Applying Möbius Inversion to $\Pi(x)$ and $\pi(x)$

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.