  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.
