Écarts entre nombres premiers successifs  [ Gaps between consecutive primes ]
Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque no. 311  (2007), Talk no. 959, p. 177-210

The Prime Number Theorem implies that the gap p n+1 -p n between consecutive prime numbers p n <p n+1 is, on average of order of magnitude logp n . Recently, D. Goldston, J. Pintz and C. Yıldırım have succeeded in proving that the normalized gap (p n+1 -p n )/log(p n ) can be arbitrarily small, improving spectacularly the previously known results. Under some assumptions which are considered as likely to be true, they manage to prove that p n+1 -p n <16 infinitely often. Their method is a beautiful application of ideas inspired by sieve methods, and it seems to offer many possibilities for further developments.

Le théorème des nombres premiers dit que la distance entre deux nombres premiers consécutifs p n <p n+1 est, en moyenne, de l’ordre de ln(p n ). Récemment, D. Goldston, J. Pintz et C. Yıldırım sont parvenus à démontrer que la distance normalisée (p n+1 -p n )/ln(p n ) pouvait devenir arbitrairement petite, améliorant spectaculairement les résultats connus auparavant. Sous des hypothèses considérées comme raisonnables, ils parviennent à montrer que p n+1 -p n <16 infiniment souvent. Leur méthode est une très jolie application d’idées inspirée par les méthodes de crible, et elle semble offrir de nombreuses possibilités de développement.

Classification:  11N05,  11N13,  11N35,  11N36,  11P32
Keywords: distribution of prime numbers, primes in arithmetic progressions, gaps between primes, sieve methods
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     author = {Kowalski, Emmanuel},
     title = {\'Ecarts entre nombres premiers successifs},
     booktitle = {S\'eminaire Bourbaki : volume 2005/2006, expos\'es 952-966},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {311},
     year = {2007},
     note = {talk:959},
     pages = {177-210},
     zbl = {1200.11074},
     mrnumber = {2359044},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2005-2006__48__177_0}
}
Kowalski, Emmanuel. Écarts entre nombres premiers successifs, in Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Talk no. 959, pp. 177-210. http://www.numdam.org/item/SB_2005-2006__48__177_0/

[1] E. Bombieri - “On twin almost primes”, Acta Arith. 28 (1975/76), p. 177-193, Correction, id., p. 457-461. | MR 396435 | Zbl 0319.10052 | Zbl 0319.10051

[2] E. Bombieri & H. Davenport - “Small differences between prime numbers”, Proc. Roy. Soc. Ser. A 293 (1966), p. 1-18. | MR 199165 | Zbl 0151.04201

[3] E. Bombieri, J. B. Friedlander & H. Iwaniec - “Primes in arithmetic progressions to large moduli”, Acta Math. 156 (1986), no. 3-4, p. 203-251. | MR 834613 | Zbl 0588.10042

[4] N. Bourbaki - Éléments de mathématique, Fonctions d'une variable réelle, Théorie élémentaire, Hermann, Paris, 1976. | Zbl 0346.26003

[5] J.-M. Deshouillers - “Progrès récents des petits cribles arithmétiques [d'après Jing Run Chen, Henryk Iwaniec, ...]”, in Séminaire Bourbaki (1977/78), Lect. Notes in Math., vol. 710, Springer, Berlin, 1979, exp. no. 520, p. 248-262. | Numdam | MR 554225 | Zbl 0406.10036

[6] É. Fouvry - “Autour du théorème de Bombieri-Vinogradov”, Acta Math. 152 (1984), no. 3-4, p. 219-244. | MR 741055 | Zbl 0552.10024

[7] -, “Cinquante ans de théorie analytique des nombres. Un point de vue parmi d'autres : celui des méthodes de crible”, in Development of mathematics 1950-2000, Birkhäuser, Basel, 2000, p. 485-514. | Zbl 0964.11041

[8] E. Fouvry & H. Iwaniec - “On a theorem of Bombieri-Vinogradov type”, Mathematika 27 (1980), no. 2, p. 135-152. | MR 610700 | Zbl 0469.10027

[9] P. X. Gallagher - “On the distribution of primes in short intervals”, Mathematika 23 (1976), no. 1, p. 4-9. | MR 409385 | Zbl 0346.10024

[10] D. A. Goldston - “On Bombieri and Davenport's theorem concerning small gaps between primes”, Mathematika 39 (1992), no. 1, p. 10-17. | MR 1176465 | Zbl 0758.11037

[11] D. A. Goldston, S. W. Graham, J. Pintz & C. Y. Yildirim - “Small gaps between primes or almost primes”, prépublication arXiv : math.NT/0506067. | Article | MR 2515812 | Zbl 1228.11148

[12] D. A. Goldston, Y. Motohashi, J. Pintz & C. Y. Yildirim - “Small gaps between primes exist”, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 4, p. 61-65. | MR 2222213 | Zbl 1168.11041

[13] D. A. Goldston, J. Pintz & C. Y. Yildirim - “The Path to Recent Progress on Small Gaps Between Primes”, prépublication arXiv : math.NT/0512436. | MR 2362197 | Zbl 1213.11168

[14] -, “Primes in tuples, I”, prépublication arXiv : math.NT/0508185. | Zbl 1207.11096

[15] H. Halberstam & H. Richert - Sieve methods, Academic Press, 1974. | Zbl 0298.10026

[16] D. Heath-Brown - “Almost-prime k-tuples”, Mathematika 44 (1997), no. 2, p. 245-266. | MR 1600529 | Zbl 0886.11052

[17] B. Host - “Progressions arithmétiques dans les nombres premiers”, in Sém. Bourbaki (2004/2005), Astérisque, vol. 307, Soc. Math. France, Paris, 2006, Exp. 944 (mars 2005). | Numdam | Zbl 1175.11052

[18] H. Iwaniec & E. Kowalski - Analytic number theory, Amer. Math. Soc. Colloquium Publications, vol. 53, Amer. Math. Soc., Providence, 2004. | Article | MR 2061214 | Zbl 1059.11001

[19] E. Kowalski - Un cours de théorie analytique des nombres, Cours Spécialisés, vol. 13, Soc. Math. France, Paris, 2004. | MR 2122960 | Zbl 1071.11001

[20] P. Michel - “Progrès recents du crible et applications [d'après Duke, Fouvry, Friedlander, Iwaniec]”, in Séminaire Bourbaki (1997/98), Astérisque, vol. 252, Soc. Math. France, Paris, 1998, Exp. no. 842, p. 185-209. | Numdam | MR 1685608 | Zbl 0940.11042

[21] J. Pintz & Y. Motohashi - “A smoothed GPY sieve”, prépublication arXiv : math.NT/0602599. | MR 2414788 | Zbl 1278.11090 | Zbl pre05309678

[22] A. Selberg - “Lectures on sieves”, in Collected papers II, Springer-Verlag, Berlin, 1991, p. 65-247. | MR 1295844

[23] J. Sivak - “Méthodes de crible appliquées aux sommes de Kloosterman et aux petits écarts entre nombres premiers”, Thèse, Université Paris-Sud, décembre 2005.