Temba Shonhiwa

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 13, 2007, Number 3, Page 1—19

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## Details

### Authors and affiliations

Temba Shonhiwa

*School of Mathematics, University of the Witwatersrand
P. Bag 3, Wits 2050, South Africa *

### Abstract

Let *A* denote the set of arithmetic functions and ∗ Dirichlet convolution. The paper presents and alternative approach to the study of arithmetic functions by introducing a homomorphism between the subgroup <*U*, ∗> of the group of units in <*A*, ∗> and the quotient ring induces through an equivalence relation. The same notion is extended to the case of unitary convolution.

### Keywords

- Dirichlet and unitary convolution
- Functional equation
- Multiplicative and completely multiplicative functions
- Equivalence relation
- Homomorphism

### AMS Classification

- 11A25

### References

- M. Apostol, Some Properties of Completely Multiplicative Functions, Amer. Math. Monthly, 78 (1971) 266-271.
- M. Chawla, On a Pair of Arithmetic Functions, J. Natur. Sci. and Math., 8 (1969), pp. 263-269. MR 39, 150. Zbl. 175, 326.
- Carlitz, Completely Multiplicative Function, Amer. Math. Monthly, 78, (1971), 1140.
- Carlitz and M. V. Subbarao, Transformation of arithmetic functions, Duke Math. J. 40, (1973), 949-958.
- D. Cashwell and C. J. Everett, The ring of arithmetic functions, Pacific J. Math., 9 (1959), 975-985.
- Cohen, Arithmetical functions associated with the unitary divisors of an integer , Math. Zeitschr. 74, 66-80 (1960).
- Cohen, Unitary products of arithmetical functions, Acta Arithmetica, vii, 29-38 (1961).
- E. Dickson, History of the Theory of Numbers, vol. I. New York, reprinted 1952.
- G. L. Dirichlet, Vorlesungen uber Zahlentheorie, 4th ed. Brunswick 1894 (edited by R. Dedekind).
- Fekete, Uber die additive Darstellung einiger zahlentheoretischer Funktionen, Math, u. naturwiss. Berichte aus Ungarn 26, 196 – 211 (1913).
- Glockner, L. G. Lucht, S. Porubsky, Solutions to Arithmetic Convolution Equations, Proc. AMS, Volume 135, Number 6, June 2007, pp. 619-1629.
- Haukkanen, A characterization of completely multiplicative arithmetical functions, Nieuw Arch. Wisk., Vierde serie Deel 14 No. 3 November 1996, pp. 325-328.
- O. LeVan, On a generlization of Chawla’s two arithmetic functions, J. Natur. Sci. and Math., Vol. 9 (1969), pp. 57-66. MR 40, 4230. Zbl. 182, 68.
- Shonhiwa, Core function based characterizations of Number Theoretic functions, Quaestiones Mathematicae Journal, 27(2004), 185 -194.
- Sivaramakrishnan, Multiplicative Function and its Dirichlet Inverse, Amer. Math. Monthly 77, 772-773.
- Sivaramakrishnan, Classical Theory of Arithmetic Functions, in Monographs and Textbooks in Pure and Applied Mathematics, Vol. 126, Marcel Dekker, Inc., New York, 1989.
- Vaidyanathaswamy, The theory of multiplicative arithmetical functions. Trans. Amer. Math. Soc 33, 579-662 (1931).

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## Cite this paper

Shonhiwa, T. (2007). Arithmetical function characterizations and identities induced through equivalence relations. Notes on Number Theory and Discrete Mathematics, 13(3), 1-19.