Let $(G, ∗)$ be the group of arithmetic functions $f : N \to C$ that satisfy $f (1)\neq 0$, with group operation given by the Dirichlet product $∗$. The identity function $I$ is the identity element of $G$ defined by $I : N \to Z$ where $I=1$ if $n=1$, $0$ if $n>1$.
An element $f \in G$ has finite order if there exists an $n\in N$ such that $f^n =I$ (here $f^n =f∗f∗\cdots ∗f$ with $n$ factors). Find all number theoretic functions $f : N \to Z$ with $f(1)\neq 0$ that have finite order in the group $G$.
Well now, I thought to start from the fact that since $f(1)\neq 0$, 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 $f*f*f\cdots *f =I$ 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.