The arithmetic derivative and Leibniz-additive functions
Abstract
An arithmetic function f is Leibniz-additive if there is a completely multiplicative function hf, i.e., hf(1)=1 and hf(mn)=hf(m)hf(n) for all positive integers m and n, satisfying f(mn)=f(m)hf(n)+f(n)hf(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 hD(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 hf 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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