Answer to Question #21294 in Combinatorics | Number Theory for Lso

Question

Let (G, ∗) be the group of all number theoretic functions f : N → C satisfying f (1) ̸= 0, with group operation given by the Dirichlet product ∗. The identity function I is the identity element of G defined by I : N → Z where I={1 if n=1, 0 if n>1}

An element f ∈ G is said to have finite order if there exists an n∈N such that f^n =I (here f^n =f∗f∗···∗f with n factors). Find all number theoretic functions f : N → Z with f(1) ≠ 0 that have finite order in the group G.

Well now, I thought to start from the fact that since f(1)=/0, then f has an inverse called g. I know that f*g=I under Dirichlet product so if I need to find f such that f*f*f...*f =I means that f^(n-1) is my inverse function but that's just what I can came up with after hours , with the info I have from my lecture notes . If anyone can hep would be much appreciated.