additive function

In number theoryMathworldPlanetmath, an additive functionMathworldPlanetmath is an arithmetic functionMathworldPlanetmath f: with the property that f(1)=0 and, for all a,b 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 functionMathworldPlanetmath is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of monoids and, because of the fundamental theorem of arithmeticMathworldPlanetmath, is completely determined by its restrictionPlanetmathPlanetmath to prime numbersMathworldPlanetmath. 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 ( 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)0. A proof of this equivalence is supplied here (

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

By exponentiating an additive function, a multiplicative function is obtained. For example, the function 2ω(n) is multiplicative. Similarly, by exponentiating a completely additive function, a completely multiplicative function is obtained. For example, the function 2Ω(n) is completely multiplicative.

Title additive function
Canonical name AdditiveFunction1
Date of creation 2013-03-22 16:07:03
Last modified on 2013-03-22 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