Dedekind sums: a combinatorial-geometric viewpoint

Details Dedekind sums: a combinatorial-geometric viewpoint Dedekind sums: a combinatorial-geometric viewpoint

2001-12-07

arxiv.org/abs/math/0112076v2

M. B. Nathanson (ed.), Unusual Applications of Number Theory, DIMACS: Series in Discrete Mathematics and Theoretical Computer

Mathematics

Number Theory

11 pages

Scientific paper

The literature on Dedekind sums is vast. In this expository paper we show that there is a common thread to many generalizations of Dedekind sums, namely through the study of lattice point enumeration of rational polytopes. In particular, there are some natural finite Fourier series which we call Fourier-Dedekind sums, and which form the building blocks of the number of partitions of an integer from a finite set of positive integers. This problem also goes by the name of the `coin exchange problem'. Dedekind sums have enjoyed a resurgence of interest recently, from such diverse fields as topology, number theory, and combinatorial geometry. The Fourier-Dedekind sums we study here include as special cases generalized Dedekind sums studied by Berndt, Carlitz, Grosswald, Knuth, Rademacher, and Zagier. Our interest in these sums stems from the appearance of Dedekind's and Zagier's sums in lattice point count formulas for polytopes. Using some simple generating functions, we show that generalized Dedekind sums are natural ingredients for such formulas. As immediate `geometric' corollaries to our formulas, we obtain and generalize reciprocity laws of Dedekind, Zagier, and Gessel. Finally, we prove a polynomial-time complexity result for Zagier's higher-dimensional Dedekind sums.

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