Mean value theorems for the S-arithmetic primitive Siegel transforms

We develop the theory and properties of primitive unimodular Sarithmetic lattices in Q^d_S 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 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.