Zeta functions of integral representations of cyclic p-groups

For a prime number p and Cpk, the cyclic group of order pk, we consider the group ring over the p-adic integers. Following L. Solomon, one can define the zeta function of the free -module , which counts submodules of finite index in . In this article we develop a recursion formula (relating submodules of to certain submodules of ), which yields some new explicit formulas for the zeta function of in the cases k=1,2 and n 1, and k=3, n=1. An important combinatorial tool for these computations is the Möbius function of a partially ordered set.