The Multiplicative Arithmetic Functions Connected with a Finite Abelian Group

An arithmetic function f(N) is multiplicative if

f(MN)=f(M)f(N)

whenever the integers M, N are relatively prime. The function F(N) defined by the equation

F(N)=Σf₁(δ)f₂(N/δ)

summed for all divisors δ of N is called the composite of the two arithmetic functions f₁, f₂. If f₁ and f₂ are multiplicative, it is easy to see that their composite F (represented by f₁, f₂) is also multiplicative.