Let $(G,*)$ be the group of number theoretic functions $f$ with $f(1)\not =0$.
1)Show that if $f$ is a multiplicative function and $f$ is not identically zero, then $f\in G$.
2) Show that the Dirichlet product of two multiplicative functions is multiplicative.
3)show that if $f$ is multiplicative and $f$ is not identically $0$, then $f^{-1}$ is also multiplicative.
4)Deduce that the set of non zero multiplicative functions forms a subgroup of $G$.
This is quite a long question, I know it is getting me to do a step by step guided proof to show that the set of non zero multiplicative functions forms a subgroup of G but wanted to write it all down to avoid confusion.
I know for 1) if $f$ is multiplicative then $f(mn)=f(m)f(n)$ but do not really know where to go from here..
A number-theoretic (or arithmetic) function is a function $f:\mathbb{N}\rightarrow \mathbb{C}$. Such functions form a factorial ring $D$ under Dirichlet convolution product and pointwise addition $(f+g)(n)=f(n)+g(n)$. The unit group of $D$ consists of all arithmetic functions with $f(1)\neq 0$. The multiplicative arithmetic functions (which satisfy $f(1)=1$) form a subgroup of the group of units of $D$. All this is proved in Tom Apostol's book on analytic number theory in detail.