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.