Arithmetic Subderivatives and Leibniz-Additive Functions

Jorma K. Merikoski; Pentti Haukkanen; Timo Tossavainen

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arXiv · 1901.02216

Arithmetic Subderivatives and Leibniz-Additive Functions

Jorma K. Merikoski·Pentti Haukkanen·Timo Tossavainen

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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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BibTeX RIS

Jorma K. Merikoski, Pentti Haukkanen, Timo Tossavainen. 2019-01-08. Arithmetic Subderivatives and Leibniz-Additive Functions. https://arxiv.org/abs/1901.02216

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