In number theory, an arithmetic function (or number-theoretic function) f(n) is a function defined for all positive integers and having values in the complex numbers (f : Z+ ≡ N → C).
Another definition of arithmetic function is a function with exactly two compositional properties:
- Ψ(m+n) = Ψ[Ψ(m)+Ψ(n)] /composition with addition/
- Ψ(m,n) = Ψ[Ψ(m)Ψ(n)] /composition with parameter/
The most important arithmetic functions are the additive and the multiplicative ones.
Note: Arithmetic function should not be confused with a function sometimes called arithmetical function which is in fact integer function (f : N → N).