Leibniz-Additive Functions on UFD’s

Abstract

Recall that an arithmetic function f is called an L-additive function with respect to a completely multiplicative function h if f(mn) = f(m)h(n) + f(n)h(m) holds for all m and n. We study L-additive functions in the fields of fractions of unique factorization domains (UFD). In particular, we describe all L-additive functions over given UFD such that these functions can be extended to its field of fractions. We find the exact formula for an L-additive function in the terms of prime elements. For a given L-additive function f(x) we study the properties of the sequence (f (k)(x))k≥1 and solutions of the equation f(x) = αx. As corollaries we obtain results about the arithmetic derivative and partial arithmetic derivatives.