completely multiplicative function

A function f(n) defined on the positive integers is completely multiplicative if f(nm)=f(n)f(m) for all pairs n and m (compare this with multiplicative functions). Three simple examples are f(n)=0, f(n)=1, and f(n)=n^c (for a fixed positive value c).

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.