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 68β€”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

Abstract

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.

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.

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