The X-Dirichlet polynomial of a finite group

For a finite group G and a subset X of G let P(G,X,t) denote the probability that t random elements of G together with X generate G. Then, as discovered by P. Hall, the function P(G,X,t) can be written as a Dirichlet polynomial and in this way can be defined for all complex values of t. As in the case of X=∅, the main result is that the factors of P(G,X,t) obtained in this way are independent of the choice of the chief series. We shall describe the Dirichlet polynomials that arise in this kind of factorization.