East Asian mathematical journal

  • 제35권1호
  • /
  • Pages.85-90
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  • 2019
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  • 1226-6973(pISSN)
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  • 2287-2833(eISSN)
영남수학회 (The Youngnam Mathematical Society)
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EVALUATION OF THE ZETA FUNCTIONS OF TOTALLY REAL NUMBER FIELDS AND ITS APPLICATION

  • Lee, Jun Ho (Department of Mathematics Education, Mokpo National University)
  • 투고 : 2019.01.06
  • 심사 : 2019.01.20
  • 발행 : 2019.01.31
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초록

In this paper, we are interested in the evaluation of special values of the Dedekind zeta function of a totally real number field. In particular, we revisit Siegel method for values of the zeta function of a totally real number field at negative odd integers and explain how this method is applied to the case of non-normal totally real number field. As one of its applications, we give divisibility property for the values in the special case

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참고문헌

  1. D. Byeon, Special values of zeta functions of the simplest cubic fields and their applications, Proc. Japan Acad., Ser. A 74 (1998), 13-15. https://doi.org/10.3792/pjaa.74.13
  2. J. Browkin, Tame kernels of cubic cyclic fields, Math. Comp. 74 (2004), 967-999. https://doi.org/10.1090/S0025-5718-04-01726-0
  3. H. Cohen, Advanced topics in computational number theory, Grad. Texts in Math. 193, Springer, New York, 2000.
  4. S. J. Cheon, H. K. Kim, J. H. Lee, Evaluation of the Dedekind zeta functions of some non-normal totally real cubic fields at negative odd integers, Manuscripta Math. 124 (2007), 551-560. https://doi.org/10.1007/s00229-007-0135-x
  5. E. M. Friedlander(editor), D. R. Grayson(editor), Handbook of K-theory, Vol. 1, Springer-Verlag, Berlin Heidelberg, 2005, pp. 139-189.
  6. Paul E. Gunnells, R. Sczech, Evaluation of the Dedekind sums, Eisenstein cocycles, and special values of L-functions, Duke Math. 118 (2003), no. 2, 229-260. https://doi.org/10.1215/S0012-7094-03-11822-0
  7. U. Halbritter, M. Pohst, On the computation of the values of zeta functions of totally real cubic fields, J. Number Thoery 36 (1990), 266-288. https://doi.org/10.1016/0022-314X(90)90090-E
  8. H. K. Kim, H. J. Hwang, Values of zeta functions and class number 1 criterion for the simplest cubic fields, Nagoya Math. J. 160 (2000), 161-180. https://doi.org/10.1017/S0027763000007741
  9. I. Kimura, Divisibility of orders of $K_2$ groups associated to quadratic fields, Demonstratio Math. 39 (2006), 277-284. https://doi.org/10.1515/dema-2006-0205
  10. H. K. Kim, J. S. Kim, Evaluation of zeta function of the simplest cubic field at negative odd integers, Math. Comp. 71 (2002), 1243-1262. https://doi.org/10.1090/S0025-5718-02-01395-9
  11. J. H. Lee, Class number one criterion for some non-normal totally real cubic fields, Taiwanese J. Math 17 (2013), no 3, 981-989. https://doi.org/10.11650/tjm.17.2013.2424
  12. J. H. Lee, Evaluation of the Dedekind zeta functions at s = -1 of the simplest quartic fields, J. Number Theory 143 (2014), 24-45. https://doi.org/10.1016/j.jnt.2014.03.011
  13. S. Louboutin, Numerical evaluation at negative integers of the Dedekind zeta functions of totally real cubic number fields, Algorithmic number theory. 318-326, Lecture Notes In Comput. Sci., 3076, Springer, Berlin, 2004).
  14. B. Mazur, A. Wiles, Class fields of abelian extensions of ${\mathbb{Q}}$, Invent. Math. 76 (1984) 179-330. https://doi.org/10.1007/BF01388599
  15. P. Ribenboim, Classical Theory of Algebraic Numbers, Universitext, Springer-Verlag, New York, 2001.
  16. J.-P. Serre, Cohomologie des groupes discretes, Prospects in mathematics(Proc. Sympos., Princeton Univ., Princeton, N.J., 1970), 77-169. Ann. of Math. Studies, No. 70. Princeton Univ. Press, Princeton, N.J.,1971.
  17. T. Shintani, On evaluation of zeta fucntions of totally real algebraic number fields at non-positive integers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23(1976), no. 2, 393-417.
  18. C. L. Siegel, Berechnung von Zetafuncktionen an ganzzahligen Stellen, Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl. II (1969), 87-102.
  19. E. Tollis, Zeros of Dedekind zeta functions in the critical strip, Math. Comp. 66 (1997),no. 219, 1295-1321. https://doi.org/10.1090/S0025-5718-97-00871-5
  20. A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990) 493-540. https://doi.org/10.2307/1971468
  21. D. B. Zagier, On the values at negative integers of the zeta function of a real quadratic field, Enseignement Math. 22 (1976), 55-95.