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

Kim, Hyun; Kim, Jung

doi:10.1090/S0025-5718-02-01395-9

Evaluation of zeta function of the simplest cubic field at negative odd integers
HTML articles powered by AMS MathViewer

by Hyun Kwang Kim and Jung Soo Kim PDF

Math. Comp. 71 (2002), 1243-1262 Request permission

Erratum: Math. Comp. 78 (2009), 617-618.

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.

References

Dongho Byeon, Special values of zeta functions of the simplest cubic fields and their applications, Proc. Japan Acad. Ser. A Math. Sci. 74 (1998), no. 1, 13–15. MR 1617691
U. Halbritter and M. Pohst, On the computation of the values of zeta functions of totally real cubic fields, J. Number Theory 36 (1990), no. 3, 266–288. MR 1077708, DOI 10.1016/0022-314X(90)90090-E
H. K. Kim and H. J. Hwang, Values of zeta functions and class number 1 criterion for the simplest cubic fields, Nagoya Math. J. 160 (2000), 161-180.
J. S. Kim, Determination of class numbers of the simplest cubic fields, to appear in Comm. of the Korean Math. Soc.
A. J. Lazarus, The class number and cyclotomy of simplest quartic fields, PhD thesis, University of California, Berkeley, 1989.
Daniel Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137–1152. MR 352049, DOI 10.1090/S0025-5718-1974-0352049-8
Carl Ludwig Siegel, Berechnung von Zetafunktionen an ganzzahligen Stellen, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1969 (1969), 87–102 (German). MR 252349
Lawrence C. Washington, Class numbers of the simplest cubic fields, Math. Comp. 48 (1987), no. 177, 371–384. MR 866122, DOI 10.1090/S0025-5718-1987-0866122-8
Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2
Don Zagier, On the values at negative integers of the zeta-function of a real quadratic field, Enseign. Math. (2) 22 (1976), no. 1-2, 55–95. MR 406957