Arithmetic subderivatives and Leibniz-additive functions

Document identifier: oai:DiVA.org:ltu-76244
Access full text here:10.33039/ami.2019.03.003
Keyword: Natural Sciences, Mathematics, Naturvetenskap, Matematik, Social Sciences, Educational Sciences, Didactics, Samhällsvetenskap, Utbildningsvetenskap, Didaktik, Arithmetic derivative, Leibniz rule, Additivity, Multiplicativity, Mathematics Education, Matematik och lärande
Publication year: 2019
Abstract:

We introduce the arithmetic subderivative of a positive integer with respect to a non-empty set of primes. This notion generalizes the concepts of the arithmetic derivative and arithmetic partial derivative. In order to generalize these notions a step further, we define that an arithmetic function ? is Leibniz-additive if there is a nonzero-valued and completely multiplicative function ℎ? satisfying ?(??) = ?(?)ℎ? (?) + ?(?)ℎ? (?) for all positive integers ? and ?. We study some basic properties of such functions. For example, we present conditions when an arithmetic function is Leibniz-additive and, generalizing the well-known bounds for the arithmetic derivative, we establish bounds for a Leibniz-additive function.

Authors

Jorma K. Merikoski

University of Tampere, Finland
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Pentti Haukkanen

University of Tampere, Finland
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Timo Tossavainen

Luleå tekniska universitet; Pedagogik, språk och Ämnesdidaktik
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