I've seen in some places an alternative version of the Mobius Inversion Formula. Instead of the usual: $$ F(x) = \sum_{n=1}^{\infty} G(x/n) \iff G(x) = \sum_{n=1}^{\infty} \mu(n) F(x/n)$$
I've seen: $$F(x)= \sum_{n=1}^{\infty} G(nx) \iff G(x) = \sum_{n=1}^{\infty} \mu(n) F(nx)$$ What I'm wondering is how to prove this second identity and what must these functions $F$ and $G$ fulfill.