On the use of explicit bounds on residues of Dedekind zeta functions taking into account the behavior of small primes

Stéphane Louboutin

doi:10.5802/jtnb.508

Stéphane Louboutin ¹

¹ Institut de Mathématiques de Luminy, UMR 6206 163, avenue de Luminy, Case 907 13288 Marseille Cedex 9, FRANCE

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 2, pp. 559-573.

Résumé
Abstract

Nous donnons des majorants explicites des résidus au point $s = 1$ des fonctions zêta $ζ_{K} (s)$ des corps de nombres tenant compte du comportement des petits nombres premiers dans $K$ . Dans le cas où $K$ est abélien, de telles majorations sont déduites de majorations de $| L (1, χ) |$ tenant compte du comportement de $χ$ sur les petits nombres premiers, pour $χ$ un caractère de Dirichlet primitif. De nombreuses applications sont données pour illustrer l’utilité de tels majorants.

Lately, explicit upper bounds on $| L (1, χ) |$ (for primitive Dirichlet characters $χ$ ) taking into account the behaviors of $χ$ on a given finite set of primes have been obtained. This yields explicit upper bounds on residues of Dedekind zeta functions of abelian number fields taking into account the behavior of small primes, and it as been explained how such bounds yield improvements on lower bounds of relative class numbers of CM-fields whose maximal totally real subfields are abelian. We present here some other applications of such bounds together with new bounds for non-abelian number fields.

MR Zbl

DOI : 10.5802/jtnb.508

Mots clés : $L$-functions, Dedekind zeta functions, number fields, class number.

Affiliations des auteurs :

Stéphane Louboutin ¹

¹ Institut de Mathématiques de Luminy, UMR 6206 163, avenue de Luminy, Case 907 13288 Marseille Cedex 9, FRANCE

@article{JTNB_2005__17_2_559_0,
     author = {St\'ephane Louboutin},
     title = {On the use of explicit bounds on residues of {Dedekind} zeta functions taking into account the behavior of small primes},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {559--573},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {2},
     year = {2005},
     doi = {10.5802/jtnb.508},
     mrnumber = {2211308},
     zbl = {1091.11039},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.508/}
}

TY  - JOUR
AU  - Stéphane Louboutin
TI  - On the use of explicit bounds on residues of Dedekind zeta functions taking into account the behavior of small primes
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2005
SP  - 559
EP  - 573
VL  - 17
IS  - 2
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.508/
DO  - 10.5802/jtnb.508
LA  - en
ID  - JTNB_2005__17_2_559_0
ER  -

%0 Journal Article
%A Stéphane Louboutin
%T On the use of explicit bounds on residues of Dedekind zeta functions taking into account the behavior of small primes
%J Journal de théorie des nombres de Bordeaux
%D 2005
%P 559-573
%V 17
%N 2
%I Université Bordeaux 1
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.508/
%R 10.5802/jtnb.508
%G en
%F JTNB_2005__17_2_559_0

Stéphane Louboutin. On the use of explicit bounds on residues of Dedekind zeta functions taking into account the behavior of small primes. Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 2, pp. 559-573. doi : 10.5802/jtnb.508. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.508/

Bibliographie
Cité par

[Bes] S. Bessassi, Bounds for the degrees of CM-fields of class number one. Acta Arith. 106 (2003), 213–245. | MR | Zbl

[BHM] Y. Bugeaud, G. Hanrot, M. Mignotte, Sur l’équation diophantienne $(x^{n} - 1) / (x - 1) = y^{q}$ . III. Proc. London Math. Soc. (3) 84 (2002), 59–78. | Zbl

[BL02] G. Boutteaux, S. Louboutin, The class number one problem for some non-normal sextic CM-fields. Part 2. Acta Math. Inform. Univ. Ostraviensis 10 (2002), 3–23. | MR | Zbl

[Boo] A. R. Booker, Quadratic class numbers and characters sums. Math. Comp., to appear. | MR | Zbl

[CL] H. Cohen, H. W. Lenstra, Heuristics on class groups of number fields. Lecture Notes in Math. 1068 (1984), 33–62. | MR | Zbl

[CW] G. Cornell, L. C. Washington, Class numbers of cyclotomic fields. J. Number Theory 21 (1985), 260–274. | MR | Zbl

[JWW] M. J. Jacobson, H. C. Williams, K. Wooding, Imaginary cyclic quartic fileds with large minus class numbers. Algorithmic Number Theory (University of Vermont, 2004), Lectures Notes in Computer Science 3076 (2004), 280–292. | MR | Zbl

[Laz1] A. J. Lazarus, Class numbers of simplest quartic fields. Number theory (Banff, AB, 1988), 313–323, Walter de Gruyter, Berlin, 1990. | MR | Zbl

[Laz2] A. J. Lazarus, On the class number and unit index of simplest quartic fields. Nagoya Math. J. 121 (1991), 1–13. | MR | Zbl

[Le] M. Le, Upper bounds for class numbers of real quadratic fields. Acta Arith. 68 (1994), 141–144. | MR | Zbl

[LK] G.-N. Lee, S.-H. Kwon, CM-fields with relative class number one. Math. Comp., to appear. | MR | Zbl

[Lou95] S. Louboutin, Determination of all non-quadratic imaginary cyclic number fields of $2$ -power degrees with ideal class groups of exponents $\leq 2$ . Math. Comp. 64 (1995), 323–340. | MR | Zbl

[Lou97] S. Louboutin, CM-fields with cyclic ideal class groups of $2$ -power orders. J. Number Theory 67 (1997), 1–10. | MR | Zbl

[Lou98] S. Louboutin, Computation of relative class numbers of imaginary abelian number fields. Experimental Math. 7 (1998), 293–303. | MR | Zbl

[Lou01] S. Louboutin, Explicit upper bounds for residues of Dedekind zeta functions and values of $L$ -functions at $s = 1$ , and explicit lower bounds for relative class numbers of CM-fields. Canad. J. Math. 53 (2001), 1194–1222. | MR | Zbl

[Lou02a] S. Louboutin, Efficient computation of class numbers of real abelian number fields. Lectures Notes in Computer Science 2369 (2002), 625–628. | MR | Zbl

[Lou02b] S. Louboutin, Computation of class numbers of quadratic number fields. Math. Comp. 71 (2002), 1735–1743. | MR | Zbl

[Lou02c] S. Louboutin, The exponent three class group problem for some real cyclic cubic number fields. Proc. Amer. Math. Soc. 130 (2002), 353–361. | MR | Zbl

[Lou03] S. Louboutin, Explicit lower bounds for residues at $s = 1$ of Dedekind zeta functions and relative class numbers of CM-fields. Trans. Amer. Math. Soc. 355 (2003), 3079–3098. | MR | Zbl

[Lou04a] S. Louboutin, Explicit upper bounds for values at $s = 1$ of Dirichlet $L$ -series associated with primitive even characters. J. Number Theory 104 (2004), 118–131. | MR | Zbl

[Lou04b] S. Louboutin, The simplest quartic fields with ideal class groups of exponents $\leq 2$ . J. Math. Soc. Japan 56 (2004), 717–727. | MR | Zbl

[Lou04c] S. Louboutin, Class numbers of real cyclotomic fields. Publ. Math. Debrecen 64 (2004), 451–461. | MR | Zbl

[Lou04d] S. Louboutin, Explicit upper bounds for $| L (1, χ) |$ for primitive characters $χ$ . Quart. J. Math. 55 (2004), 57–68. | MR | Zbl

[Mos] C. Moser, Nombre de classes d’une extension cyclique réelle de $Q$ de degré $4$ ou $6$ et de conducteur premier. Math. Nachr. 102 (1981), 45–52. | Zbl

[MP] C. Moser, J.-J. Payan, Majoration du nombre de classes d’un corps cubique de conducteur premier. J. Math. Soc. Japan 33 (1981), 701–706. | Zbl

[MR] M. Mignotte, Y. Roy, Minorations pour l’équation de Catalan. C. R. Acad. Sci. Paris 324 (1997), 377–380. | Zbl

[Odl] A. Odlyzko, Some analytic estimates of class numbers and discriminants. Invent. Math. 29 (1975), 275–286. | MR | Zbl

[Ram01] O. Ramaré, Approximate formulae for $L (1, χ)$ . Acta Arith. 100 (2001), 245–266. | MR | Zbl

[Ram04] O. Ramaré, Approximate formulae for $| L (1, χ) |$ . II. Acta Arith. 112 (2004), 141–149. | MR | Zbl

[SSW] R. G. Stanton, C. Sudler, H. C. Williams, An upper bound for the period of the simple continued fraction for $\sqrt{D}$ . Pacific J. Math. 67 (1976), 525–536. | MR | Zbl

[Sta] H. M. Stark, Some effective cases of the Brauer-Siegel Theorem. Invent. Math. 23 (1974), 135–152. | MR | Zbl

[Ste] R. Steiner, Class number bounds and Catalan’s equation. Math. Comp. 67 (1998), 1317–1322. | Zbl

[SWW] E. Seah, L. C. Washington, H. C. Williams, The calculation of a large cubic class number with an application to real cyclotomic fields. Math. Comp. 41 (1983), 303–305. | MR | Zbl

[Wa] L. C. Washington, Class numbers of the simplest cubic fields. Math. Comp. 48 (1987), 371–384. | MR | Zbl

[WB] H. C. Williams, J. Broere, A computational technique for evaluating $L (1, χ)$ and the class number of a real quadratic field. Math. Comp. 30 (1976), 887–893. | MR | Zbl

Cité par Sources :