Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Small gaps between primes or almost primes

Authors: D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim
Journal: Trans. Amer. Math. Soc. 361 (2009), 5285-5330
MSC (2000): Primary 11N25; Secondary 11N36
Published electronically: May 27, 2009
MathSciNet review: 2515812
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ p_n$ denote the $ n^{{\rm th}}$ prime. Goldston, Pintz, and Yıldırım recently proved that

$\displaystyle \liminf_{n\to \infty} \frac{(p_{n+1}-p_n)}{\log p_n} =0. $

We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let $ q_n$ denote the $ n^{{\rm th}}$ number that is a product of exactly two distinct primes. We prove that

$\displaystyle \liminf_{n\to \infty} (q_{n+1}-q_n) \le 26. $

If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to $ 6$.

References [Enhancements On Off] (What's this?)

  • 1. E. Bombieri, On the large sieve, Mathematika 12 (1965), 201–225. MR 0197425
  • 2. E. Bombieri and H. Davenport, Small differences between prime numbers, Proc. Roy. Soc. Ser. A 293 (1966), 1–18. MR 0199165
  • 3. Jing Run Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), 157–176. MR 0434997
  • 4. Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR 606931
  • 5. L.E. Dickson, History of the Theory of Numbers, Vol. I, Chelsea, New York.
  • 6. P. D. T. A. Elliott and H. Halberstam, A conjecture in prime number theory, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London, 1970, pp. 59–72. MR 0276195
  • 7. P. Erdös, The difference of consecutive primes, Duke Math. J. 6 (1940), 438–441. MR 0001759
  • 8. P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976), no. 1, 4–9. MR 0409385
  • 9. D.A. Goldston and C.Y. Yıldırım, Higher correlations of divisor sums related to primes III: Small gaps between primes, Proc. London Math. Soc., to appear.
  • 10. D. A. Goldston, J. Pintz, and C.Y. Yıldırım, Primes in tuples I, Annals of Mathematics, to appear.
  • 11. D. A. Goldston, J. Pintz, and C.Y. Yıldırım, Primes in tuples II, preprint.
  • 12. H. Halberstam and H.-E. Richert, Sieve methods, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1974. London Mathematical Society Monographs, No. 4. MR 0424730
  • 13. Heini Halberstam and Klaus Friedrich Roth, Sequences, 2nd ed., Springer-Verlag, New York-Berlin, 1983. MR 687978
  • 14. G. H. Hardy and J. E. Littlewood, Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes, Acta Math. 44 (1923), no. 1, 1–70. MR 1555183, 10.1007/BF02403921
  • 15. G. H. Hardy and J. E. Littlewood, Some problems of `Partitio Numerorum': VII, unpublished manuscript; see [24].
  • 16. D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika 31 (1984), no. 1, 141–149. MR 762186, 10.1112/S0025579300010743
  • 17. D. R. Heath-Brown, Almost-prime 𝑘-tuples, Mathematika 44 (1997), no. 2, 245–266. MR 1600529, 10.1112/S0025579300012584
  • 18. Adolf Hildebrand, Über die punktweise Konvergenz von Ramanujan-Entwicklungen zahlentheoretischer Funktionen, Acta Arith. 44 (1984), no. 2, 109–140 (German). MR 774094
  • 19. Martin Huxley, An application of the Fouvry-Iwaniec theorem, Acta Arith. 43 (1984), no. 4, 441–443. MR 756293
  • 20. Helmut Maier, Small differences between prime numbers, Michigan Math. J. 35 (1988), no. 3, 323–344. MR 978303, 10.1307/mmj/1029003814
  • 21. Hugh L. Montgomery and Robert C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. MR 2378655
  • 22. Yoichi Motohashi, An induction principle for the generalization of Bombieri’s prime number theorem, Proc. Japan Acad. 52 (1976), no. 6, 273–275. MR 0422179
  • 23. A. de Polignac, Six propostions arithmologiques déduites de crible d'Ératosthène, Nouv. Ann. Math. 8 (1849), 423-429.
  • 24. R. A. Rankin, The difference between consecutive prime numbers. II, Proc. Cambridge Philos. Soc. 36 (1940), 255–266. MR 0001760
  • 25. J.-C. Schlage-Puchta, The equation 𝜔(𝑛)=𝜔(𝑛+1), Mathematika 50 (2003), no. 1-2, 99–101 (2005). MR 2136354, 10.1112/S0025579300014820
  • 26. Atle Selberg, Collected papers. Vol. II, Springer-Verlag, Berlin, 1991. With a foreword by K. Chandrasekharan. MR 1295844
  • 27. R. C. Vaughan, An elementary method in prime number theory, Acta Arith. 37 (1980), 111–115. MR 598869
  • 28. A.I. Vinogradov, On the density hypothesis for Dirichlet L-functions, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 903-934. Corrigendum, loc. cit. 30 (1966), 719-720.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11N25, 11N36

Retrieve articles in all journals with MSC (2000): 11N25, 11N36

Additional Information

D. A. Goldston
Affiliation: Department of Mathematics, San Jose State University, San Jose, California 95192

S. W. Graham
Affiliation: Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan 48859

J. Pintz
Affiliation: Rényi Mathematical Institute of the Hungarian Academy of Sciences, H-1053 Budapest, Realtanoda u. 13–15, Hungary

C. Y. Yildirim
Affiliation: Department of Mathematics, Bogaziçi University, Istanbul 34342, Turkey – and – Feza Gürsey Enstitüsü, Çengelköy, Istanbul, P.K. 6, 81220, Turkey

Keywords: Primes, almost primes, gaps, Selberg's sieve, applications of sieve methods
Received by editor(s): September 17, 2007
Published electronically: May 27, 2009
Additional Notes: The first author was supported by NSF grant DMS-0300563, the NSF Focused Research Group grant 0244660, and the American Institute of Mathematics.
The second author was supported by a sabbatical leave from Central Michigan University and by NSF grant DMS-070193.
The third author was supported by OTKA grants No. 43623, 49693, 67676 and the Balaton program.
The fourth author was supported by TÜBİTAK
Article copyright: © Copyright 2009 American Mathematical Society