Applying Möbius Inversion to $\Pi(x)$ and $\pi(x)$ I would appreciate help as to how to apply the Möbius inversion theorem to prime counting $\Pi (x)$ and $\pi (x)$, where:
$$\Pi(x) := \sum_{n = 1}^\infty \frac{1}{n} \pi(x^{1/n})$$
and $\pi (x) = \sum_{n = 1}^\infty \frac{\mu (n)}{n} \Pi(x^{1/n})$.
What I am having difficulty with, to begin with, is putting the $\Pi (x)$ relation in the form that makes the Möbius inversion theorem applicable:
$$f(n) = \sum_{d\mid n} g(d)$$
(I hope if I can see that, then I can figure out the $g(n)$ inverse.)
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Edwards, "Riemann's Zeta Function" page 34 has a nice algorithm replacing $f(x)$ with $f(x) - (1/p)f(x^{1/p})$ on each side of the $\Pi (x)$ equation above which I tried. Of course it works. 
I am also wondering if this works for all Möbius inversion applications. And also how this ties in with basic relations of the Möbius inversion theorem as stated above.
EDIT: In that I am asking two questions in one, perhaps any responders might like to give separate answers so that I may fully acknowledge their kind efforts.
Thanks very much.
 A: Let $F(x)$ and $G(x)$ (here) be real-valued functions defined on the positive real axis such that $F(x)$ and $G(x)$ are $0$ for $0< x< 1.$
Let $\alpha(n) = 1/n.$ This function is called completely multiplicative since for all m,n
$$\alpha(m)\cdot\alpha(n) = \alpha(mn),$$ which is true since 
$\frac{1}{n}\cdot\frac{1}{m} = \frac{1}{mn}.$
The Dirichlet inverse $\alpha^{-1}(n) $ of a completely multiplicative function $\alpha(n) $ is
$\alpha^{-1}(n) = \mu(n)\alpha(n),~~~ n\geq 1.$
Given that $\alpha^{-1}(n) $ exists,
$$ G(x) = \sum_{n\leq x}\alpha(n)F(x/n) \leftrightarrow F(x) = \sum_{n\leq x}\alpha^{-1}(n)G(x/n) = \mu(n)\alpha(n)G(x/n)$$
and conversely. This is the generalized Möbius inversion formula. 
The application to Edwards' inversion is straightforward taking $G(x) = \Pi(x), F(x) = \pi(x)$ and $\alpha(n) = 1/n $ as the completely multiplicative function 
required for the application of the general inversion theorem. 
Writing out the terms of Edwards' inversion gives some intuition about how $\mu(n)$ works here. It is surely possible to work backwards and relate the simpler statements of the inversion formula to the general one, though that is a separate question and I have to leave off here.
This material is covered generally in Apostol's Intro. to Analytic Number Theory pp. 30-40.    
