multiplicative function

A function f(n) defined on the positive integers is multiplicative if f(nm)=f(n)f(m) whenever n and m are relatively prime. Clearly f(1) must be 0 or 1. If f(1)=0, then f(n)=0 for all positive integers n. So some authors require that f(1) be non-zero.

If f(n) is multiplicative and we factor n into distinct primes as n=p₁^a₁. p₂^a₂. ...^.p_k^a_k, then

f(n) = f(p₁^a₁)^. f(p₂^a₂)^. ...^.f(p_k^a_k).

Finally, if f(n) is multiplicative, then so is the function F(n) = sum of f(i) (where the sum is taken over the divisors i of n).

See Also: CompletelyMultiplicative, EulersPhi