Ok so from the Möbius inversion formula, the book just assumes that its:
$$\sum_{d|n} \mu(\frac{n}{d})F(d) = \sum_{d|n} \mu(d)F(\frac{n}{d})$$
- But why is this the case? Also some further confusions regarding Möbius inversion, in the book it says:
$\sum_{d|n} \mu(d)F(\frac{n}{d})=\sum_{d|n} \mu(d)\sum_{k|(n/d)}f(k)=\sum_{dk|n} \mu(d)f(k)$, then we can reverse the roles of $d$ and $k$ to write: $\sum_{d|(n/k)} \mu(d)\sum_{k|n}f(k)=f(n)$
- Why does the sum suddenly become over $dk|n$? Also why is it possible to reverse the roles so that $d|n, k|(n/d)$ becomes $k|n, d|(n/k)$?
Using properties of Dirichlet Convolution: $$f*g=\sum_{nk=m}f(n)g(k)=\sum_{kn=m}f(k)g(n)=\sum_{d|m}f(d)g(\frac{m}{d})=\sum_{d|m}f(\frac{m}{d})g(d)=g*f$$
We see that switching $n$ and $k$ does not matter since multiplication is commutative. So it does not matter which function has which divisor evaluated.