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The Pick Theorem and the Proof of the Reciprocity Law for Dedekind Sums
Beifang Chen
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Annals of Combinatorics 7 (4) p.425-439 December, 2003
AMS Subject Classification: 52C05, 11H06, 57N05, 57N15, 57N35
This paper is to provide some new generalizations of the Pick Theorem. We first derive a point-set version of the Pick Theorem for an arbitrary bounded lattice polyhedron. Then, we use the idea of a weight function of [2] to obtain a weighted version. Other Pick type theorems known to the author for the integral lattice Z2 are reduced to some special cases of this generalization. Finally, using an idea of Ehrhart [6] and the Pick Theorem, we give a direct proof of the reciprocity law for Dedekind sums. The ideas and methods presented here may be pushed to higher dimensions.
Keywords: Pick theorem, lattice points, winding number, rotation number, Dedekind sums


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