Zeta functions of integral representations of cyclic \(p\)-groups.
Let \(p\) be a prime number and \(C_{p^k}\) be a cyclic group of order \(p^k\). Let \(L\) be a finitely generated torsion free \(\mathbb{Z}[C_{p^k}]\)-module. \textit{L. Solomon} [Adv. Math. 26, 306-326 (1977; Zbl 0374.20007)] introduced a zeta function attached to \(L\). Calculation of this zeta function can be reduced to calculation of the zeta function of the free \(\mathbb{Z}_p[C_{p^k}]\)-module \(\mathbb{Z}_p[C_{p^k}]^n\). It is developed a recursion formula, which yields some new explicit formulas for the zeta function of \(\mathbb{Z}_p[C_{p^k}]^n\) in the cases \(k=1\), \(2\) and \(n\geq 1\), and \(k=3\), \(n=1\). An important combinatorial tool for these computations is the Möbius function of a partially ordered set.
scientific article; zbMATH DE number 2066834