The Pick Theorem and the Proof of the Reciprocity Law for Dedekind Sums

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.AMS Subject Classification: 52C05, 11H06, 57N05, 57N15, 57N35.