Mathematics of Computation

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Localization of the first zero of the Dedekind zeta function

Author: Sami Omar
Journal: Math. Comp. 70 (2001), 1607-1616
MSC (2000): Primary 11R42
Published electronically: March 7, 2001
MathSciNet review: 1836922
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Abstract | References | Similar Articles | Additional Information


Using Weil's explicit formula, we propose a method to compute low zeros of the Dedekind zeta function. As an application of this method, we compute the first zero of the Dedekind zeta function associated to totally complex fields of degree less than or equal to 30 having the smallest known discriminant.

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  • [CDO] H. Cohen, F. Diaz y Diaz and M. Olivier, A table of totally complex number fields of small discriminants, Algorithmic Number Theory, (Lectures Notes in Computer Science; Springer Verlag, 1998), pp. 381-391. CMP 2000:05
  • [Co] Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206
  • [DaR] Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers] New York-London, 1975. Computer Science and Applied Mathematics. MR 0448814
  • [EM] William John Ellison, Les nombres premiers, Hermann, Paris, 1975 (French). En collaboration avec Michel Mendès France; Publications de l’Institut de Mathématique de l’Université de Nancago, No. IX; Actualités Scientifiques et Industrielles, No. 1366. MR 0417077
  • [F] Eduardo Friedman, Hecke’s integral formula, Séminaire de Théorie des Nombres, 1987–1988 (Talence, 1987–1988), Univ. Bordeaux I, Talence, 19??, pp. Exp. No. 5, 23. MR 993106
  • [GR] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. Corrected and enlarged edition edited by Alan Jeffrey; Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tseĭtlin]; Translated from the Russian. MR 582453
  • [L] Serge Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0282947
  • [La] Edmund Landau, Vorlesungen über Zahlentheorie. Erster Band, zweiter Teil; zweiter Band; dritter Band, Chelsea Publishing Co., New York, 1969 (German). MR 0250844
  • [O] 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), no. 1, 119–141 (English, with French summary). MR 1061762
  • [Ph] M. Pohst, The minimum discriminant of seventh degree totally real algebraic number fields, Number theory and algebra, Academic Press, New York, 1977, pp. 235–240. MR 0466069
  • [Pt1] G. Poitou, Minorations de discriminants, Séminaire Bourbaki, 28e année, 1975/76, n 479.
  • [Pt2] Georges Poitou, Minorations de discriminants (d’après A. M. Odlyzko), Séminaire Bourbaki, Vol. 1975/76 28ème année, Exp. No. 479, Springer, Berlin, 1977, pp. 136–153. Lecture Notes in Math., Vol. 567. MR 0435033
  • [T] Emmanuel Tollis, Zeros of Dedekind zeta functions in the critical strip, Math. Comp. 66 (1997), no. 219, 1295–1321. MR 1423079, 10.1090/S0025-5718-97-00871-5

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Additional Information

Sami Omar
Affiliation: Laboratoire d’Algorithmique Arithmétique, Université Bordeaux I, 351 cours de la Libération, F-33405 Talence Cedex France

Keywords: Dedekind zeta function, zeros, discriminants
Received by editor(s): June 4, 1999
Received by editor(s) in revised form: February 1, 2000
Published electronically: March 7, 2001
Article copyright: © Copyright 2001 American Mathematical Society