E. Detomi and the author had studied [in J. Algebra 265, No. 2, 651-668 (2003; Zbl 1072.20031)] the factorization in the ring of Dirichlet polynomials with integer coefficients of \(P_G(X,s)\), when \(G\) is a finite group, and \(X\) is the empty subset of \(G\). The factorization depends on the knowledge of the chief factors of \(G\). The goal of the paper under review is to prove similar results when \(X\) is an arbitrary subset of \(G\). This requires in particular a refinement of an equivalence relation on the set of irreducible \(G\)-groups, which was introduced in the previous paper.
Recall that \(P_G(X,s)=\sum_{n\in\mathbb{N}}a_n/n^s\), where \(a_n=\sum\mu(H,G)\). Here \(\mu\) is the Möbius function associated with the subgroup lattice of \(G\), and the summation ranges over all subgroups \(H\), of index \(n\) in \(G\), which contain \(X\).