  There is a well-known asymptotic formula, due to W. M. Schmidt (1968) for the
number of full-rank integer lattices of index at most $V$ in $\mathbb{Z}^n$.
This set of lattices $L$ can naturally be partitioned with respect to the
factor group $\mathbb{Z}^n/L$. Accordingly, we count the number of full-rank
integer lattices $L \subseteq \mathbb{Z}^n$ such that $\mathbb{Z}^n/L$ is
cyclic and of order at most $V$, and deduce that these co-cyclic lattices are
dominant among all integer lattices: their natural density is $\left(\zeta(6)
\prod_{k=4}^n \zeta(k)\right)^{-1} \approx 85\%$. The problem is motivated by
complexity theory, namely worst-case to average-case reductions for lattice
problems.
