Abstract
The problem of counting the number of lattice points inside a convex lattice polytope in R' (a polytope whose vertices have integer coordinates) has been studied from a variety of perspectives. Here we use the Poisson summation formula and related techniques in Fourier analysis to obtain a formula for the number of lattice points inside all integral dilates of a lattice polytope. We show that an associated generating function has an explicit representation in terms of sums of cotangents, which are higher-dimensional generalizations of Dedekind sums. Let En denote the n-dimensional integer lattice in R', and let P be an n-dimensional lattice polytope in R', which is a compact simplicial complex of pure dimension n whose vertices lie on the lattice. Consider the function of an integer-valued variable t that describes the number of lattice points that lie inside the dilated polytope tP:
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