EDITED TO ACCOMMODATE COMMENTS:
I'm trying (self-taught) to understand more about Möbius inversion. Take two arithmetic functions $f$ and $g$ defined by
$$g(n)=\sum_{d|n}f(d)$$
(Presumably $d$ refers to divisors $>0$ and $\le n$). As I understand it, the inverse via Möbius inversion is then
$$f(n)=\sum_{d|n}\mu(d)g\biggl(\frac{n}{d}\biggr)$$
My question is this: is it possible to modify this to invert sums that run across all natural numbers? In other words, can I invert the following function?
$$g(x)=\sum_{n=1}^\infty f\bigl(h(x)\bigr)$$
where $h(x)$ is a third arithmetic function that applies $n$ as a variable. An example might be $h(x)=x^n$ - but please note that this is only an example, I am after a general solution.
Presumably I need to find a bijective relationship between $d$ and $\mathbb{N}$. If so, how? Or is there a better approach?