Duke Mathematical Journal

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

Paul E. Gunnells and Robert Sczech

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Abstract

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.

Article information

Source
Duke Math. J., Volume 118, Number 2 (2003), 229-260.

Dates
First available in Project Euclid: 23 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1082744647

Digital Object Identifier
doi:10.1215/S0012-7094-03-11822-0

Mathematical Reviews number (MathSciNet)
MR1980994

Zentralblatt MATH identifier
1047.11041

Subjects
Primary: 11F20: Dedekind eta function, Dedekind sums
Secondary: 11F75: Cohomology of arithmetic groups 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]

Citation

Gunnells, Paul E.; Sczech, Robert. Evaluation of Dedekind sums, Eisenstein cocycles, and special values of L -functions. Duke Math. J. 118 (2003), no. 2, 229--260. doi:10.1215/S0012-7094-03-11822-0. https://projecteuclid.org/euclid.dmj/1082744647


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