Rationality of the zeta function of the subgroups of abelian p-groups
Résumé
Given a finite abelian p-group F, we prove an efficient recursive formula for sigma(alpha)(F) = Sigma(H <= F) vertical bar H vertical bar(alpha) where H ranges over the subgroups of F. We infer from this formula that the p-component of the corresponding zeta-function on groups of p-rank bounded by some constant r is rational with a simple denominator. We also provide two explicit examples in rank r = 3 and r = 4, as well as, a closed formula for sigma(alpha)(F).