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

## TL;DR

This paper establishes a recursive formula for subgroup sums in finite abelian p-groups, demonstrating the rationality of their zeta functions' p-components and providing explicit examples and formulas.

## Contribution

It introduces an efficient recursive method for computing subgroup sum functions and proves the rationality of the associated zeta functions for bounded p-rank groups.

## Key findings

- Recursive formula for (F)  subgroup sums
- Rationality of p-component of zeta functions for bounded p-rank groups
- Explicit examples and a closed-form expression for (F)

## Abstract

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