``A finitely generated group has only a finite number of subgroups of each finite index. How does this number vary with the index?'' Thus the authors introduce this beautiful paper. The question is tackled for finitely generated torsion-free nilpotent groups, referred to as \({\mathcal T}\)-groups, via the zeta function associated to the arithmetical function implicitly defined above. Some interesting and rather deep results are obtained, pointing to a number of intriguing open questions. For a family \({\mathcal X}\) of subgroups of a group \(G\), let \(a_ n({\mathcal X})\) be the number of subgroups in \({\mathcal X}\) of index \(n\) in \(G\), and let \(\zeta_{{\mathcal X}}(s)=\sum^{\infty}_{n=1}a_ n({\mathcal X})n^{- s}=\sum_{H\in {\mathcal X}}| G\) \(:H|^{-s}\). This is a Dirichlet series which, by general principles, will converge on some half plane \(\{\) \(s\in {\mathbb{C}}:\) \(Re(s)>\alpha_{{\mathcal X}}\}\), where the number \(\alpha_{{\mathcal X}}\) is called the abscissa of convergence of the series; it could be infinite in general. This series contains information about the question above. For example, let \(s_ n({\mathcal X})=\sum_{j\leq n}a_ j({\mathcal X})\). Then it is known that \(\alpha_{{\mathcal X}}=\inf \{\alpha \geq 0:\) \(\exists c>0\) with \(s_ n({\mathcal X})\alpha\) \(*_ G\), where \(\alpha_ G*=\alpha_ G\), \(\alpha_ G^{\triangleleft}\) or \(\alpha_ G^{\wedge}\) respectively. This focuses attention on the ``local factors'' \(\zeta\) \(*_{G,p}\). These are now shown to be rational functions of \(p^{-s}\). Theorem 1. Let \(G\) be a \({\mathcal T}\)-group, and \(\zeta\) * stand for one of \(\zeta\), \(\zeta^{\triangleleft}\) or \(\zeta^{\wedge}\). Then for each prime \(p\) there exist polynomials \(\Phi\) \(*_ p({\mathcal X})\) and \(\Psi\) \(*_ p({\mathcal X})\) over \({\mathbb{Z}}\) such that \(\zeta\) \(*_{G,p}(s)=\Phi\) \(*_ p(p^{-s})/\Psi\) \(*_ p(p^{-s}).\) The degrees of \(\Phi\) \(*_ p\) and \(\Psi\) \(*_ p\) are bounded independently of \(p\). This implies that, for example, that if \(a_{p^n}\) is the number of subgroups of index \(p^n\) in the \({\mathcal T}\)-group \(G\), then the sequence \((a_{p^n})_{n\geq 1}\) satisfies a linear recurrence relation. The authors then take up the question of how the rational functions above vary with \(p\). They conjecture, roughly, that these rational functions are given by one of a finite number of rational functions over \({\mathbb{Q}}\) in \(p\) and \({\mathcal X}\). The rest of the paper is taken up with the case of free nilpotent groups. A much more specific conjecture than that mentioned above is made, and this rather technical conjecture is verified in some cases. We refer the reader to the paper for the details, which are quite lengthy and become involved with decomposition of primes in number fields, and so on. The techniques of the paper are quite varied and elaborate. In particular, the proof of the rationality of \(\zeta\) \(*_{G,p}\) as a function of \(p^{-s}\), involves expressing it first as a \(p\)-adic integral and then using results of Denef and Macintyre to deduce the rationality. Many avenues remain open for exploration in this developing field.