The cotype zeta function of [formula omitted

Abstract

We give an asymptotic formula for the number of sublattices Λ⊆Zd of index at most X for which Zd/Λ has rank at most m, answering a question of Nguyen and Shparlinski. We compare this result to 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 Zd, a multivariable generalization of the subgroup growth zeta function of Zd.