Arithmetic Subderivatives and Leibniz-Additive Functions

2019/01/08 by Merikoski, Jorma K., Haukkanen, Pentti, Tossavainen, Timo
#11A25 (Primary) 11A05 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1901.02216

Abstract

We first 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. More generally, we then define that an arithmetic function f is Leibniz-additive if there is a nonzero-valued and completely multiplicative function h_f satisfying f(mn)=f(m)h_f(n)+f(n)h_f(m) for all positive integers m and n. We study some basic properties of such functions. For example, we present conditions when an arithmetic function is Leibniz-additive and, generalizing well-known bounds for the arithmetic derivative, establish bounds for a Leibniz-additive function.

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