Computing the residue of the Dedekind zeta function

Belabas, Karim; Friedman, Eduardo

doi:10.1090/S0025-5718-2014-02843-3

Computing the residue of the Dedekind zeta function
HTML articles powered by AMS MathViewer

by Karim Belabas and Eduardo Friedman;

Math. Comp. 84 (2015), 357-369

DOI: https://doi.org/10.1090/S0025-5718-2014-02843-3

Published electronically: May 7, 2014

PDF | Request permission

Abstract:

Assuming the Generalized Riemann Hypothesis, Bach has shown that one can calculate the residue of the Dedekind zeta function of a number field $K$ by a clever use of the splitting of primes $p<X$, with an error asymptotically bounded by $8.33\log \Delta _K/(\sqrt {X}\log X)$, where $\Delta _K$ is the absolute value of the discriminant of $K$. Guided by Weil’s explicit formula and still assuming GRH, we make a different use of the splitting of primes and thereby improve Bach’s constant to $2.33$. This results in substantial speeding of one part of Buchmann’s class group algorithm.

References

Eric Bach, Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), no. 191, 355–380. MR 1023756, DOI 10.1090/S0025-5718-1990-1023756-8
Hermann Kober, Transformationen von algebraischem Typ, Ann. of Math. (2) 40 (1939), 549–559 (German). MR 96, DOI 10.2307/1968939
Karim Belabas, Francisco Diaz y Diaz, and Eduardo Friedman, Small generators of the ideal class group, Math. Comp. 77 (2008), no. 262, 1185–1197. MR 2373197, DOI 10.1090/S0025-5718-07-02003-0
Johannes Buchmann, A subexponential algorithm for the determination of class groups and regulators of algebraic number fields, Séminaire de Théorie des Nombres, Paris 1988–1989, Progr. Math., vol. 91, Birkhäuser Boston, Boston, MA, 1990, pp. 27–41. MR 1104698
Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR 606931, DOI 10.1007/978-1-4757-5927-3
Edmund Landau, Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale, Chelsea Publishing Co., New York, 1949 (German). MR 31002
Jacques Martinet, Petits discriminants des corps de nombres, Number theory days, 1980 (Exeter, 1980) NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., vol. 91, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 151–193 (French). MR 697261
PARI/GP, version 2.6.0, Bordeaux, 2012, http://pari.math.u-bordeaux.fr/.
Georges Poitou, Sur les petits discriminants, Séminaire Delange-Pisot-Poitou, 18e année: 1976/77, Théorie des nombres, Fasc. 1, Secrétariat Math., Paris, 1977, pp. Exp. No. 6, 18 (French). MR 551335
R. J. Schoof, Class groups of complex quadratic fields, Math. Comp. 41 (1983), no. 163, 295–302. MR 701640, DOI 10.1090/S0025-5718-1983-0701640-0
H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135–152. MR 342472, DOI 10.1007/BF01405166
André Weil, Sur les “formules explicites” de la théorie des nombres premiers, Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (1952), no. Tome Supplémentaire, 252–265 (French). MR 53152