In number theory  , an is an arithmetic function  $f\colon\mathbb{N}\to\mathbb{C}$ with the property that $f(1)=0$ and, for all $a,b\in\mathbb{N}$ with $\gcd(a,b)=1$, $f(ab)=f(a)+f(b)$.

An arithmetic function $f$ is said to be completely additive if $f(1)=0$ and $f(ab)=f(a)+f(b)$ holds for all positive integers $a$ and $b$, 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 additive is usually used for all functions with the property $f(a+b)=f(a)+f(b)$ for all arguments $a$ and $b$. (For instance, see the other entry titled additive function (http://planetmath.org/AdditiveFunction2).) This entry discusses number theoretic additive functions.

Additive functions cannot have convolution inverses since an arithmetic function $f$ has a convolution inverse if and only if $f(1)\neq 0$. A proof of this equivalence is supplied here (http://planetmath.org/ConvolutionInversesForArithmeticFunctions).

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

• $\omega(n)$, the number of distinct prime factors function

• $\Omega(n)$, the number of (nondistinct) prime factors function (http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction)

By exponentiating an additive function, a multiplicative function is obtained. For example, the function $\displaystyle 2^{\omega(n)}$ is multiplicative. Similarly, by exponentiating a completely additive function, a completely multiplicative function is obtained. For example, the function $\displaystyle 2^{\Omega(n)}$ is completely multiplicative.