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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Evaluation of zeta function of the simplest cubic field at negative odd integers


Authors: Hyun Kwang Kim and Jung Soo Kim
Journal: Math. Comp. 71 (2002), 1243-1262
MSC (2000): Primary 11R42; Secondary 11R16
Published electronically: January 11, 2002
Erratum: Math. Comp. 78 (2009), 617-618
MathSciNet review: 1898754
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Abstract: In this paper, we are interested in the evaluation of the zeta function of the simplest cubic field. We first introduce Siegel's formula for values of the zeta function of a totally real number field at negative odd integers. Next, we will develop a method of computing the sum of a divisor function for ideals, and will give a full description for a Siegel lattice of the simplest cubic field. Using these results, we will derive explicit expressions, which involve only rational integers, for values of a zeta function of the simplest cubic field. Finally, as an illustration of our method, we will give a table for zeta values for the first one hundred simplest cubic fields.


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Additional Information

Hyun Kwang Kim
Affiliation: Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, Korea; and School of Mathematics, Korea Institute for Advanced Study, Seoul 130-012, Korea
Email: hkkim@postech.ac.kr

Jung Soo Kim
Affiliation: Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, Korea
Email: integer@euclid.postech.ac.kr

DOI: http://dx.doi.org/10.1090/S0025-5718-02-01395-9
PII: S 0025-5718(02)01395-9
Keywords: The simplest cubic field, zeta function, Siegel lattice
Received by editor(s): July 25, 2000
Received by editor(s) in revised form: September 26, 2000
Published electronically: January 11, 2002
Additional Notes: The present studies were supported by the Korea Research Foundation Grant (KRF-97-001-D00011-D1101) and Com$^{2}$MaC-KOSEF
Article copyright: © Copyright 2002 American Mathematical Society