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 $.