Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



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
Full-text PDF

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.

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

  • [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] H. Cohen, A course in computational algebraic number theory, (Graduate Texts in Maths; Springer Verlag, 1993). MR 94i:11105
  • [DaR] P. Davis and P. Rabinowitz, Methods of numerical integration, (Academic Press, 1975). MR 56:7119
  • [EM] W. J. Ellison and M. Mendès France, Les nombres premiers, (Hermann, 1975). MR 54:5138
  • [F] E. Friedman, Hecke's integral formula, Semin. Theor. Nombres Bordeaux (1987-1988), Exp 5, 23 pp. MR 90i:11136
  • [GR] I. S. Gradshteyn and I. M. Ryzhic, Table of integrals, series and products, (Academic Press, Inc., 1980). MR 81g:33001
  • [L] S. Lang, Algebraic number theory, (Addision Wesley, 1970). MR 44:181
  • [La] E. Landau, Vorlesungen über Zahlentheorie, (Chelsea Publishing Company, 1969). MR 40:4076
  • [O] A. M. Odlyzko, Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: A survey of recent results., Semin. Theor. Nombres Bordeaux 2 (1990), 119-141. MR 91i:11154
  • [Ph] M. Pohst, The minimum discriminant of seventh degree totally real algebraic number fields, Number Theory and Algebra (Academic Press) (1977), 235-240. MR 57:5952
  • [Pt1] G. Poitou, Minorations de discriminants, Séminaire Bourbaki, 28e année, 1975/76, n 479.
  • [Pt2] G. Poitou, Sur les petits discriminants, Séminaire Delange-Pisot-Poitou, 18e année, 1976/77, n 6 = Lecture Notes in Math. 567 (1977), 136-153. MR 55:7995
  • [T] E. Tollis, Zeros of Dedekind zeta functions in the critical strip, Math. Comp. 66 (1997), 1295-1321. MR 98d:11140

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11R42

Retrieve articles in all journals with MSC (2000): 11R42

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

American Mathematical Society