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

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 Petrogradskyu0027s formulas for the cotype zeta function of $mathbb{Z}^d$, a multivariable generalization of the subgroup growth zeta function of $mathbb{Z}^d$.