Rationality of the zeta function of the subgroups of abelian p-groups

Abstract : 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).
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Submitted on : Thursday, June 29, 2017 - 4:47:21 PM
Last modification on : Monday, March 4, 2019 - 2:04:22 PM

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Olivier Ramaré. Rationality of the zeta function of the subgroups of abelian p-groups. Publicationes Mathematicae Debrecen, 2017, 90 (1-2), pp.91 - 105. ⟨10.5486/PMD.2017.7466⟩. ⟨hal-01550584⟩

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