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

We give an asymptotic formula for the number of sublattices $\Lambda \subseteq $\mathbb{Z}$^d$ofindexatmost$X$forwhich$$\mathbb{Z}$^d/\Lambda$hasrankatmost$m$, 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 ${\mathbb{Z}}^d$.