On the density of prime vectors in lattices (Q1310441)

Let \(\mathbb{Z}^ k A+b\) be a lattice in \(\mathbb{Z}^ n\) with \(1\leq k\leq n\). \(A\) is an integral \(k\times n\) matrix of rank \(k\) and \(b\) is a vector of \(\mathbb{Z}^ n\). It is assumed that the matrix \(A\) has non-zero columns. Let \(A'\) be the matrix \(A\) with the additional row \(b\). Furthermore, it is supposed that each two columns of \(A'\) are linearly independent and the elements of any column of \(A'\) are coprime. With respect to such a lattice the number of points \(x\in\mathbb{Z}^ k\) with \(| x-x_ 0|\leq N/2\), \(xA+b\) are prime vectors (all coordinates are primes), are estimated for large \(N\).