# The arithmetic derivative and Leibniz-additive functions

Pentti Haukkanen, Jorma K. Merikoski and Timo Tossavainen
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310β5132, Online ISSN 2367β8275
Volume 24, 2018, Number 3, Pages 68β76
DOI: 10.7546/nntdm.2018.24.3.68-76

## 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(ππ) = 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 multiplicative function
• Leibniz rule
• Dirichlet convolution

• 11A25
• 11A41

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## Cite this paper

APA

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.

Chicago

Haukkanen, Pentti, Jorma K. Merikoski and Timo Tossavainen. “The Arithmetic Derivative and Leibniz-additive Functions.” Notes on Number Theory and Discrete Mathematics 24, no. 3 (2018): 68-76, doi: 10.7546/nntdm.2018.24.3.68-76.

MLA

Haukkanen, Pentti, Jorma K. Merikoski and Timo Tossavainen. “The Arithmetic Derivative and Leibniz-additive Functions.” Notes on Number Theory and Discrete Mathematics 24.3 (2018): 68-76. Print, doi: 10.7546/nntdm.2018.24.3.68-76.