P 2 in short intervals

Henryk Iwaniec; M. Laborde

Annales de l'institut Fourier (1981)

  • Volume: 31, Issue: 4, page 37-56
  • ISSN: 0373-0956

Abstract

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For any sufficiently large real number x , the interval [ x , x + x 0 , 45 ] contains at least one integer having at most two prime factors .

How to cite

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Iwaniec, Henryk, and Laborde, M.. "$P_2$ in short intervals." Annales de l'institut Fourier 31.4 (1981): 37-56. <http://eudml.org/doc/74517>.

@article{Iwaniec1981,
abstract = {For any sufficiently large real number $x$, the interval $[x,x+x^\{0,45\}]$ contains at least one integer having at most two prime factors .},
author = {Iwaniec, Henryk, Laborde, M.},
journal = {Annales de l'institut Fourier},
keywords = {number with at most two prime factors; short intervals; Iwaniec's bilinear form of sieve remainder term; Laborde's weights},
language = {eng},
number = {4},
pages = {37-56},
publisher = {Association des Annales de l'Institut Fourier},
title = {$P_2$ in short intervals},
url = {http://eudml.org/doc/74517},
volume = {31},
year = {1981},
}

TY - JOUR
AU - Iwaniec, Henryk
AU - Laborde, M.
TI - $P_2$ in short intervals
JO - Annales de l'institut Fourier
PY - 1981
PB - Association des Annales de l'Institut Fourier
VL - 31
IS - 4
SP - 37
EP - 56
AB - For any sufficiently large real number $x$, the interval $[x,x+x^{0,45}]$ contains at least one integer having at most two prime factors .
LA - eng
KW - number with at most two prime factors; short intervals; Iwaniec's bilinear form of sieve remainder term; Laborde's weights
UR - http://eudml.org/doc/74517
ER -

References

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  1. [1] A. A. BUCHSTAB, Combinatorial strengthening of the sieve method of Eratosthenes (Russian), Uspehi Math. Nauk., 22 (1967), n° 3 (135), 199-226. Zbl0199.09001
  2. [2] Jing-run CHEN, On the distribution of almost primes in an interval, Scientia Sinica, 18 (1975), 611-627. Zbl0381.10033MR56 #15584
  3. [3] Jing-run CHEN, On the distribution of almost primes in an interval (II), Scientia Sinica, 22 (1979), 253-275. Zbl0408.10030MR82d:10065
  4. [4] H. HALBERSTAM and H.-E. RICHERT, Sieve Methods, London 1974. Zbl0298.10026MR54 #12689
  5. [5] H. HALBERSTAM, D. R. HEATH-BROWN and H.-E. RICHERT, Almost-primes in short intervals, to appear. Zbl0461.10041
  6. [6] H. IWANIEC, A new form of the error term in the linear sieve, Acta Arith., 27 (1980), 307-320. Zbl0444.10038MR82d:10069
  7. [7] W. B. JURKAT and H.-E. RICHERT, An improvement of Selberg sieve method, I, Acta Arith., 11 (1965), 217-240. Zbl0128.26902MR34 #2540
  8. [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. [9] M. LABORDE, Buchstab's sifting weights, Mathematika, 26 (1979), 250-257. Zbl0429.10028MR82m:10070
  10. [10] R. A. RANKIN, Van der Corput's method and the theory of exponent pairs, Quart. J. Oxford, (2) 6 (1955), 147-153. Zbl0065.27802MR17,240a
  11. [11] H.-E. RICHERT, Selberg's sieve with weights, Mathematika, 16 (1969), 1-22. Zbl0192.39703MR40 #119
  12. [12] E. C. TITCHMARSH, The theory of the Riemann Zeta-Function, Oxford 1951. Zbl0042.07901MR13,741c

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