Computing the residue of the Dedekind zeta function
Authors:
Karim Belabas and Eduardo Friedman
Journal:
Math. Comp. 84 (2015), 357-369
MSC (2010):
Primary 11R42; Secondary 11Y40
DOI:
https://doi.org/10.1090/S0025-5718-2014-02843-3
Published electronically:
May 7, 2014
MathSciNet review:
3266965
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Assuming the Generalized Riemann Hypothesis, Bach has shown that one can calculate the residue of the Dedekind zeta function of a number field by a clever use of the splitting of primes
, with an error asymptotically bounded by
, where
is the absolute value of the discriminant of
. Guided by Weil's explicit formula and still assuming GRH, we make a different use of the splitting of primes and thereby improve Bach's constant to
. This results in substantial speeding of one part of Buchmann's class group algorithm.
- [1] Eric Bach, Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), no. 191, 355–380. MR 1023756, https://doi.org/10.1090/S0025-5718-1990-1023756-8
- [2] Eric Bach, Improved approximations for Euler products, Number theory (Halifax, NS, 1994) CMS Conf. Proc., vol. 15, Amer. Math. Soc., Providence, RI, 1995, pp. 13–28. MR 1353917, https://doi.org/10.1016/0009-2614(95)01161-4
- [3] Karim Belabas, Francisco Diaz y Diaz, and Eduardo Friedman, Small generators of the ideal class group, Math. Comp. 77 (2008), no. 262, 1185–1197. MR 2373197, https://doi.org/10.1090/S0025-5718-07-02003-0
- [4] Johannes Buchmann, A subexponential algorithm for the determination of class groups and regulators of algebraic number fields, Séminaire de Théorie des Nombres, Paris 1988–1989, Progr. Math., vol. 91, Birkhäuser Boston, Boston, MA, 1990, pp. 27–41. MR 1104698
- [5] Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR 606931
- [6] Edmund Landau, Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale, Chelsea Publishing Company, New York, N. Y., 1949 (German). MR 0031002
- [7] Jacques Martinet, Petits discriminants des corps de nombres, Number theory days, 1980 (Exeter, 1980) London Math. Soc. Lecture Note Ser., vol. 56, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 151–193 (French). MR 697261
- [8] PARI/GP, version 2.6.0, Bordeaux, 2012, http://pari.math.u-bordeaux.fr/.
- [9] Georges Poitou, Sur les petits discriminants, Séminaire Delange-Pisot-Poitou, 18e année: (1976/77), Théorie des nombres, Fasc. 1 (French), Secrétariat Math., Paris, 1977, pp. Exp. No. 6, 18 (French). MR 551335
- [10] R. J. Schoof, Class groups of complex quadratic fields, Math. Comp. 41 (1983), no. 163, 295–302. MR 701640, https://doi.org/10.1090/S0025-5718-1983-0701640-0
- [11] H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135–152. MR 342472, https://doi.org/10.1007/BF01405166
- [12] André Weil, Sur les “formules explicites” de la théorie des nombres premiers, Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (1952), no. Tome Supplémentaire, 252–265 (French). MR 53152
Retrieve articles in Mathematics of Computation with MSC (2010): 11R42, 11Y40
Retrieve articles in all journals with MSC (2010): 11R42, 11Y40
Additional Information
Karim Belabas
Affiliation:
Université Bordeaux, IMB, UMR 5251, F-33400 Talence; France; CNRS, IMB, UMR 5251, F-33400 Talence, France; INRIA, F-33400 Talence, France
Email:
Karim.Belabas@math.u-bordeaux1.fr
Eduardo Friedman
Affiliation:
Departamento de Matemática, Universidad de Chile, Casilla 653, Santiago, Chile
Email:
friedman@uchile.cl
DOI:
https://doi.org/10.1090/S0025-5718-2014-02843-3
Keywords:
Dedekind zeta function,
Buchmann's algorithm
Received by editor(s):
June 18, 2012
Received by editor(s) in revised form:
April 30, 2013
Published electronically:
May 7, 2014
Additional Notes:
The first author was supported by the ANR projects ALGOL (07-BLAN-0248) and PEACE (ANR-12-BS01-0010-01).
The second author was partially supported by the Chilean Programa Iniciativa Científica Milenio grant ICM P07-027-F and Fondecyt grant 1110277.
Article copyright:
© Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.