We define higher-dimensional Dedekind sums that generalize the
classical Dedekind-Rademacher sums as well as Zagier's sums, and we
show how to compute them effectively using a generalization of the
continued-fraction algorithm.
¶ We present two applications. First, we show how to express special
values of partial zeta functions associated to totally real number
fields in terms of these sums via the Eisenstein cocycle introduced by
R. Sczech. Hence we obtain a polynomial time algorithm for computing
these special values. Second, we show how to use our techniques to
compute certain special values of the Witten zeta function, and we
compute some explicit examples.