Small gaps between primes

Abstract

We introduce a refinement of the GPY sieve method for studying prime $k$-tuples and small gaps between primes. This refinement avoids previous limitations of the method and allows us to show that for each $k$, the prime $k$-tuples conjecture holds for a positive proportion of admissible $k$-tuples. In particular, $\liminf_{n}(p_{n+m}-p_n)<\infty$ for every integer $m$. We also show that $\liminf(p_{n+1}-p_n)\le 600$ and, if we assume the Elliott-Halberstam conjecture, that $\liminf_n(p_{n+1}-p_n)\le 12$ and $\liminf_n (p_{n+2}-p_n)\le 600$.

  • [ElliottHalberstam] P. D. T. A. Elliott and H. Halberstam, "A conjecture in prime number theory," in Symposia Mathematica, Vol. IV, London: Academic Press, 1970, pp. 59-72.
    @incollection {ElliottHalberstam, MRKEY = {0276195},
      AUTHOR = {Elliott, P. D. T. A. and Halberstam, H.},
      TITLE = {A conjecture in prime number theory},
      BOOKTITLE = {Symposia {M}athematica, {V}ol. {IV}},
      VENUE={{INDAM},
      {R}ome, 1968/69},
      PAGES = {59--72},
      PUBLISHER = {Academic Press},
      ADDRESS = {London},
      YEAR = {1970},
      MRCLASS = {10.64},
      MRNUMBER = {0276195},
      MRREVIEWER = {K. Thanigasalam},
      zblnumber = {0238.10030},
      }
  • [FriedlanderGranville] Go to document J. Friedlander and A. Granville, "Limitations to the equi-distribution of primes. I," Ann. of Math., vol. 129, iss. 2, pp. 363-382, 1989.
    @article {FriedlanderGranville, MRKEY = {0986796},
      AUTHOR = {Friedlander, John and Granville, Andrew},
      TITLE = {Limitations to the equi-distribution of primes. {I}},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {129},
      YEAR = {1989},
      NUMBER = {2},
      PAGES = {363--382},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11N05},
      MRNUMBER = {0986796},
      MRREVIEWER = {S. W. Graham},
      DOI = {10.2307/1971450},
      zblnumber = {0671.10041},
      }
  • [GGPY] Go to document D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yildirim, "Small gaps between products of two primes," Proc. Lond. Math. Soc., vol. 98, iss. 3, pp. 741-774, 2009.
    @article {GGPY, MRKEY = {2500871},
      AUTHOR = {Goldston, D. A. and Graham, S. W. and Pintz, J. and Yıldırım, C. Y.},
      TITLE = {Small gaps between products of two primes},
      JOURNAL = {Proc. Lond. Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {98},
      YEAR = {2009},
      NUMBER = {3},
      PAGES = {741--774},
      ISSN = {0024-6115},
      MRCLASS = {11N25 (11N36)},
      MRNUMBER = {2500871},
      MRREVIEWER = {Frank Henry Thorne},
      DOI = {10.1112/plms/pdn046},
      zblnumber = {1213.11171},
      }
  • [GPY:ManyPrimes] Go to document D. A. Goldston, J. Pintz, and C. Y. Yildirim, "Primes in tuples. III. On the difference $p_{n+\nu}-p_n$," Funct. Approx. Comment. Math., vol. 35, pp. 79-89, 2006.
    @article {GPY:ManyPrimes, MRKEY = {2271608},
      AUTHOR = {Goldston, Daniel A. and Pintz, J{á}nos and Yıldırım, Cem Yal{ç}ım},
      TITLE = {Primes in tuples. {III}. {O}n the difference {$p\sb {n+\nu}-p\sb n$}},
      JOURNAL = {Funct. Approx. Comment. Math.},
      FJOURNAL = {Uniwersytet im. Adama Mickiewicza w Poznaniu. Wydzia\l Matematyki i Informatyki. Functiones et Approximatio Commentarii Mathematici},
      VOLUME = {35},
      YEAR = {2006},
      PAGES = {79--89},
      ISSN = {0208-6573},
      MRCLASS = {11N05 (11N13)},
      MRNUMBER = {2271608},
      MRREVIEWER = {S. W. Graham},
      DOI = {10.7169/facm/1229442618},
      zblnumber = {1196.11123},
      }
  • [GPY] Go to document D. A. Goldston, J. Pintz, and C. Y. Yildirim, "Primes in tuples. I," Ann. of Math., vol. 170, iss. 2, pp. 819-862, 2009.
    @article {GPY, MRKEY = {2552109},
      AUTHOR = {Goldston, Daniel A. and Pintz, J{á}nos and Yıldırım, Cem Y.},
      TITLE = {Primes in tuples. {I}},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {170},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {819--862},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11N05 (11N36)},
      MRNUMBER = {2552109},
      MRREVIEWER = {S. W. Graham},
      DOI = {10.4007/annals.2009.170.819},
      zblnumber = {1207.11096},
      }
  • [GoldstonYildirim] Go to document D. A. Goldston and C. Y. Yildirim, "Higher correlations of divisor sums related to primes. III. Small gaps between primes," Proc. Lond. Math. Soc., vol. 95, iss. 3, pp. 653-686, 2007.
    @article {GoldstonYildirim, MRKEY = {2368279},
      AUTHOR = {Goldston, D. A. and Yıldırım, C. Y.},
      TITLE = {Higher correlations of divisor sums related to primes. {III}. {S}mall gaps between primes},
      JOURNAL = {Proc. Lond. Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {95},
      YEAR = {2007},
      NUMBER = {3},
      PAGES = {653--686},
      ISSN = {0024-6115},
      MRCLASS = {11N05 (11N37)},
      MRNUMBER = {2368279},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.1112/plms/pdm021},
      zblnumber = {1118.11040},
      }
  • [Polymath] D. H. J. Polymath, New equidistribution estimates of Zhang type, and bounded gaps between primes.
    @misc{Polymath,
      author={Polymath, D. H. J.},
      TITLE={New equidistribution estimates of {Z}hang type, and bounded gaps between primes},
      NOTE={preprint},
      ARXIV = {1402.0811},
      }
  • [Selberg:CollectedWorks] A. Selberg, Collected Papers. Vol. II, New York: Springer-Verlag, 1991.
    @book {Selberg:CollectedWorks, MRKEY = {1295844},
      AUTHOR = {Selberg, Atle},
      TITLE = {Collected Papers. {V}ol. {II}},
      NOTE = {with a foreword by K. Chandrasekharan},
      PUBLISHER = {Springer-Verlag},
      YEAR = {1991},
      PAGES = {viii+253},
      ISBN = {3-540-50626-8},
      MRCLASS = {01A75},
      MRNUMBER = {1295844},
      MRREVIEWER = {Matti Jutila},
      ADDRESS = {New York},
      zblnumber = {0729.11001},
      }
  • [Zhang] Go to document Y. Zhang, "Bounded gaps between primes," Ann. of Math., vol. 179, iss. 3, pp. 1121-1174, 2014.
    @article {Zhang, MRKEY = {3171761},
      AUTHOR = {Zhang, Yitang},
      TITLE = {Bounded gaps between primes},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {179},
      YEAR = {2014},
      NUMBER = {3},
      PAGES = {1121--1174},
      ISSN = {0003-486X},
      MRCLASS = {11N05 (11N36)},
      MRNUMBER = {3171761},
      DOI = {10.4007/annals.2014.179.3.7},
      zblnumber = {06302171},
      }

Authors

James Maynard

Magdalen College, Oxford, Oxford OX1 4AU, United Kingdom