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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
Full-text PDF Free Access
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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
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