Equidistribution of primitive lattices in $\mathbb{R}^n$

We count primitive lattices of rank $d$ inside $\mathbb{Z}^{n}$ as their covolume tends to infinity, with respect to certain parameters of such lattices. These parameters include, for example, the subsapce that a lattice spans, namely its projection to the Grassmannian; its homothety class; and its equivalence class modulo rescaling and rotation, often referred to as a shape. We add to a prior work of Schmidt by allowing sets in the spaces of parameters that are general enough to conclude joint equidistribution of these parameters. In addition to the primitive $d$-lattices themselves, we also consider their orthogonal complements in $\mathbb{Z}^{n}$, and show that the equidistribution occurs jointly for primitive lattices and their orthogonal complements. Finally, our asymptotic formulas for the number of primitive lattices include an explicit error term.