Let $f \in \mathbb{A}$, and let $F \in \mathbb{A}$ such that, for all $n \in \mathbb{N}^*, F(n) = \sum_{d \mid n} f(d)$. Show that, for all $n \in \mathbb{N}^*$, $$f(n) = \sum_{d \mid n} \mu(d) F\left(\frac{n}{d}\right)$$
Let $f \in \mathbb{A}$, and let $F \in \mathbb{A}$ such that, for all $n \in \mathbb{N}^*, F(n) = \sum_{d \mid n} f(d)$. Show that, for all $n \in \mathbb{N}^*$,
$$f(n) = \sum_{d \mid n} \mu(d) F\left(\frac{n}{d}\right)$$