P2 in short intervals
Annales de l'Institut Fourier, Tome 31 (1981) no. 4, pp. 37-56.

On démontre que l’intervalle [x,x+x0,45] contient un entier ayant au plus deux facteurs premiers dès que x est un nombre réel suffisamment grand.

For any sufficiently large real number x, the interval [x,x+x0,45] contains at least one integer having at most two prime factors .

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Iwaniec, Henryk; Laborde, M. $P_2$ in short intervals. Annales de l'Institut Fourier, Tome 31 (1981) no. 4, pp. 37-56. doi : 10.5802/aif.848. http://www.numdam.org/articles/10.5802/aif.848/

[1] A. A. Buchstab, Combinatorial strengthening of the sieve method of Eratosthenes (Russian), Uspehi Math. Nauk., 22 (1967), n° 3 (135), 199-226. | Zbl

[2] Jing-Run Chen, On the distribution of almost primes in an interval, Scientia Sinica, 18 (1975), 611-627. | MR | Zbl

[3] Jing-Run Chen, On the distribution of almost primes in an interval (II), Scientia Sinica, 22 (1979), 253-275. | MR | Zbl

[4] H. Halberstam and H.-E. Richert, Sieve Methods, London 1974. | MR | Zbl

[5] H. Halberstam, D. R. Heath-Brown and H.-E. Richert, Almost-primes in short intervals, to appear. | Zbl

[6] H. Iwaniec, A new form of the error term in the linear sieve, Acta Arith., 27 (1980), 307-320. | MR | Zbl

[7] W. B. Jurkat and H.-E. Richert, An improvement of Selberg sieve method, I, Acta Arith., 11 (1965), 217-240. | MR | Zbl

[8] M. Laborde, Les sommes trigonométriques de Chen et les poids de Buchstab en théorie du crible, Thèse de 3e cycle, Université de Paris-Sud.

[9] M. Laborde, Buchstab's sifting weights, Mathematika, 26 (1979), 250-257. | MR | Zbl

[10] R. A. Rankin, Van der Corput's method and the theory of exponent pairs, Quart. J. Oxford, (2) 6 (1955), 147-153. | MR | Zbl

[11] H.-E. Richert, Selberg's sieve with weights, Mathematika, 16 (1969), 1-22. | MR | Zbl

[12] E. C. Titchmarsh, The theory of the Riemann Zeta-Function, Oxford 1951. | MR | Zbl

  • Wu, Liuying On the least almost-prime in arithmetic progressions., Czechoslovak Mathematical Journal, Volume 74 (2024) no. 2, pp. 535-548 | DOI:10.21136/cmj.2024.0459-23 | Zbl:7893397
  • Narkiewicz, Władysław The Last Period, Rational Number Theory in the 20th Century (2012), p. 307 | DOI:10.1007/978-0-85729-532-3_6
  • Wu, Jie Almost primes in short intervals, Science China. Mathematics, Volume 53 (2010) no. 9, pp. 2511-2524 | DOI:10.1007/s11425-010-4039-y | Zbl:1221.11196
  • Baker, Roger Numbers in a given set with (or without) a large prime factor, The Ramanujan Journal, Volume 20 (2009) no. 3, pp. 275-295 | DOI:10.1007/s11139-009-9179-8 | Zbl:1213.11173
  • Alkan, Emre; Zaharescu, Alexandru B-free numbers in short arithmetic progressions, Journal of Number Theory, Volume 113 (2005) no. 2, pp. 226-243 | DOI:10.1016/j.jnt.2004.10.003 | Zbl:1138.11339
  • Guy, Richard K. Prime Numbers, Unsolved Problems in Number Theory, Volume 1 (2004), p. 3 | DOI:10.1007/978-0-387-26677-0_2
  • Brüdern, Jörg; Kawada, Koichi Ternary problems in additive prime number theory, Analytic number theory. Proceedings of the 1st China-Japan seminar on number theory, Beijing, China, September 13–17, 1999 and the annual conference on analytic number theory, Kyoto, Japan, November 29–December 3, 1999, Dordrecht: Kluwer Academic Publishers, 2002, pp. 39-91 | Zbl:1028.11062
  • Baker, Roger C.; Harman, Glyn Sparsely totient numbers, Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI, Volume 5 (1996) no. 2, pp. 183-190 | DOI:10.5802/afst.826 | Zbl:0871.11060
  • Baker, R. C.; Harman, G. The sequence x/n and its subsequences, Rocky Mountain Journal of Mathematics, Volume 26 (1996) no. 3, pp. 795-814 | DOI:10.1216/rmjm/1181072023 | Zbl:0881.11060
  • Liu, H.-Q. A special triple exponential sum, Mathematika, Volume 42 (1995) no. 1, pp. 137-143 | DOI:10.1112/s0025579300011414 | Zbl:0829.11042
  • Guy, Richard K. Prime Numbers, Unsolved Problems in Number Theory, Volume 1 (1994), p. 3 | DOI:10.1007/978-1-4899-3585-4_2
  • Wu, Jie P 2 Dans Les Petits Intervalles, Séminaire de Théorie des Nombres, Paris, 1989–90, Volume 102 (1992), p. 233 | DOI:10.1007/978-1-4757-4269-5_16
  • Wu, Jie P2 dans les petits intervalles. (P2 in short intervals), Séminaire de théorie des nombres, Paris, France, 1989-90, Boston, MA etc.: Birkhäuser, 1992, pp. 233-267 | Zbl:0743.11050
  • Fouvry, Etienne Nombres presque premiers dans les petits intervalles, Analytic Number Theory, Volume 1434 (1990), p. 65 | DOI:10.1007/bfb0097125
  • Luo, Wenzhi Bilinear forms of remainder terms in short intervals, Acta Mathematica Sinica, Volume 32 (1989) no. 1, pp. 86-90 | Zbl:0662.10032
  • Fouvry, Etienne; Iwaniec, Henryk Exponential sums with monomials, Journal of Number Theory, Volume 33 (1989) no. 3, pp. 311-333 | DOI:10.1016/0022-314x(89)90067-x | Zbl:0687.10028
  • Bantle, G.; Grupp, F. On a problem of Erdős and Szemerédi, Journal of Number Theory, Volume 22 (1986), pp. 280-288 | DOI:10.1016/0022-314x(86)90012-0 | Zbl:0578.10057
  • Halberstam, H.; Richert, H.-E. A weighted sieve of Greaves' type. II, Elementary and analytic theory of numbers, Banach Cent. Publ. 17, 183-215, 1985 | Zbl:0592.10041
  • Bantle, Gerhard Obere Abschätzung für die Anzahl der B-Zwillinge auf kurzen Intervallen, Mathematische Zeitschrift, Volume 189 (1985), pp. 561-570 | DOI:10.1007/bf01168160 | Zbl:0545.10029
  • Halberstam, H.; Richert, H.-E. Weighted sieves, Journees arithmetiques, Orsay 1982, Publ. Math. Orsay 83.04, 97-114, 1983 | Zbl:0514.10037

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