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Mathematics of Computation

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



Zeros of Dedekind zeta functions
in the critical strip

Author: Emmanuel Tollis
Journal: Math. Comp. 66 (1997), 1295-1321
MSC (1991): Primary 11R42
MathSciNet review: 1423079
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Abstract: In this paper, we describe a computation which established the GRH to height $92$ (resp. $40$) for cubic number fields (resp. quartic number fields) with small discriminant. We use a method due to E. Friedman for computing values of Dedekind zeta functions, we take care of accumulated roundoff error to obtain results which are mathematically rigorous, and we generalize Turing's criterion to prove that there is no zero off the critical line. We finally give results concerning the GRH for cubic and quartic fields, tables of low zeros for number fields of degree $5$ and $6$, and statistics about the smallest zero of a number field.

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Additional Information

Emmanuel Tollis
Affiliation: U.M.R. 9936 du C.N.R.S., U.F.R. de Mathématiques et Informatique, Université Bordeaux 1, 351 Cours de la Libération, 33405 Talence Cedex, France

Keywords: Dedekind zeta function, Generalized Riemann Hypothesis
Received by editor(s): January 20, 1996
Received by editor(s) in revised form: March 10, 1996
Article copyright: © Copyright 1997 American Mathematical Society