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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Computing the residue of the Dedekind zeta function


Authors: Karim Belabas and Eduardo Friedman
Journal: Math. Comp.
MSC (2010): Primary 11R42; Secondary 11Y40
Published electronically: May 7, 2014
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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.


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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: Karim.Belabas@math.u-bordeaux1.fr

Eduardo Friedman
Affiliation: Departamento de Matemática, Universidad de Chile, Casilla 653, Santiago, Chile
Email: friedman@uchile.cl

DOI: http://dx.doi.org/10.1090/S0025-5718-2014-02843-3
PII: S 0025-5718(2014)02843-3
Keywords: Dedekind zeta function, Buchmann's algorithm
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.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.