Vichian Laohakosol and Pinthira Tangsupphathawat

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 24, 2018, Number 4, Pages 11—17

DOI: 10.7546/nntdm.2018.24.4.11-17

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## Details

### Authors and affiliations

Vichian Laohakosol

*Department of Mathematics, Faculty of Science
Kasetsart University, Bangkok 10900, Thailand*

Pinthira Tangsupphathawat

*Department of Mathematics, Faculty of Science and Technology
Phranakorn Rajabhat University, Bangkok 10220, Thailand*

### Abstract

An identical equation for arithmetic functions is proved generalizing the 2-variable case due to Venkataraman. It is then applied to characterize multiplicative functions which are variable-separated, and to deduce interesting properties of generalized Ramanujan sums.

### Keywords

- Arithmetic function of several variables
- Identical equation
- Multiplicative functions
- Completely multiplicative functions

### 2010 Mathematics Subject Classification

- 11A25

### References

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- Haukkanen, P. (2018) Derivation of arithmetical functions under the Dirichlet convolution, Int. J. Number Theory, 14(05), 1257–1264.
- Laohakosol, V., Ruengsinsub, P., & Pabhapote, N. (2006) Ramanujan sums via generalized M¨obius functions and applications, Int. J. Math. Math. Sci., Volume 2006, Article ID 60528, 34 pages.
- Sivaramakrishnan, R. (1989) Classical Theory of Arithmetic Functions, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 126, Marcel Dekker, New York.
- Souriau, J.-M. (1944) Généralisation de certaines formules arithmétiques d’inversion. Applications, Revue Scientific (Rev. Rose Illus.), 82, 204–211.
- Tóth, L. (2014) Multiplicative arithmetic functions of several vatiables: a survey. Mathematics Without Boundaries: Surveys in Pure Mathematics, edited by Themistocles M. Rassias, Panos M. Pardalos, pp. 483-514, Springer, New York.
- Venkataraman, C. S. (1946) A new identical equation for multiplicative functions of two arguments and its applications to Ramanujan’s sum
*C*(_{M}*N*), Proc. Ind. Acad. Sci., XXIV(Ser. A), 518–529.

## Related papers

## Cite this paper

APALaohakosol, V., & Tangsupphathawat, P. (2018). An identical equation for arithmetic functions of several variables and applications. *Notes on Number Theory and Discrete Mathematics*, 24(4), 11-17, doi: 10.7546/nntdm.2018.24.4.11-17.

Laohakosol, Vichian and Pinthira Tangsupphathawat. “An Identical Equation for Arithmetic Functions of Several Variables and Applications.” Notes on Number Theory and Discrete Mathematics 24, no. 4 (2018): 11-17, doi: 10.7546/nntdm.2018.24.4.11-17.

MLALaohakosol, Vichian and Pinthira Tangsupphathawat. “An Identical Equation for Arithmetic Functions of Several Variables and Applications.” Notes on Number Theory and Discrete Mathematics 24.4 (2018): 11-17. Print, doi: 10.7546/nntdm.2018.24.4.11-17.