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Zeros of Dedekind zeta functions in the critical strip. (English) Zbl 0877.11061

The author reports on his computation concerning the generalized Riemann hypothesis for the Dedekind zeta function \(\zeta_K(s)\) of a number field \(K\). He chose 50 cubic fields and 30 quartic fields with varying signature and small discriminant and verified that the zeros \(s=\sigma +it\) of \(\zeta_K(s)\) have \(\sigma= \frac 12\) up to a certain bound of \(|t|\). This bound was about 92 in the cubic case and about 40 in the quartic case. The results, given in tables, also contain the lowest zero and the least and greatest gap between two successive zeros as well as some related statistics. Moreover, low zeros of \(\zeta_K(s)\) were computed for a number of fields \(K\) of degree 5 and 6. The author’s method for computing values of \(\zeta_K(s)\) is due to E. Friedman [Sémin. Théor. Nombres, Univ. Bordeaux I 1987/1988, Exp. No. 5 (1988; Zbl 0697.12010)]. The proof that there is no zero off the critical line is based on a generalization of Turing’s criterion; here the author largely follows a work by R. S. Lehman [Proc. Lond. Math. Soc., III. Ser. 20, 303-320 (1970; Zbl 0203.35502)]. Special care is taken to control the round-off errors.

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
11Y35 Analytic computations
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References:

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