Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Upper bounds for residues of Dedekind zeta functions and class numbers of cubic and quartic number fields


Author: Stéphane R. Louboutin
Journal: Math. Comp. 80 (2011), 1813-1822
MSC (2010): Primary 11R42; Secondary 11R16, 11R29
DOI: https://doi.org/10.1090/S0025-5718-2011-02457-9
Published electronically: January 25, 2011
MathSciNet review: 2785481
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ K$ be an algebraic number field. Assume that $ \zeta_K(s)/\zeta (s)$ is entire. We give an explicit upper bound for the residue at $ s=1$ of the Dedekind zeta function $ \zeta_K(s)$ of $ K$. We deduce explicit upper bounds on class numbers of cubic and quartic number fields.


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

  • [Cus] T. W. Cusick.
    Lower bounds for regulators.
    Lecture Notes in Math., 1068, Springer, Berlin (1984), 63-73. MR 756083 (85k:11052)
  • [Dai] R. Daileda.
    Non-abelian number fields with very large class numbers.
    Acta Arith. 125 (2006), 215-255. MR 2276192 (2007k:11186)
  • [Lan] S. Lang.
    Algebraic number theory (Second Edition).
    Graduate Texts in Mathematics 110, Springer-Verlag, New York, 1994. MR 1282723 (95f:11085)
  • [Le] M. Le.
    Upper bounds for class numbers of real quadratic fields.
    Acta Arith. 68 (1994), 141-144. MR 1305196 (95j:11101)
  • [Lou93] S. Louboutin.
    Majorations explicites de $ \vert L(1,\chi )\vert$.
    C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), 11-14. MR 1198740 (93m:11084)
  • [Lou96] S. Louboutin.
    Majorations explicites de $ \vert L(1,\chi )\vert$. (Suite).
    C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), 443-446. MR 1408973 (97k:11123)
  • [Lou00] S. Louboutin.
    Explicit bounds for residues of Dedekind zeta functions, values of $ L$-functions at $ s=1$, and relative class numbers.
    J. Number Theory 85 (2000), 263-282. MR 1802716 (2002i:11111)
  • [Lou01] S. Louboutin.
    Explicit upper bounds for residues of Dedekind zeta functions and values of $ L$-functions at $ s=1$, and explicit lower bounds for relative class numbers of CM-fields.
    Canad. J. Math. 53 (2001), 1194-1222. MR 1863848 (2003d:11167)
  • [Lou05] S. Louboutin.
    Explicit upper bounds for the residues at $ s=1$ of the Dedekind zeta functions of some totally real number fields.
    Séminaires & Congrès 11 (2005), 171-178; (AGCT 2003) SMF.
  • [MM] R. M. Murty and V. K. Murty.
    Non-vanishing of $ L$-functions and applications.
    Progress in Mathematics 157. Birkhäuser-Verlag, Basel, 1997. MR 1482805 (98h:11106)
  • [MP] C. Moser and J. J. Payan.
    Majoration du nombre de classes d'un corps cubique cyclique de conducteur premier.
    J. Math. Soc. Japan 33 (1981), 701-706. MR 630633 (83a:12006)
  • [Nak] K. Nakamula.
    Certain quartic fields with small regulators.
    J. Number Theory 57 (1996), 1-21. MR 1378570 (97h:11128)
  • [Odl] A. M. Odlyzko.
    Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results.
    Sém Théor. Nombres Bordeaux (2) 2 (1990), 119-141. MR 1061762 (91i:11154)
  • [Ram] O. Ramaré.
    Approximate formulae for $ L(1,\chi )$.
    Acta Arith. 100 (2001), 245-266. MR 1865385 (2002k:11144)
  • [Sil] J. H. Silverman.
    An inequality relating the regulator and discriminant of a number field.
    J. Number Theory 19 (1984), 437-442. MR 769793 (86c:11094)
  • [Uch] K. Uchida.
    On Artin $ L$-functions.
    Tohoku Math. J. 27 (1975), 75-81. MR 0369323 (51:5558)
  • [vdW] R. W. van der Waall.
    On a conjecture of Dedekind on zeta functions.
    Indag. Math. 37 (1975), 83-86. MR 0379439 (52:344)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11R42, 11R16, 11R29

Retrieve articles in all journals with MSC (2010): 11R42, 11R16, 11R29


Additional Information

Stéphane R. Louboutin
Affiliation: Institut de Mathématiques de Luminy, UMR 6206, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France
Email: loubouti@iml.univ-mrs.fr

DOI: https://doi.org/10.1090/S0025-5718-2011-02457-9
Keywords: Dedekind zeta function, number field, class number
Received by editor(s): November 25, 2009
Received by editor(s) in revised form: June 15, 2010
Published electronically: January 25, 2011
Article copyright: © Copyright 2011 American Mathematical Society

American Mathematical Society