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Values of zeta functions at negative integers, Dedekind sums and toric geometry

Author(s): Stavros Garoufalidis; James E. Pommersheim
Journal: J. Amer. Math. Soc. 14 (2001), 1-23.
MSC (1991): Primary 11M06; Secondary 14M25, 11F20
Posted: September 18, 2000
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Abstract:

We study relations among special values of zeta functions, invariants of toric varieties, and generalized Dedekind sums. In particular, we use invariants arising in the Todd class of a toric variety to give a new explicit formula for the values of the zeta function of a real quadratic field at nonpositive integers. We also express these invariants in terms of the generalized Dedekind sums studied previously by several authors. The paper includes conceptual proofs of these relations and explicit computations of the various zeta values and Dedekind sums involved.

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Additional Information:

Stavros Garoufalidis
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
Email: stavros@math.gatech.edu

James E. Pommersheim
Affiliation: Department of Mathematics, Pomona College, 610 North College Ave., Claremont, California 91711
Email: jpommersheim@pomona.edu

DOI: 10.1090/S0894-0347-00-00352-0
PII: S 0894-0347(00)00352-0
Keywords: Zeta functions, Dedekind sums, toric varieties
Received by editor(s): June 1, 1999
Received by editor(s) in revised form: May 24, 2000
Posted: September 18, 2000
Additional Notes: The authors were partially supported by NSF grants DMS-95-05105 and DMS-95-08972, respectively.
Dedicated: Dedicated to our teacher, W. Fulton
Copyright of article: Copyright 2000, American Mathematical Society

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