Pentti Haukkanen, Jorma K. Merikoski and Timo Tossavainen

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310β5132, Online ISSN 2367β8275

Volume 24, 2018, Number 3, Pages 68β76

DOI: 10.7546/nntdm.2018.24.3.68-76

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## Details

### 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 π is Leibniz-additive if there is a completely multiplicative function β_{π} such that π(ππ) = π(π)β_{π}(π) + π(π)β_{π}(π) for all positive integers π and π. A motivation for the present study is the fact that Leibniz-additive functions are generalizations of the arithmetic derivative π·; namely, π· is Leibniz-additive with β_{π·}(π) = π. We study the basic properties of Leibniz-additive functions and, among other things, show that a Leibniz-additive function π is totally determined by the values of π and β_{π} 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

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## Cite this paper

Haukkanen, P., Merikoski,Β J. K., & Tossavainen, T. (2018). The arithmetic derivative and Leibniz-additive functions. Notes on Number Theory and Discrete Mathematics, 24(3), 68-76, doi: 10.7546/nntdm.2018.24.3.68-76.