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Computing the residue of the Dedekind zeta function

Authors: Karim Belabas and Eduardo Friedman
Journal: Math. Comp. 84 (2015), 357-369
MSC (2010): Primary 11R42; Secondary 11Y40
Published electronically: May 7, 2014
MathSciNet review: 3266965
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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

Eduardo Friedman
Affiliation: Departamento de Matemática, Universidad de Chile, Casilla 653, Santiago, Chile
MR Author ID: 69455

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.