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arXiv · 1708.08547

The cotype zeta function of $\mathbb{Z}^d$

Gautam Chinta·Nathan Kaplan·Shaked Koplewitz
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Abstract

We give an asymptotic formula for the number of sublattices $\Lambda \subseteq \mathbb{Z}^d$ of index at most $X$ for which $\mathbb{Z}^d/\Lambda$ has rank at most $m$, answering a question of Nguyen and Shparlinski. We compare this result to recent work of Stanley and Wang on Smith Normal Forms of random integral matrices and discuss connections to the Cohen-Lenstra heuristics. Our arguments are based on Petrogradsky's formulas for the cotype zeta function of $\mathbb{Z}^d$, a multivariable generalization of the subgroup growth zeta function of $\mathbb{Z}^d$.

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Gautam Chinta, Nathan Kaplan, Shaked Koplewitz. 2017-08-28. The cotype zeta function of $\mathbb{Z}^d$. https://arxiv.org/abs/1708.08547

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