Counting singular matrices with primitive row vectors (Q1769062)

Let \(M_n(T)\) be the set of all \(n\times n\) singular integral matrices whose row vectors have Euclidean length at most \(T\). \textit{Y. R. Katznelson} [``Singular matrices and a uniform bound for congruence groups of \(\text{SL}_n(\mathbb{Z})\)'', Duke Math. J. 69, 121--136 (1993; Zbl 0785.11050)] gave an asymptotic formula for the number of these matrices as \(T\) tends to infinity. Here the author studies the number of those matrices of \(M_n(T)\) whose rows are primitive, i.e., they are not an integral multiple of an integer vector. For \(T\to\infty\) he proves that this number is asymptotically of order \((n-1)\mu_n/(\zeta(n)\zeta(n-1))T^{n^2-n}\log(T)\), where \(\zeta(n)\) denotes the \(\zeta\)-function and \(\mu_n\) depends, roughly speaking, on the parity of \(n\).