Additive Function

In mathematics the term additive function has two different definitions, depending on the specific field of application.

In algebra an additive function (or additive map) is a function that preserves the addition operation:

f(x + y) = f(x) + f(y)

for any two elements x and y in the domain. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation.

In number theory, an additive function is an arithmetic function f(n) of the positive integer n such that whenever a and b are coprime, the function of the product is the sum of the functions:

f(ab) = f(a) + f(b).

The remainder of this article discusses number theoretic additive functions, using the second definition. For a specific case of the first definition see additive polynomial. Note also that any homomorphism f between Abelian groups is "additive" by the first definition.