The Pick theorem and the proof of the reciprocity law for Dedekind sums (Q1424254)

The paper presents some generalizations of the Pick theorem. One of them says that for a bounded closed lattice polyhedron \(X\) we have \(\; \text{area}(X) = i(X) - \chi(X) - {1\over 2}i(\text{fr\,} X)\) with equality if and only if \(X\) is a manifold with boundary. The symbols \(i(X)\) and \(i(\text{fr}\, X)\) denote the numbers of lattice points of \(X\) and of its frontier \(\text{fr}\, X\), respectively. The symbol \(\chi (X)\) designates the Euler characteristic of \(X\). Continuing his earlier research [Proc. Natl. Acad. Sci. USA 95, No.16, 9093--9098 (1998; Zbl 0902.57026)], the author also presents a weighted version of the Pick theorem. Moreover, the reciprocity law for Dedekind sums is deduced from the Pick theorem.