The arithmetic derivative and Leibniz-additive functions

Pentti Haukkanen, Jorma K. Merikoski and Timo Tossavainen
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 3, Pages 68–76
DOI: 10.7546/nntdm.2018.24.3.68-76
Full paper (PDF, 162 Kb)

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Authors and affiliations

Pentti Haukkanen
Faculty of Natural Sciences
FI-33014 University of Tampere, Finland

Jorma K. Merikoski
Faculty of Natural Sciences
FI-33014 University of Tampere, Finland

Timo Tossavainen
Department of Arts, Communication and Education
Lulea University of Technology, SE-97187 Lulea, Sweden

Abstract

An arithmetic function 𝑓 is Leibniz-additive if there is a completely multiplicative function ℎ_𝑓 such that 𝑓(𝑚𝑛) = 𝑓(𝑚)ℎ_𝑓(𝑛) + 𝑓(𝑛)ℎ_𝑓(𝑚) for all positive integers 𝑚 and 𝑛. A motivation for the present study is the fact that Leibniz-additive functions are generalizations of the arithmetic derivative 𝐷; namely, 𝐷 is Leibniz-additive with ℎ_𝐷(𝑛) = 𝑛. We study the basic properties of Leibniz-additive functions and, among other things, show that a Leibniz-additive function 𝑓 is totally determined by the values of 𝑓 and ℎ_𝑓 at primes. We also find connections of Leibniz-additive functions to the usual product, composition and Dirichlet convolution of arithmetic functions. The arithmetic partial derivative is also considered.

Keywords

• Arithmetic derivative
• Arithmetic partial derivative
• Arithmetic function
• Completely additive function
• Completely multiplicative function
• Leibniz rule
• Dirichlet convolution

2010 Mathematics Subject Classification

• 11A25
• 11A41

References

1. Apostol, T. M. (1976) Introduction to Analytic Number Theory, Springer-Verlag, New York.
2. Barbeau, E. J. (1961) Remarks on an arithmetic derivative, Canad. Math. Bull. 4(2), 117–122
3. Chawla, L. M. (1973) A note on distributive arithmetical functions, J. Nat. Sci. Math. 13(1), 11–17.
4. Haukkanen, P. (1992) A note on generalized multiplicative and generalized additive arithmetic functions, Math. Stud. 61, 113–116.
5. Haukkanen, P., Merikoski, J. K. & Tossavainen, T. (2016) On arithmetic partial differential equations, J. Integer Seq. 19, Article 16.8.6.
6. Kiuchi, I. & Minamide, M. (2014) On the Dirichlet convolution of completely additive functions. J. Integer Seq. 17, Article 14.8.7.
7. Kovic, J. (2012) The arithmetic derivative and antiderivative, J. Integer Seq. 15, Article 12.3.8.
8. Laohakosol, V. & Tangsupphathawat, P. (2016) Characterizations of additive functions. Lith. Math. J. 56(4), 518–528.
9. McCarthy, P. J. (1986) Introduction to Arithmetical Functions, Springer-Verlag, New York.
10. Sándor, J. & Crstici, B. (2004) Handbook of Number Theory II, Kluwer Academic, Dordrecht.
11. Schwab, E. D. (1995) Dirichlet product and completely additive arithmetical functions, Nieuw Arch. Wisk. 13(2), 187–193.
12. Shapiro, H. N. (1983) Introduction to the Theory of Numbers, Wiley InterScience, New York.
13. Ufnarovski, V. & Åhlander, B. (2003) How to differentiate a number, J. Integer Seq. 6, Article 03.3.4.