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arXiv · 1703.00684

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

Olivier Ramaré
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Abstract

Given a finite abelian $p$-group $F$, we prove an efficient recursive formula for $σ_a(F)=\sum_{\substack{H\leq F}}|H|^a$ 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 $σ_a(F)$.

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Olivier Ramaré. 2017-03-02. Rationality of the zeta function of the subgroups of abelian $p$-groups. https://arxiv.org/abs/1703.00684

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