Characterizations of additive functions

Abstract

An arithmetic function is said to be additive if f(mn) = f(m) + f(n) for any relatively prime positive integers m, n and is said to be completely additive if the equality holds for any positive integers m, n. We derive various characterizations of additive and completely additive functions based on the concepts of distributivity, discriminative product, and the use of Souriau–Hsu–Mobius function.