I'm teaching myself Möbius inversion.
From Wikipedia it appears if $F$ and $G$ are complex-valued then
$G(x)=\sum\limits_{1 \le n \le x} F(x/n)$
implies
$F(x)=\sum\limits_{1 \le n \le x} \mu(n) G(x/n)$
I was wondering if the implication works the other way around? That is, can we replace the "implies" with "iff"? If not, are there any simple counterexamples?