained in this paper which is applicable to a variety of sets . Also the formula given by Bordellès for the summatory function of can be derived from our result. Further, depending on the structure of , 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.