Computing the residue of the Dedekind zeta function

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Belabas , Karim ; Friedman , Eduardo (2015)
  • Publisher: American Mathematical Society
  • Subject: [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT] | 11R42 (Primary), 11Y40 (Secondary) | Mathematics - Number Theory
    arxiv: Mathematics::Number Theory

16 pages; International audience; 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 D_K/(\sqrt{X}\log X), where D_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 (5)

    1. E. Bach, Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), no. 191, 355-380. MR 91m:11096

    2. , Improved approximations for Euler products, Number theory (Halifax, NS, 1994), Amer. Math. Soc., 1995, pp. 13-28. MR 96i:11124

    3. 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 MR2373197

    4. Johannes Buchmann, A subexponential algorithm for the determination of class groups and regulators of algebraic number fields, S´eminaire de Th´eorie des Nombres, Paris 1988-1989, Progr. Math., vol. 91, Birkh¨auser, 1990, pp. 27-41. MR 92g:11125

    5. Harold Davenport, Multiplicative number theory, second ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 1980, Revised by Hugh L. Montgomery. MR 606931 (82m:10001)

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