This paper investigates the local zeta functions counting the \(S\)-congruence subgroups in \(S\)-arithmetic groups; more precisely: Let \(\Gamma\) be an \(S\)-arithmetic subgroup of a \(k\)-linear algebraic group \(G\), \(k\) a number field and \[ c_n (\Gamma):= \text{card} \{H\leq \Gamma \mid [\Gamma :H] =n \text{ and \(H\) is an \(S\)-congruence subgroup}\}. \] Then the local zeta-function in question is the function \(\zeta_{\Gamma, p}^c (s)= \sum^\infty_{n=0} c_{p^n} (\Gamma) p^{-ns}\). The following theorem is proved: Theorem. Let \(\Gamma\) be an \(S\)-arithmetic subgroup of the \(k\)-linear algebraic group \(G\). Suppose that (i) \(G\) is simply connected, (ii) \(R(G)\), the radical of \(G\), is unipotent, and (iii) \(G(k_\nu)\) is not compact for some \(\nu\in S\). If \(p\) is an odd prime, then \(\zeta_{\Gamma, p}^c (s)\) is rational in \(p^{-s}\). The proof of the theorem uses the rationality of \(\zeta_{G, p} (s)\) for compact \(p\)-adic analytic groups \(G\), a classification of Guralnick of subgroups of \(p\)-power index in finite simple groups and some finiteness properties on the solutions of some exponential diophantine equations due to Shorey and Tijdeman. For \(p=2\) there are some interesting links with the yet unsolved problem of whether there are infinitely many Mersenne primes: If there were only finitely many Mersenne primes one can add the case \(p=2\) to the above theorem.