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The arithmetic derivative and Leibniz-additive functions
University of Tampere, Finland.
University of Tampere, Finland.
LuleΓ₯ University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.ORCID iD: 0000-0002-7494-4632
2018 (English)In: Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132, Vol. 24, no 3, p. 68-76Article in journal (Refereed) Published
Abstract [en]

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 Leibnizadditive functions are generalizations of the arithmetic derivative 𝐷; namely, 𝐷 is Leibnizadditive 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.

Place, publisher, year, edition, pages
β€œMarin Drinov” Academic Publishing House of the Bulgarian Academy of Sciences , 2018. Vol. 24, no 3, p. 68-76
Keywords [en]
Arithmetic derivative, Arithmetic partial derivative, Arithmetic function, Completely additive function, Completely multiplicative function, Leibniz rule, Dirichlet convolution
National Category
Mathematics Didactics
Research subject
Mathematics Education
Identifiers
URN: urn:nbn:se:ltu:diva-71166DOI: 10.7546/nntdm.2018.24.3.68-76ISI: 000448478500008OAI: oai:DiVA.org:ltu-71166DiVA, id: diva2:1254712
Note

Validerad;2018;NivΓ₯ 2;2018-11-04 (svasva)

Available from: 2018-10-10 Created: 2018-10-10 Last updated: 2019-01-18Bibliographically approved

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Publisher's full texthttp://nntdm.net/papers/nntdm-24/NNTDM-24-3-068-076.pdf

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Tossavainen, Timo

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