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Almost primes in almost all short intervals

Published online by Cambridge University Press:  13 April 2016

JONI TERÄVÄINEN
JONI TERÄVÄINEN*
Affiliation:
Department of Mathematics, University of Turku, 20014 Turku, Finland. e-mail: [email protected]
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Abstract

Let Ek be the set of positive integers having exactly k prime factors. We show that almost all intervals [x, x + log1+ϵx] contain E 3 numbers, and almost all intervals [x,x + log3.51x] contain E 2 numbers. By this we mean that there are only o(X) integers 1 ⩽ x ⩽ X for which the mentioned intervals do not contain such numbers. The result for E 3 numbers is optimal up to the ϵ in the exponent. The theorem on E 2 numbers improves a result of Harman, which had the exponent 7 + ϵ in place of 3.51. We also consider general Ek numbers, and find them on intervals whose lengths approach log x as k → ∞.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 2 , September 2016 , pp. 247 - 281
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Baker, R. C., Harman, G. and Pintz, J. The difference between consecutive primes. II. Proc. London Math. Soc. (3), 83 (3) (2001), 532–562.CrossRefGoogle Scholar
[2] Bourgain, J. On large values estimates for Dirichlet polynomials and the density hypothesis for the Riemann zeta function. Internat. Math. Res. Notices (3) (2000), 133–146.CrossRefGoogle Scholar
[3] Freiberg, T. Short intervals with a given number of primes. J. Number Theory. 163 (2016), 159–171.CrossRefGoogle Scholar
[4] Friedlander, J. and Iwaniec, H. Opera de cribro. Amer. Math. Soc. Colloq. Pub. vol. 57 (American Mathematical Society, Providence, RI, 2010).CrossRefGoogle Scholar
[5] Gallagher, P. X. On the distribution of primes in short intervals. Mathematika. 23 (1) (1976), 4–9.CrossRefGoogle Scholar
[6] Goldston, D. A., Pintz, J. and Yildirim, C. Y. Positive proportion of small gaps between consecutive primes. Publ. Math. Debrecen. 79 (3–4) (2011), 433–444.CrossRefGoogle Scholar
[7] Goldston, D. A., Pintz, J. and Yildirim, C. Y. Primes in tuples IV: Density of small gaps between consecutive primes. Acta Arith. 160 (1) (2013), 37–53.CrossRefGoogle Scholar
[8] Hardy, G. H. and Ramanujan, S. The normal number of prime factors of a number n [Quart. J. Math. 48 (1917), 76–92]. In Collected papers of Srinivasa Ramanujan. (AMS Chelsea Publ., Providence, RI, 2000), pages 262–275.Google Scholar
[9] Harman, G. Almost-primes in short intervals. Math. Ann. 258 (1) (1981/82), 107–112.CrossRefGoogle Scholar
[10] Harman, G. Prime-detecting sieves. London Math. Soc. Monog. Series. vol. 33 (Princeton University Press, Princeton, NJ, 2007).Google Scholar
[11] Heath-Brown, D. R. Prime numbers in short intervals and a generalised Vaughan identity. Canad. J. Math. 34 (6) (1982), 1365–1377.CrossRefGoogle Scholar
[12] Iwaniec, H. and Kowalski, E. Analytic number theory. Amer. Math. Soc. Colloq. Pub. vol. 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
[13] Jia, C. Almost all short intervals containing prime numbers. Acta Arith. 76 (1) (1996), 21–84.CrossRefGoogle Scholar
[14] Jutila, M. Zero-density estimates for L-functions. Acta Arith. 32 (1) (1977), 55–62.CrossRefGoogle Scholar
[15] Matomäki, K. and Radziwiłł, M. Multiplicative functions in short intervals. To appear in Ann. of Math. Google Scholar
[16] Mikawa, H. Almost-primes in arithmetic progressions and short intervals. Tsukuba J. Math. 13 (2) (1989), 387–401.CrossRefGoogle Scholar
[17] Montgomery, H. L. Ten lectures on the interface between analytic number theory and harmonic analysis. CBMS Regional Conference Series in Mathematics. vol. 84 Published for the Conference Board of the Mathematical Sciences, Washington, DC (American Mathematical Society, Providence, RI, 1994).CrossRefGoogle Scholar
[18] Selberg, A. On the normal density of primes in small intervals, and the difference between consecutive primes. Arch. Math. Naturvid. 47 (6) (1943), 87–105.Google Scholar
[19] Watt, N. Kloosterman sums and a mean value for Dirichlet polynomials. J. Number Theory 53 (1) (1995), 179–210.CrossRefGoogle Scholar
[20] Watt, N. Short intervals almost all containing primes. Acta Arith. 72 (2) (1995), 131–167.CrossRefGoogle Scholar
[21] Wolke, D. Fast-Primzahlen in kurzen Intervallen. Math. Ann. 244 (3) (1979), 233–242.CrossRefGoogle Scholar