Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
Published electronically: May 7, 2014
MathSciNet review: 3266965
Full-text PDF

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 $ K$ by a clever use of the splitting of primes $ p<X$, with an error asymptotically bounded by $ 8.33\log \Delta _K/(\sqrt {X}\log X)$, where $ \Delta _K$ is the absolute value of the discriminant of $ K$. 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 $ 2.33$. This results in substantial speeding of one part of Buchmann's class group algorithm.

References [Enhancements On Off] (What's this?)

  • [1] Eric Bach, Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), no. 191, 355-380. MR 1023756 (91m:11096),
  • [2] Eric Bach, Improved approximations for Euler products, Number theory (Halifax, NS, 1994), Amer. Math. Soc., 1995, pp. 13-28. MR 96i:11124
  • [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 (2009c:11179),
  • [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 (92g:11125)
  • [5] Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 1980. Revised by Hugh L. Montgomery. MR 606931 (82m:10001)
  • [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 (11,85d)
  • [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, 1982, pp. 151-193 (French). MR 697261 (84g:12009)
  • [8] PARI/GP, version 2.6.0, Bordeaux, 2012,
  • [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 (81i:12007)
  • [10] R. J. Schoof, Class groups of complex quadratic fields, Math. Comp. 41 (1983), no. 163, 295-302. MR 701640 (84h:12005),
  • [11] H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135-152. MR 0342472 (49 #7218)
  • [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 Supplementaire, 252-265 (French). MR 0053152 (14,727e)

Similar Articles

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

Eduardo Friedman
Affiliation: Departamento de Matemática, Universidad de Chile, Casilla 653, Santiago, Chile

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.

American Mathematical Society