Mean value theorems for the S-arithmetic primitive Siegel transforms

We develop the theory and properties of primitive unimodular $S$-arithmetic lattices in $\mathbb{Q}_S^d$ by giving integral formulas in the spirit of Siegel's primitive mean value formula and Rogers' and Schmidt's second moment formulas. When $d=2$, unlike in the real case, functions arising from the $S$-primitive Siegel transform are unbounded, requiring a careful analysis to establish their integrability. We then use mean value and second moment formulas in three applications. First, we obtain quantitative estimates for counting primitive $S$-arithmetic lattice points. We next establish a quantitative Khintchine--Groshev theorem, which, in the real case, involves counting primitive integer points in $\mathbb{Z}^d$ subject to congruence conditions. Finally, we derive an $S$-arithmetic logarithm law for unipotent flows in the spirit of Athreya--Margulis. These applications follow the spirit of the real case, but require new technical aspects of the proofs, particularly when $d=2$.