additive function
In number theory^{}, an additive function^{} is an arithmetic function^{} $f:\mathbb{N}\to \u2102$ with the property that $f(1)=0$ and, for all $a,b\in \mathbb{N}$ with $\mathrm{gcd}(a,b)=1$, $f(ab)=f(a)+f(b)$.
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 $f(1)\ne 0$. 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:

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

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

Canonical name  AdditiveFunction1 
Date of creation  20130322 16:07:03 
Last modified on  20130322 16:07:03 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  12 
Author  Wkbj79 (1863) 
Entry type  Definition 
Classification  msc 11A25 
Related topic  MultiplicativeFunction 
Defines  additive 
Defines  completely additive 
Defines  completely additive function 