  We consider generalized Dedekind sums in dimension $n$, for fixed $n$-tuple
of natural numbers, defined as sum of products of values of periodic Bernoulli
functions. This includes the higher dimensional Dedekind sums of Zagier and
Apostol-Carlitz' generalized Dedekind sums as well as the original Dedekind
sums. These are realized as coefficients of Todd series of lattice cones and
satisfy reciprocity law from the cocycle property of Todd series. Using
iterated residue formula, we compute the coefficient of the decomposition of of
the Todd series corresponding to a nonsingular decomposition of the lattice
cone defining the Dedekind sums. We associate a Laurent polynomial which is
added to generalized Dedekind sums of fixed index to make their denominators
bounded. We give explicitly the denominator in terms of Bernoulli numbers. This
generalizes the role played by the rational function given by the difference of
the Rademacher function and the classical Dedekind sums. We associate an
exponential sum to the generalized Dedekind sums using the integrality of the
generalized Rademacher function. We show that this exponential sum has a
nontrivial bound that is sufficient to fulfill Weyl's equidistribution
criterion and thus the fractional part of the generalized Dedekind sums are
equidistributed. As an example, for a 3 dimensional case and Zagier's higher
dimensional generalization of Dedekind sums, we compute the Laurent polynomials
associated.
