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
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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


An arithmetic function 𝑓 is Leibniz-additive if there is a completely multiplicative function β„Žπ‘“ such that 𝑓(π‘šπ‘›) = 𝑓(π‘š)β„Žπ‘“(𝑛) + 𝑓(𝑛)β„Žπ‘“(π‘š) for all positive integers π‘š 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 β„Žπ·(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 β„Žπ‘“ 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.


  • 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.

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