Lattice Points, Dedekind Sums, and Ehrhart Polynomials of Lattice Polyhedra

Let σ be a simplex of RN with vertices in the integral lattice ZN . The number of lattice points of mσ(={mα : α ∈ σ}) is a polynomial function L(σ,m) of m ≥ 0 . In this paper we present: (i) a formula for the coefficients of the polynomial L(σ,t) in terms of the elementary symmetric functions; (ii) a hyperbolic cotangent expression for the generating functions of the sequence L(σ,m) , m ≥ 0 ; (iii) an explicit formula for the coefficients of the polynomial L(σ,t) in terms of torsion. As an application of (i), the coefficient for the lattice n -simplex of Rn with the vertices (0,. . ., 0, aj, 0,. . . ,0)(1≤ j≤ n) plus the origin is explicitly expressed in terms of Dedekind sums; and when n=2 , it reduces to the reciprocity law about Dedekind sums. The whole exposition is elementary and self-contained.