Dedekind sums: a combinatorial-geometric viewpoint

TL;DR

This paper explores Dedekind sums through a combinatorial-geometric lens, connecting them to lattice point enumeration in rational polytopes and introducing Fourier-Dedekind sums as fundamental components.

Contribution

It unifies various generalizations of Dedekind sums via lattice point enumeration and introduces Fourier-Dedekind sums, providing new formulas and reciprocity laws.

Findings

Fourier-Dedekind sums form the basis of partition counting.

Derived generalized reciprocity laws for Dedekind sums.

Proved polynomial-time complexity for Zagier's higher-dimensional sums.

Abstract

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.