Theorem

For a multiplicative function $f$, we have $$\sum_{d|n}f(d)=\prod_{p^{a}\|n}\left( \sum_{j=0}^{a} f(p^{a})\right)$$where $$p^{a}\|n\iff p^{a}|n \ \& \ p^{a}+1\not\mid n$$