The arithmetic derivative and Leibniz-additive functions

TL;DR

This paper explores Leibniz-additive functions, generalizing the arithmetic derivative, by analyzing their properties, how they are determined by prime values, and their behavior under various arithmetic operations.

Contribution

It introduces Leibniz-additive functions, characterizes their properties, and shows they are determined by their prime values and associated multiplicative functions.

Findings

Leibniz-additive functions generalize the arithmetic derivative.

Such functions are determined by their values at primes.

They exhibit specific behaviors under product, composition, and Dirichlet convolution.

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.