Dedekind sums: a geometric viewpoint

We define a *generalized Dedekind sum* as an expression of the form

Sum_{ra = 1 }r^{t} /( 1 - r^{a1})
... ( 1 - r^{a n})

Here the sum is taken over all *a*-th roots of unity for which the summand
is not singular. Sums of this type have intrigued mathematicians from
various areas such as Number Theory, Topology, Computational Complexity,
and Combinatorial Geometry since their introduction by Dedekind in 1892.
Our definition, which is due to Gessel, includes as special cases the
classical Dedekind sum (essentially the case *n=2*, *t=0*) and its
generalizations due to Rademacher (*n=2*, arbitrary *t*), and Zagier
(*t=0*, arbitrary *n*). Our interest in these sums stems from the
appearance of Dedekind's and Zagier's sums in lattice point count formulas
for polytopes. Through an interplay of complex integration and generating
functions, we show that generalized Dedekind sums are natural ingredients
for such formulas. As corollaries to our formulas, we recover and extend
obtain ``reciprocity theorems'' of Dedekind, Zagier, and Gessel.