A generalization of a series for the density of abundant numbers (Q2800151)

It is known due to the author of the present paper that the natural density of the set of abundant numbers can be expressed as a series of the form NEWLINE\[NEWLINE \sum_{a\in\mathcal{P}}\frac{1}{a}\prod_{p|c(a)}\left(1-\frac{1}{p}\right), NEWLINE\]NEWLINE where \(\mathcal{P}\) is the set of primitive nondeficient numbers and \(c(a)\) is a number whose prime factorization can be found easily, given by prime factorization of \(a\). Generalizing the above result, in this paper the author presents a class of multiplicative functions for which the similar series result holds. As an application, the author considers computations for the function \(n/\varphi(n)\).