Intuitive basis of Mobius inversion? If we're given $f(n)= \sum_{d|n}g\left(\frac{n}{d}\right),n \in \mathbb{N},$ then Mobius inversion gives
$$g(n)=\sum_{d|n}\mu \left( d\right) f \left( \frac{n}{d}\right).$$
Also, the generalised Mobius inversion formula states that the above is correct for $n \in \mathbb{R}$ if we change $d|n \to1\le d\le n$.
I'm familiar with the standard proof of the vanilla and generalised inversion formulae, but it relies on blind algebraic manipulation of Dirichlet convolution (and a generalised convolution between arithmetic and non-arithmetic functions), which I find unsatisfactory. 

Is there a more intuitive way of looking at the (generalised/ungeneralised) Mobius inversion formula, perhaps with knowledge garnered from more advanced theory?


Best approach would be to compute the cyclotomic polynomial for $\Phi_n(X)$ by  attempting to factorize $X^n-1$, using inclusion-exclusion. You want to get hold of a polynomial with integer coefficients satisfied by, say, 30th roots of unity, which are not roots of unity of lower order (i.e., they are primitive). Then it has to be a factor of $X^{30}-1$. We have to remove 15th roots of unity: So divide out by $X^{15}-1$. But $-1$ is still an undesirable root. So divide out by $X+1$ and continue. You will rediscover this principle.
