For a finite group G and a non-negative integer s, let P(G,s) be the probability that a randomly chosen s-tuple generates G. Philip Hall gave an explicit formula for P(G,s), exhibiting the latter as a finite Dirichlet series. One can therefore speak of P(G,s) for an arbitrary complex number s. The reciprocal of this function of s is called the zeta function of G. The present work arose from an ...