Zeros of Dedekind zeta functions in the critical strip

Tollis, Emmanuel

doi:10.1090/S0025-5718-97-00871-5

Zeros of Dedekind zeta functions in the critical strip
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by Emmanuel Tollis PDF

Math. Comp. 66 (1997), 1295-1321 Request permission

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.

References

Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1966. MR 0208797
R. Backlund, Sur les zeros de la fonction $\zeta (s)$ de Riemann, C. R. Acad. Sci. Paris 158 (1914), 1979-1982.
B. L. J. Braaksma, Asymptotic expansions and analytic continuations for a class of Barnes-integrals, Compositio Math. 15 (1964), 239–341 (1964). MR 167651
Richard P. Brent, On the zeros of the Riemann zeta function in the critical strip, Math. Comp. 33 (1979), no. 148, 1361–1372. MR 537983, DOI 10.1090/S0025-5718-1979-0537983-2
J. L. Duras, Etude de la fonction nombre de façons de représenter un entier comme produit de $k$ facteurs, Doctorat de l’Université de Limoges (1993).
H. M. Edwards, Riemann’s zeta function, Pure and Applied Mathematics, Vol. 58, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0466039
Eduardo Friedman, Hecke’s integral formula, Séminaire de Théorie des Nombres, 1987–1988 (Talence, 1987–1988) Univ. Bordeaux I, Talence, [1988?], pp. Exp. No. 5, 23. MR 993106
Eduardo Friedman, Analytic formulas for the regulator of a number field, Invent. Math. 98 (1989), no. 3, 599–622. MR 1022309, DOI 10.1007/BF01393839
E. Hecke, Über die Zetafunktion beliebiger algebraisher Zahlkörper, Gött. Nachr. (1917), 159-171.
J. C. Lagarias and A. M. Odlyzko, On computing Artin $L$-functions in the critical strip, Math. Comp. 33 (1979), no. 147, 1081–1095. MR 528062, DOI 10.1090/S0025-5718-1979-0528062-9
Serge Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0282947
R. Sherman Lehman, On the distribution of zeros of the Riemann zeta-function, Proc. London Math. Soc. (3) 20 (1970), 303–320. MR 258768, DOI 10.1112/plms/s3-20.2.303
R. Sherman Lehman, Separation of zeros of the Riemann zeta-function, Math. Comp. 20 (1966), 523–541. MR 203909, DOI 10.1090/S0025-5718-1966-0203909-5
N. Nielsen, Der Theorie der Gamma-Funktion, Chelsea Publishing Company, 1965.
N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer, Berlin, 1924.
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, DOI 10.5802/jtnb.22
Andrew M. Odlyzko, Analytic computations in number theory, Mathematics of Computation 1943–1993: a half-century of computational mathematics (Vancouver, BC, 1993) Proc. Sympos. Appl. Math., vol. 48, Amer. Math. Soc., Providence, RI, 1994, pp. 451–463. MR 1314883, DOI 10.1090/psapm/048/1314883
Hans Rademacher, On the Phragmén-Lindelöf theorem and some applications, Math. Z 72 (1959/1960), 192–204. MR 0117200, DOI 10.1007/BF01162949
J. B. Rosser, L. Schonfeld and J. M. Yohe, Rigorous computations and the zeros of the Riemann zeta-function, Proceeding of IFIP Congress 68, 13-18.
Robert Rumely, Numerical computations concerning the ERH, Math. Comp. 61 (1993), no. 203, 415–440, S17–S23. MR 1195435, DOI 10.1090/S0025-5718-1993-1195435-0
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14