An explicit upper bound for the least prime ideal in the Chebotarev density theorem

Ahn, Jeoung-Hwan; Kwon, Soun-Hi

doi:10.5802/aif.3274

An explicit upper bound for the least prime ideal in the Chebotarev density theorem
[Une borne explicite pour le plus petit idéal premier dans le théorème de densité de Chebotarev]

Ahn, Jeoung-Hwan ¹ ; Kwon, Soun-Hi ¹

¹ Department of Mathematics Education Korea University 02841, Seoul (Korea)

Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1411-1458.

Résumé
Abstract

Lagarias, Montgomery, et Odlyzko ont démontré qu’il existe une constante absolue effectivement calculable $A_{1}$ telle que pour chaque extension finie $K$ de $ℚ$ , chaque extension galoisienne finie $L$ de $K$ à groupe de Galois $G$ , et chaque classe de conjugaison $C$ de $G$ , il existe un idéal premier $𝔭$ de $K$ qui est nonramifié dans $L$ , pour lequel $[\frac{L / K}{𝔭}] = C$ , pour lequel $N_{K / ℚ} 𝔭$ est un nombre premier rationel, et qui satisfait $N_{K / ℚ} 𝔭 \leq 2 {d_{L}}^{A_{1}}$ . Dans cet article nous démontrons sans aucune restriction que $N_{K / ℚ} 𝔭 \leq {d_{L}}^{12577}$ si $L \neq ℚ$ , en suivant la méthode developpée par Lagarias, Montgomery, et Odlyzko.

Lagarias, Montgomery, and Odlyzko proved that there exists an effectively computable absolute constant $A_{1}$ such that for every finite extension $K$ of $ℚ$ , every finite Galois extension $L$ of $K$ with Galois group $G$ and every conjugacy class $C$ of $G$ , there exists a prime ideal $𝔭$ of $K$ which is unramified in $L$ , for which $[\frac{L / K}{𝔭}] = C$ , for which $N_{K / ℚ} 𝔭$ is a rational prime, and which satisfies $N_{K / ℚ} 𝔭 \leq 2 {d_{L}}^{A_{1}}$ . In this paper we show without any restriction that $N_{K / ℚ} 𝔭 \leq {d_{L}}^{12577}$ if $L \neq ℚ$ , using the approach developed by Lagarias, Montgomery, and Odlyzko.

Reçu le : 2017-06-23
Révisé le : 2018-02-22
Accepté le : 2018-06-13
Publié le : 2019-06-03

Zbl

DOI : 10.5802/aif.3274

Classification : 11R44, 11R42, 11M41, 11R45
Keywords: The Chebotarev density theorem, Dedekind zeta functions, the Deuring–Heilbronn phenomenon
Mot clés : théorème de densité de Chebotarev, fonction de zêta de Dedekind, phénomène de Deuring–Heilbronn

Affiliations des auteurs :

Ahn, Jeoung-Hwan ¹ ; Kwon, Soun-Hi ¹

¹ Department of Mathematics Education Korea University 02841, Seoul (Korea)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{AIF_2019__69_3_1411_0,
     author = {Ahn, Jeoung-Hwan and Kwon, Soun-Hi},
     title = {An explicit upper bound for the least prime ideal in the {Chebotarev} density theorem},
     journal = {Annales de l'Institut Fourier},
     pages = {1411--1458},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {3},
     year = {2019},
     doi = {10.5802/aif.3274},
     zbl = {1275.11144},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3274/}
}

TY  - JOUR
AU  - Ahn, Jeoung-Hwan
AU  - Kwon, Soun-Hi
TI  - An explicit upper bound for the least prime ideal in the Chebotarev density theorem
JO  - Annales de l'Institut Fourier
PY  - 2019
SP  - 1411
EP  - 1458
VL  - 69
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3274/
DO  - 10.5802/aif.3274
LA  - en
ID  - AIF_2019__69_3_1411_0
ER  -

%0 Journal Article
%A Ahn, Jeoung-Hwan
%A Kwon, Soun-Hi
%T An explicit upper bound for the least prime ideal in the Chebotarev density theorem
%J Annales de l'Institut Fourier
%D 2019
%P 1411-1458
%V 69
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3274/
%R 10.5802/aif.3274
%G en
%F AIF_2019__69_3_1411_0

Ahn, Jeoung-Hwan; Kwon, Soun-Hi. An explicit upper bound for the least prime ideal in the Chebotarev density theorem. Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1411-1458. doi : 10.5802/aif.3274. https://aif.centre-mersenne.org/articles/10.5802/aif.3274/

Bibliographie
Cité par

[1] Ahn, Jeoung-Hwan; Kwon, Soun-Hi Some explicit zero-free regions for Hecke $L$ -functions, J. Number Theory, Volume 145 (2014), pp. 433-473 | DOI | MR | Zbl

[2] Ankeny, Nesmith C. The least quadratic non residue, Ann. Math., Volume 55 (1952), pp. 65-72 | DOI | MR | Zbl

[3] Bach, Eric Explicit bounds for primality testing and related problems, Math. Comput., Volume 55 (1990) no. 191, pp. 355-380 | DOI | MR | Zbl

[4] Bach, Eric; Sorenson, Jonathan Explicit bounds for primes in residue classes, Math. Comput., Volume 65 (1996) no. 216, pp. 1717-1735 | DOI | MR | Zbl

[5] Bombieri, Enrico Le grand crible dans la théorie analytique des nombres, Astérisque, 18, Société Mathématique de France, 1974, i+87 pages (Avec une sommaire en anglais) | MR | Zbl

[6] Burgess, D. A. The distribution of quadratic residues and non-residues, Mathematika, Volume 4 (1957), pp. 106-112 | DOI | MR | Zbl

[7] Chen, Jing Run On the least prime in an arithmetical progression, Sci. Sin., Volume 14 (1965), pp. 1868-1871 | MR

[8] Chen, Jing Run; Liu, Jianmin On the least prime in an arithmetical progression and theorems concerning the zeros of Dirichlet’s $L$ -functions. V, International Symposium in Memory of Hua Loo Keng, Vol. I (Beijing, 1988), Springer, 1991, pp. 19-42 | DOI | MR | Zbl

[9] Delange, Hubert Une remarque sur la dérivée logarithmique de la fonction zêta de Riemann, Colloq. Math., Volume 53 (1987) no. 2, pp. 333-335 | DOI | MR | Zbl

[10] Deuring, Max Über den Tschebotareffschen Dichtigkeitssatz, Math. Ann., Volume 110 (1935) no. 1, pp. 414-415 | DOI | MR | Zbl

[11] Ellison, William John Les nombres premiers, Publications de l’Université de Nancago, IX, Hermann, 1975, xiv+442 pages (En collaboration avec Michel Mendès France) | MR | Zbl

[12] Graham, Sidney On Linnik’s constant, Acta Arith., Volume 39 (1981) no. 2, pp. 163-179 | DOI | MR | Zbl

[13] Granville, Andrew; Pomerance, Carl On the least prime in certain arithmetic progressions, J. Lond. Math. Soc., Volume 41 (1990) no. 2, pp. 193-200 | DOI | MR | Zbl

[14] Heath-Brown, David R. Zero-free regions for Dirichlet $L$ -functions, and the least prime in an arithmetic progression, Proc. Lond. Math. Soc., Volume 64 (1992) no. 2, pp. 265-338 | DOI | MR | Zbl

[15] Heilbronn, Hans Zeta-functions and $L$ -functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Academic Press Inc., 1967, pp. 204-230 | MR

[16] Hoffstein, Jeffrey On the Siegel–Tatuzawa theorem, Acta Arith., Volume 38 (1980) no. 2, pp. 167-174 | DOI | MR | Zbl

[17] Hooley, Christopher The distribution of sequences in arithmetic progressions, Proceedings of the International Congress of Mathematicians (Vancouver, 1974), Vol. 1, Canadian Mathematical Congress, 1975, pp. 357-364 | MR | Zbl

[18] Jutila, Matti On Linnik’s constant, Math. Scand., Volume 41 (1977) no. 1, pp. 45-62 | DOI | MR | Zbl

[19] Kadiri, Habiba Une région explicite sans zéros pour la fonction $ζ$ de Riemann, Acta Arith., Volume 117 (2005) no. 4, pp. 303-339 | DOI | MR | Zbl

[20] Kadiri, Habiba Explicit zero-free regions for Dedekind zeta functions, Int. J. Number Theory, Volume 8 (2012) no. 1, pp. 125-147 | DOI | MR | Zbl

[21] Kadiri, Habiba; Ng, Nathan Explicit zero density theorems for Dedekind zeta functions, J. Number Theory, Volume 132 (2012) no. 4, pp. 748-775 | DOI | MR | Zbl

[22] Kumar Murty, Vijaya The least prime in a conjugacy class, C. R. Math. Acad. Sci., Soc. R. Can., Volume 22 (2000) no. 4, pp. 129-146 | MR | Zbl

[23] Lagarias, Jeffrey C.; Montgomery, Hugh L.; Odlyzko, Andrew M. A bound for the least prime ideal in the Chebotarev density theorem, Invent. Math., Volume 54 (1979) no. 3, pp. 271-296 | DOI | MR | Zbl

[24] Lagarias, Jeffrey C.; Odlyzko, Andrew M. Effective versions of the Chebotarev density theorem, Algebraic number fields: $L$ -functions and Galois properties (Durham, 1975), Academic Press Inc., 1977, pp. 409-464 | MR | Zbl

[25] Lamzouri, Youness; Li, Xiannan; Soundararajan, Kannan Conditional bounds for the least quadratic non-residue and related problems, Math. Comput., Volume 84 (2015) no. 295, pp. 2391-2412 | DOI | MR | Zbl

[26] Lamzouri, Youness; Li, Xiannan; Soundararajan, Kannan Corrigendum to “Conditional bounds for the least quadratic non-residue and related problems”, Math. Comput., Volume 86 (2017) no. 307, pp. 2551-2554 | DOI | MR | Zbl

[27] Landau, Edmund Algebraische Zahlen, 1927

[28] Lang, Serge Algebraic number theory, Graduate Texts in Mathematics, 110, Springer, 1994, xiv+357 pages | DOI | MR | Zbl

[29] Linnik, Yuri V. On the least prime in an arithmetic progression. I. The basic theorem, Mat. Sb., N. Ser., Volume 15(57) (1944), pp. 139-178 | MR | Zbl

[30] Linnik, Yuri V. On the least prime in an arithmetic progression. II. The Deuring–Heilbronn phenomenon, Mat. Sb., N. Ser., Volume 15(57) (1944), pp. 347-368 | MR | Zbl

[31] Louboutin, Stéphane Minoration au point $1$ des fonctions $L$ et détermination des corps sextiques abéliens totalement imaginaires principaux, Acta Arith., Volume 62 (1992) no. 2, pp. 109-124 | DOI | MR | Zbl

[32] Louboutin, Stéphane An explicit lower bound on moduli of Dirichlet $L$ -functions at $s = 1$ , J. Ramanujan Math. Soc., Volume 30 (2015) no. 1, pp. 101-113 | MR | Zbl

[33] Louboutin, Stéphane Explicit upper bounds for residues of Dedekind zeta functions, Mosc. Math. J., Volume 15 (2015) no. 4, pp. 727-740 | DOI | MR | Zbl

[34] Louboutin, Stéphane Real zeros of Dedekind zeta functions, Int. J. Number Theory, Volume 11 (2015) no. 3, pp. 843-848 | DOI | MR | Zbl

[35] MacCluer, Charles R. A reduction of the Čebotarev density theorem to the cyclic case, Acta Arith., Volume 15 (1968), pp. 45-47 | DOI | MR | Zbl

[36] McCurley, Kevin S. Explicit zero-free regions for Dirichlet $L$ -functions, J. Number Theory, Volume 19 (1984) no. 1, pp. 7-32 | DOI | MR | Zbl

[37] Mossinghoff, Michael J.; Trudgian, Timothy S. Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function, J. Number Theory, Volume 157 (2015), pp. 329-349 | DOI | MR | Zbl

[38] Murty, V. Kumar The least prime which does not split completely, Forum Math., Volume 6 (1994) no. 5, pp. 555-565 | DOI | MR | Zbl

[39] Neukirch, Jürgen Class field theory, Grundlehren der Mathematischen Wissenschaften, 280, Springer, 1986, viii+140 pages | DOI | MR | Zbl

[40] Odlyzko, Andrew M. On conductors and discriminants, Algebraic number fields: $L$ -functions and Galois properties (Durham, 1975), Academic Press Inc., 1977, pp. 377-407 | MR | Zbl

[41] Oesterlé, Joseph Versions effectives du théorème de Chebotarev sous l’hypothèse de Riemann généralisée, Journees arithmÃ©tiques de Luminy (1978) (Astérisque), Volume 61, Société Mathématique de France, 1979, pp. 165-167 | Zbl

[42] Pan, Cheng Dong On the least prime in an arithmetical progression, Sci. Record, New Ser., Volume 1 (1957), pp. 311-313 | MR | Zbl

[43] Pomerance, Carl A note on the least prime in an arithmetic progression, J. Number Theory, Volume 12 (1980) no. 2, pp. 218-223 | DOI | MR | Zbl

[44] Rosser, J. Barkley; Schoenfeld, Lowell Approximate formulas for some functions of prime numbers, Ill. J. Math., Volume 6 (1962), pp. 64-94 | DOI | MR | Zbl

[45] Rosser, J. Barkley; Schoenfeld, Lowell Sharper bounds for the Chebyshev functions $θ (x)$ and $ψ (x)$ , Math. Comput., Volume 29 (1975), pp. 243-269 (Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday) | DOI | MR | Zbl

[46] Serre, Jean-Pierre Corps locaux, Publications de l’Université de Nancago, VIII, Hermann, 1968, 245 pages | MR | Zbl

[47] Serre, Jean-Pierre Quelques applications du théorème de densité de Chebotarev, Publ. Math., Inst. Hautes Étud. Sci. (1981) no. 54, pp. 323-401 | MR | Zbl

[48] Stark, Harold M. Some effective cases of the Brauer–Siegel theorem, Invent. Math., Volume 23 (1974), pp. 135-152 | DOI | MR | Zbl

[49] Stečkin, S. B. The zeros of the Riemann zeta-function, Mat. Zametki, Volume 8 (1970), pp. 419-429 | MR | Zbl

[50] Stevenhagen, Peter; Lenstra, Hendrik W. Jr. Chebotarëv and his density theorem, Math. Intell., Volume 18 (1996) no. 2, pp. 26-37 | DOI | MR | Zbl

[51] Thorner, Jesse; Zaman, Asif An explicit bound for the least prime ideal in the Chebotarev density theorem, Algebra Number Theory, Volume 11 (2017) no. 5, pp. 1135-1197 | DOI | MR | Zbl

[52] Trudgian, Timothy S. An improved upper bound for the error in the zero-counting formulae for Dirichlet $L$ -functions and Dedekind zeta-functions, Math. Comput., Volume 84 (2015) no. 293, pp. 1439-1450 | DOI | MR | Zbl

[53] Tschebotareff, Nikolaĭ Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören, Math. Ann., Volume 95 (1926) no. 1, pp. 191-228 | DOI | MR

[54] Vinogradov, Ivan M. On a general theorem concerning the distribution of the residues and non-residues of powers, Trans. Am. Math. Soc., Volume 29 (1927) no. 1, pp. 209-217 | DOI | MR | Zbl

[55] Wang, Wei On the least prime in an arithmetic progression, Acta Math. Sin., Volume 29 (1986) no. 6, pp. 826-836 | MR | Zbl

[56] Wang, Wei On the least prime in an arithmetic progression, Acta Math. Sin., New Ser., Volume 7 (1991) no. 3, pp. 279-289 A Chinese summary appears in Acta Math. Sin. 8 (1992), no. 4, p. 575 | DOI | MR | Zbl

[57] Weiss, Alfred The least prime ideal, J. Reine Angew. Math., Volume 338 (1983), pp. 56-94 | DOI | MR | Zbl

[58] Whittaker, Edmund T.; Watson, George N. A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, Cambridge Mathematical Library, Cambridge University Press, 1996, vi+608 pages | DOI | MR | Zbl

[59] Winckler, Bruno Théorème de Chebotarev effectif (2013) (https://arxiv.org/abs/1311.5715v1)

[60] Winckler, Bruno Intersection arithmétique et problème de Lehmer elliptique, Ph. D. Thesis, Université de Bordeaux (France) (2015)

[61] Xylouris, Triantafyllos On the least prime in an arithmetic progression and estimates for the zeros of Dirichlet $L$ -functions, Acta Arith., Volume 150 (2011) no. 1, pp. 65-91 | DOI | MR | Zbl

[62] Zaman, Asif Explicit estimates for the zeros of Hecke $L$ -functions, J. Number Theory, Volume 162 (2016), pp. 312-375 | DOI | MR | Zbl

[63] Zaman, Asif Bounding the least prime ideal in the Chebotarev density theorem, Funct. Approximatio, Comment. Math., Volume 57 (2017) no. 1, pp. 115-142 | DOI | MR | Zbl

Cité par Sources :

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT