additive function (original) (raw)

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)≠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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  Ω⁢(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 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.