Search

Your search keyword '"Dedekind zeta function"' showing total 7 results
Start Over Descriptor "Dedekind zeta function" Publisher hal ccsd
7 results on '"Dedekind zeta function"'

1. Upper bounds on L (1, χ ) taking into account ramified prime ideals

Author

Stéphane Louboutin, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU)

Subjects
Discrete mathematics, Dedekind zeta function, Algebra and Number Theory, L-function, 11R42 (11M20), 010102 general mathematics, Residues, Algebraic number field, 01 natural sciences, Prime (order theory), [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Conductor, Character (mathematics), Quadratic equation, 0103 physical sciences, Ray class group, 010307 mathematical physics, 0101 mathematics, Abelian group, Number field, Mathematics
Abstract

Let χ range over the non-trivial primitive characters associated with the abelian extensions L / K of a given number field K , i.e. over the non-trivial primitive characters on ray class groups of K . Let f χ be the norm of the finite part of the conductor of such a character. It is known that | L ( 1 , χ ) | ≤ 1 2 Res s = 1 ( ζ K ( s ) ) log ⁡ f χ + O ( 1 ) , where the implied constants in this O ( 1 ) are effective and depend on K only. The proof of this result suggests that one can expect better upper bounds by taking into account prime ideals of K dividing the conductor of χ, i.e. ramified prime ideals. This has already been done only in the case that K = Q . This paper is devoted to giving for the first time such improvements for any K . As a non-trivial example, we give fully explicit bounds when K is an imaginary quadratic number field.

Published
2017

2. Real zeros of Dedekind zeta functions

Author

Stéphane Louboutin, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU)

Subjects
Discrete mathematics, Dedekind zeta function, Algebra and Number Theory, Dedekind sum, real zero, Algebraic number field, Siegel zero, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], symbols.namesake, Arithmetic zeta function, symbols, Dedekind eta function, 11R42, 11R29, Dedekind cut, Complex plane, Mathematics
Abstract

International audience; Building on Stechkin and Kadiri's ideas we derive an explicit zero-free region of the real axis for Dedekind zeta functions of number fields. We then explain how this new region enables us to improve upon the previously known explicit lower bounds for class numbers of number fields and relative class numbers of CM-fields.

Published
2015

3. Computing the residue of the Dedekind zeta function

Author

Karim Belabas, Eduardo Friedman, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Lithe and fast algorithmic number theory (LFANT), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Universidad de Santiago de Chile [Santiago] (USACH), European Project: 278537,EC:FP7:ERC,ERC-2011-StG_20101014,ANTICS(2012), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Discrete mathematics, Algebra and Number Theory, Mathematics - Number Theory, Applied Mathematics, Mathematics::Number Theory, 010102 general mathematics, Dedekind sum, 010103 numerical & computational mathematics, Algebraic number field, 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Computational Mathematics, Arithmetic zeta function, Riemann hypothesis, symbols.namesake, Discriminant, 11R42 (Primary), 11Y40 (Secondary), Bounded function, symbols, FOS: Mathematics, Dedekind eta function, Number Theory (math.NT), 0101 mathematics, Dedekind zeta function, Mathematics
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 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., 16 pages

Published
2015

4. Note sur les valeurs moyennes criblées de certaines fonctions arithmétiques

Author

Armand Lachand, Gérald Tenenbaum, Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics::General Mathematics, General Mathematics, Mathematics::Number Theory, 010102 general mathematics, Mathematical analysis, Algebraic number field, 16. Peace & justice, 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Combinatorics, symbols.namesake, [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], 0103 physical sciences, symbols, Arithmetic function, 010307 mathematical physics, 0101 mathematics, Dirichlet series, Dedekind zeta function, Mathematics
Abstract

International audience; We provide, in a wide range of the parameters, an estimate for the mean-value over sifted integers of certain arithmetic functions with Dirichlet series analytically close to $1/\zeta_{\mathbb{K}}(s)$, where $\mathbb{K}$ is a number field and $\zeta_{\mathbb{K}}$ its Dedekind zeta function.

Published
2015

5. Upper bounds for residues of Dedekind zeta functions and class numbers of cubic and quartic number fields

Author

Stéphane Louboutin, Institut de mathématiques de Luminy (IML), Centre National de la Recherche Scientifique (CNRS)-Université de la Méditerranée - Aix-Marseille 2, and Université de la Méditerranée - Aix-Marseille 2-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discrete mathematics, class number, Dedekind zeta function, Pure mathematics, Algebra and Number Theory, Explicit formulae, Applied Mathematics, 010102 general mathematics, Dedekind sum, number field, 010103 numerical & computational mathematics, Algebraic number field, 01 natural sciences, Class number formula, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Riemann zeta function, Computational Mathematics, Arithmetic zeta function, symbols.namesake, symbols, Primary 11R42, Secondary 11R16, 11R29, Dedekind eta function, 0101 mathematics, Mathematics
Abstract

International audience; Let K be an algebraic number field. Assume that ζ K (s)/ζ(s) is entire. We give an explicit upper bound for the residue at s = 1 of the Dedekind zeta function ζ K (s) of K. We deduce explicit upper bounds on class numbers of cubic and quartic number fields.

Published
2011

6. Moyennes de certaines fonctions multiplicatives sur les entiers friables, 2

Author

Gérald Tenenbaum, Jie Wu, Guillaume Hanrot, Curves, Algebra, Computer Arithmetic, and so On (CACAO), INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Dedekind zeta function, Pure mathematics, friable integers, General Mathematics, 010102 general mathematics, 010103 numerical & computational mathematics, 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], symbols.namesake, saddle-point method, symbols, AMS Classification: 11N37 (11N25, 11M41, 11C08), mean value of arithmetic function, 0101 mathematics, Dirichlet series, Mathematics
Abstract

International audience; Nous évaluons les moyennes indiquées dans le titre sous des hypothèses analytiques concernant la série de Dirichlet associée

Published
2008

7. Une minoration pour l'exposant du groupe des classes d'un corps engendré par un nombre de Salem

Author

Francesco Amoroso, Laboratoire de Mathématiques Nicolas Oresme ( LMNO ), Université de Caen Normandie ( UNICAEN ), Normandie Université ( NU ) -Normandie Université ( NU ) -Centre National de la Recherche Scientifique ( CNRS ), Amoroso, Francesco, Laboratoire de Mathématiques Nicolas Oresme (LMNO), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), and Normandie Université (NU)-Normandie Université (NU)

Subjects
Class (set theory), Algebra and Number Theory, Group (mathematics), 010102 general mathematics, Field (mathematics), Algebraic number field, 16. Peace & justice, 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], 010101 applied mathematics, Combinatorics, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Riemann hypothesis, symbols.namesake, Salem number, symbols, Exponent, 0101 mathematics, Dedekind zeta function, Mathematics, [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
Abstract

International audience; In this article we extend the main result of F. Amoroso and R. Dvornicich ``A Lower Bound for the Height in Abelian Extensions", J. Number Theory 80 , no. 2, 260-272 (2000), concerning lower bounds for the exponent of the class group of CM-fields. We consider a number field K generated by a Salem number z. If k denotes the field fixed by z->z^{-1}, we prove, under the generalized Riemann hypothesis for the Dedekind zeta function of K, lower bounds for the relative exponent e_{K/k} and the relative size h_{K/k} of the class group of K with respect to the class group of k.

Published
2007

Catalog

Books, media, physical & digital resources
See catalog results