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

Completely additive

An additive function f(n) is said to be completely additive if holds for all positive integers a and b, even when they are not coprime. Totally additive is also used in this sense by analogy with totally multiplicative functions. If f is a completely additive function then f(1) = 0. Every completely additive function is additive, but not vice versa.

Examples

Examples of arithmetic functions which are completely additive are: Examples of arithmetic functions which are additive but not completely additive are:

Multiplicative functions

From any additive function f(n) it is possible to create a related g(n), which is a function with the property that whenever a and b are coprime then: One such example is Likewise if f(n) is completely additive, then is completely multiplicative. More generally, we could consider the function, where c is a nonzero real constant.

Summatory functions

Given an additive function f, let its summatory function be defined by. The average of f is given exactly as The summatory functions over f can be expanded as where The average of the function f^2 is also expressed by these functions as There is always an absolute constant C_f > 0 such that for all natural numbers x \geq 1, Let Suppose that f is an additive function with such that as , Then where G(z) is the Gaussian distribution function Examples of this result related to the prime omega function and the numbers of prime divisors of shifted primes include the following for fixed z \in \R where the relations hold for x \gg 1: