Computing special values of partial zeta functions

We discuss computation of the special values of partial zeta functions associated to totally real number fields. The main tool is the \emph{Eisenstein cocycle} $\Psi$, a group cocycle for $GL_{n} (\Z )$; the special values are computed as periods of $\Psi$, and are expressed in terms of generalized Dedekind sums. We conclude with some numerical examples for cubic and quartic fields of small discriminant.Comment: 10 p

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Evaluation of Dedekind sums, Eisenstein cocycles, and special values of L-functions

We define higher-dimensional Dedekind sums that generalize the classical Dedekind-Rademacher sums as well as Zagier\u27s 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