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http://dx.doi.org/10.7858/eamj.2019.011

EVALUATION OF THE ZETA FUNCTIONS OF TOTALLY REAL NUMBER FIELDS AND ITS APPLICATION  

Lee, Jun Ho (Department of Mathematics Education, Mokpo National University)
Publication Information
East Asian mathematical journal / v.35, no.1, 2019 , pp. 85-90 More about this Journal
Abstract
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
Keywords
totally real number field; zeta function;
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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.   DOI
2 J. Browkin, Tame kernels of cubic cyclic fields, Math. Comp. 74 (2004), 967-999.   DOI
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.   DOI
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.   DOI
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.   DOI
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.   DOI
9 I. Kimura, Divisibility of orders of $K_2$ groups associated to quadratic fields, Demonstratio Math. 39 (2006), 277-284.   DOI
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.   DOI
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.   DOI
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.   DOI
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.   DOI
15 P. Ribenboim, Classical Theory of Algebraic Numbers, Universitext, Springer-Verlag, New York, 2001.
16 E. Tollis, Zeros of Dedekind zeta functions in the critical strip, Math. Comp. 66 (1997),no. 219, 1295-1321.   DOI
17 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.
18 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.
19 C. L. Siegel, Berechnung von Zetafuncktionen an ganzzahligen Stellen, Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl. II (1969), 87-102.
20 A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990) 493-540.   DOI
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.