additive function

In number theory , an additive function is an arithmetic function f : ℕ → ℂ with the property that f ⁢ ( 1 ) = 0 and, for all a , b ∈ ℕ with gcd ⁡ ( a , b ) = 1 , f ⁢ ( a ⁢ b ) = f ⁢ ( a ) + f ⁢ ( b ) .

An arithmetic function f is said to be completely additive if f ⁢ ( 1 ) = 0 and f ⁢ ( a ⁢ b ) = 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 ) ≠ 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:

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