In number theory, an additive function is an arithmetic function with the property that and, for all with , .

An arithmetic function is said to be completely additive if and holds for all positive integers and , even when they are not relatively prime. In this case, the function is a homomorphism of monoids and, because of the fundamental theorem of arithmetic, is completely determined by its restriction to prime numbers. Every completely additive function is additive.

Outside of number theory, the term additive is usually used for all functions with the property for all arguments and . (For instance, see the other entry titled additive function.) This entry discusses number theoretic additive functions.

Additive functions cannot have convolution inverses since an arithmetic function has a convolution inverse if and only if . A proof of this equivalence is supplied here.

The most common type of additive function in all of mathematics is the logarithm. Other additive functions that are useful in number theory are:

• , the number of distinct prime factors function
• , the number of (nondistinct) prime factors function

By exponentiating an additive function, a multiplicative function is obtained. For example, the function is multiplicative. Similarly, by exponentiating a completely additive function, a completely multiplicative function is obtained. For example, the function is completely multiplicative.