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

Paul E. Gunnells, Robert Sczech

Research output: Contribution to journalArticle

21 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)229-260
Number of pages32
JournalDuke Mathematical Journal
Volume118
Issue number2
DOIs
StatePublished - Jun 1 2003

Fingerprint

Dedekind Sums
Cocycle
L-function
Riemann zeta function
Evaluation
Continued fraction
Number field
Polynomial-time Algorithm
High-dimensional
Express
Partial
Generalise
Computing

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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Evaluation of Dedekind sums, Eisenstein cocycles, and special values of L-functions. / Gunnells, Paul E.; Sczech, Robert.

In: Duke Mathematical Journal, Vol. 118, No. 2, 01.06.2003, p. 229-260.

Research output: Contribution to journalArticle

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