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arXiv · 1803.06849

The arithmetic derivative and Leibniz-additive functions

Pentti Haukkanen·Jorma K. Merikoski·Timo Tossavainen
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Abstract

An arithmetic function $f$ is Leibniz-additive if there is a completely multiplicative function $h_f$, i.e., $h_f(1)=1$ and $h_f(mn)=h_f(m)h_f(n)$ for all positive integers $m$ and $n$, satisfying $$ f(mn)=f(m)h_f(n)+f(n)h_f(m) $$ for all positive integers $m$ 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 $h_D(n)=n$. In this paper, 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 $h_f$ at primes. We also consider properties of Leibniz-additive functions with respect to the usual product, composition and Dirichlet convolution of arithmetic functions.

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Pentti Haukkanen, Jorma K. Merikoski, Timo Tossavainen. 2018-03-19. The arithmetic derivative and Leibniz-additive functions. https://arxiv.org/abs/1803.06849

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