Coprime Ehrhart theory and counting free segments

A lattice polytope is "free" (or "empty") if its vertices are the only lattice points it contains. In the context of valuation theory, Klain (1999) proposed to study the functions $\alpha_i(P;n)$ that count the number of free polytopes in $nP$ with $i$ vertices. For $i=1$, this is the famous Ehrhart polynomial. For $i > 3$, the computation is likely impossible and for $i=2,3$ computationally challenging. In this paper, we develop a theory of coprime Ehrhart functions, that count lattice points with relatively prime coordinates, and use it to compute $\alpha_2(P;n)$ for unimodular simplices. We show that the coprime Ehrhart function can be explicitly determined from the Ehrhart polynomial and we give some applications to combinatorial counting.