Specialized Möbius inversion

Let Φ denote the set of arithmetic functions $\text{f}\text{(n)}$ such that $\text{f}\text{(n) = f(e}{}_1\text{, e}{}_2\text{,…, e}{}_{\mathrm{r}}\text{)}$, where n = p₁^e₁p₂^e₂…p_r^e_r and f(e₁, e₂,…,e_r) depends on the e_j but is independent of the primes p_j. The general Möbius inversion theorem then becomes the equivalence of (1.11) and (1.12) below. This in turn leads to relations between exponential generating functions as well as several applications.