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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Computing the residue of the Dedekind zeta function
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by Karim Belabas and Eduardo Friedman PDF
Math. Comp. 84 (2015), 357-369 Request permission


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.
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Additional Information
  • Karim Belabas
  • Affiliation: Université Bordeaux, IMB, UMR 5251, F-33400 Talence; France; CNRS, IMB, UMR 5251, F-33400 Talence, France; INRIA, F-33400 Talence, France
  • Email:
  • Eduardo Friedman
  • Affiliation: Departamento de Matemática, Universidad de Chile, Casilla 653, Santiago, Chile
  • MR Author ID: 69455
  • Email:
  • Received by editor(s): June 18, 2012
  • Received by editor(s) in revised form: April 30, 2013
  • Published electronically: May 7, 2014
  • Additional Notes: The first author was supported by the ANR projects ALGOL (07-BLAN-0248) and PEACE (ANR-12-BS01-0010-01).
    The second author was partially supported by the Chilean Programa Iniciativa Científica Milenio grant ICM P07-027-F and Fondecyt grant 1110277.
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 84 (2015), 357-369
  • MSC (2010): Primary 11R42; Secondary 11Y40
  • DOI:
  • MathSciNet review: 3266965