Zeros of Dedekind zeta functions in the critical strip

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. 0. Introduction and notations The Riemann zeta function and its generalization to number fields, the Dedekind zeta function, have been for well over a hundred years one of the central tools in number theory. It is recognized that the deepest single open problem in mathematics is the settling of the Riemann Hypothesis, and number theorists know that its generalization to number fields and algebraic varieties is almost equally important. Much energy has been devoted to the numerical investigation of the zeta function (see [4] for example). There has been some investigations of its closest cousins, the Dirichlet L-functions (see [20]). However, the case of a general number field has remained totally unexplored territory. We give here the first numerical evidence in favour of the Generalized Riemann Hypothesis for a number field where calculations cannot be reduced to the classical L-functions. We explain how we have transformed a formula proved by Eduardo Friedman (in 1987) into an efficient algorithm for computing values of Dedekind zeta functions. The program (now included in the package Pari/GP) needs a few seconds to compute a single value of ζK for a number field of small degree. We also generalize Turing’s criterion for Dedekind zeta functions to check the GRH. In all the investigation, great care is taken to obtain results which are mathematically rigorous (estimate of error terms, of roundoff error...). We finally give numerical results concerning the GRH. We have verified this hypothesis for 50 cubic number fields up to height 92 and for 30 quartic number fields up to height 40. We give statistics about the gaps between zeros and about the height of the first zero of a number field. Received by the editor January 20, 1996 and, in revised form, March 10, 1996. 1991 Mathematics Subject Classification. Primary 11R42.