Counting Co-Cyclic Lattices

There is a well-known asymptotic formula, due to W. M. Schmidt [Duke Math. J., 35 (1968), pp. 327-339], for the number of full-rank integer lattices of index at most V in Z(n). This set of lattices L can naturally be partitioned with respect to the factor group Z(n)/L. Accordingly, we count the number of full-rank integer lattices L subset of Z(n) such that 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 (zeta(6) Pi(n)(k=4) zeta(k))(-1) approximate to 85%. The problem is motivated by complexity theory, namely worst-case to average-case reductions for lattice problems.