Bounded gaps between primes

Abstract

It is proved that $$ \liminf_{n\to\infty}(p_{n+1}-p_n)<7\times 10^7, $$ where $p_n$ is the $n$-th prime.
Our method is a refinement of the recent work of Goldston, Pintz and Yıldırım on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only, but it is adequate for our purpose.

  • [1] Go to document E. Bombieri, J. B. Friedlander, and H. Iwaniec, "Primes in arithmetic progressions to large moduli," Acta Math., vol. 156, iss. 3-4, pp. 203-251, 1986.
    @article {1, MRKEY = {0834613},
      AUTHOR = {Bombieri, E. and Friedlander, J. B. and Iwaniec, H.},
      TITLE = {Primes in arithmetic progressions to large moduli},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {156},
      YEAR = {1986},
      NUMBER = {3-4},
      PAGES = {203--251},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {11N13},
      MRNUMBER = {0834613},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.1007/BF02399204},
      ZBLNUMBER = {0588.10042},
     }
  • [2] Go to document E. Bombieri, J. B. Friedlander, and H. Iwaniec, "Primes in arithmetic progressions to large moduli. II," Math. Ann., vol. 277, iss. 3, pp. 361-393, 1987.
    @article {2, MRKEY = {0891581},
      AUTHOR = {Bombieri, E. and Friedlander, J. B. and Iwaniec, H.},
      TITLE = {Primes in arithmetic progressions to large moduli. {II}},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {277},
      YEAR = {1987},
      NUMBER = {3},
      PAGES = {361--393},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {11N13},
      MRNUMBER = {0891581},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.1007/BF01458321},
      ZBLNUMBER = {0625.10036},
     }
  • [3] Go to document E. Bombieri, J. B. Friedlander, and H. Iwaniec, "Primes in arithmetic progressions to large moduli. III," J. Amer. Math. Soc., vol. 2, iss. 2, pp. 215-224, 1989.
    @article {3, MRKEY = {0976723},
      AUTHOR = {Bombieri, E. and Friedlander, J. B. and Iwaniec, H.},
      TITLE = {Primes in arithmetic progressions to large moduli. {III}},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {2},
      YEAR = {1989},
      NUMBER = {2},
      PAGES = {215--224},
      ISSN = {0894-0347},
      MRCLASS = {11N13},
      MRNUMBER = {0976723},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.2307/1990976},
      ZBLNUMBER = {0674.10036},
      }
  • [4] Go to document P. Deligne, "La conjecture de Weil. I," Inst. Hautes Études Sci. Publ. Math., vol. 43, pp. 273-307, 1974.
    @article {4, MRKEY = {0340258},
      AUTHOR = {Deligne, Pierre},
      TITLE = {La conjecture de {W}eil. {I}},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      VOLUME = {43},
      YEAR = {1974},
      PAGES = {273--307},
      ISSN = {0073-8301},
      MRCLASS = {14G13},
      MRNUMBER = {0340258},
      MRREVIEWER = {Nicholas M. Katz},
      ZBLNUMBER = {0287.14001},
      DOI = {10.1007/BF02684373},
      }
  • [5] Go to document E. Fouvry and H. Iwaniec, "Primes in arithmetic progressions," Acta Arith., vol. 42, iss. 2, pp. 197-218, 1983.
    @article {5, MRKEY = {0719249},
      AUTHOR = {Fouvry, E. and Iwaniec, H.},
      TITLE = {Primes in arithmetic progressions},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
      VOLUME = {42},
      YEAR = {1983},
      NUMBER = {2},
      PAGES = {197--218},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {10H15 (10H20)},
      MRNUMBER = {0719249},
      MRREVIEWER = {Matti Jutila},
      ZBLNUMBER = {0517.10045},
      URL = {http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-aav42i2p197bwm?q=bwmeta1.element.bwnjournal-number-aa-1982-1983-42-2;5&qt=CHILDREN-STATELESS},
     }
  • [6] Go to document J. B. Friedlander and H. Iwaniec, "Incomplete Kloosterman sums and a divisor problem," Ann. of Math., vol. 121, iss. 2, pp. 319-350, 1985.
    @article {6, MRKEY = {0786351},
      AUTHOR = {Friedlander, John B. and Iwaniec, Henryk},
      TITLE = {Incomplete {K}loosterman sums and a divisor problem},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {121},
      YEAR = {1985},
      NUMBER = {2},
      PAGES = {319--350},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11N25 (11L40 11N69)},
      MRNUMBER = {0786351},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.2307/1971175},
      ZBLNUMBER = {0572.10029},
     }
  • [7] 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 {7, 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},
     }
  • [8] Go to document D. A. Goldston, J. Pintz, and C. Y. Yildirim, "Primes in tuples. II," Acta Math., vol. 204, iss. 1, pp. 1-47, 2010.
    @article {8, MRKEY = {2600432},
      AUTHOR = {Goldston, Daniel A. and Pintz, J{á}nos and Yıldırım, Cem Yal{ç}in},
      TITLE = {Primes in tuples. {II}},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {204},
      YEAR = {2010},
      NUMBER = {1},
      PAGES = {1--47},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {11N05 (11N36)},
      MRNUMBER = {2600432},
      MRREVIEWER = {S. W. Graham},
      DOI = {10.1007/s11511-010-0044-9},
      ZBLNUMBER = {1207.11097},
     }
  • [9] Go to document D. R. Heath-Brown, "Prime numbers in short intervals and a generalized Vaughan identity," Canad. J. Math., vol. 34, iss. 6, pp. 1365-1377, 1982.
    @article {9, MRKEY = {0678676},
      AUTHOR = {Heath-Brown, D. R.},
      TITLE = {Prime numbers in short intervals and a generalized {V}aughan identity},
      JOURNAL = {Canad. J. Math.},
      FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques},
      VOLUME = {34},
      YEAR = {1982},
      NUMBER = {6},
      PAGES = {1365--1377},
      ISSN = {0008-414X},
      CODEN = {CJMAAB},
      MRCLASS = {10H15 (10H25)},
      MRNUMBER = {0678676},
      MRREVIEWER = {Kenneth A. Jukes},
      DOI = {10.4153/CJM-1982-095-9},
      ZBLNUMBER = {0478.10024},
     }
  • [10] Go to document D. R. Heath-Brown, "The divisor function $d_3(n)$ in arithmetic progressions," Acta Arith., vol. 47, iss. 1, pp. 29-56, 1986.
    @article {10, MRKEY = {0866901},
      AUTHOR = {Heath-Brown, D. R.},
      TITLE = {The divisor function {$d\sb 3(n)$} in arithmetic progressions},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
      VOLUME = {47},
      YEAR = {1986},
      NUMBER = {1},
      PAGES = {29--56},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {11N25 (11N37)},
      MRNUMBER = {0866901},
      MRREVIEWER = {A. J. Hildebrand},
      ZBLNUMBER = {0549.10034},
      URL = {http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-aav47i1p29bwm?q=bwmeta1.element.bwnjournal-number-aa-1986-47-1;2&qt=CHILDREN-STATELESS},
      }
  • [11] Go to document H. Iwaniec, "A new form of the error term in the linear sieve," Acta Arith., vol. 37, pp. 307-320, 1980.
    @article {11, MRKEY = {0598883},
      AUTHOR = {Iwaniec, Henryk},
      TITLE = {A new form of the error term in the linear sieve},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
      VOLUME = {37},
      YEAR = {1980},
      PAGES = {307--320},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {10H30},
      MRNUMBER = {0598883},
      MRREVIEWER = {H.-E. Richert},
      ZBLNUMBER = {0444.10038},
      URL = {http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-aav37i1p307bwm?q=bwmeta1.element.bwnjournal-number-aa-1980-37-1;26&qt=CHILDREN-STATELESS},
      }
  • [12] Go to document P. Shiu, "A Brun-Titchmarsh theorem for multiplicative functions," J. Reine Angew. Math., vol. 313, pp. 161-170, 1980.
    @article {12, MRKEY = {0552470},
      AUTHOR = {Shiu, P.},
      TITLE = {A {B}run-{T}itchmarsh theorem for multiplicative functions},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {313},
      YEAR = {1980},
      PAGES = {161--170},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {10H25},
      MRNUMBER = {0552470},
      MRREVIEWER = {A. I. Vinogradov},
      DOI = {10.1515/crll.1980.313.161},
      ZBLNUMBER = {0412.10030},
      }
  • [13] Go to document K. Soundararajan, "Small gaps between prime numbers: the work of Goldston-Pintz-Y\i ld\i r\i m," Bull. Amer. Math. Soc., vol. 44, iss. 1, pp. 1-18, 2007.
    @article {13, MRKEY = {2265008},
      AUTHOR = {Soundararajan, K.},
      TITLE = {Small gaps between prime numbers: the work of {G}oldston-{P}intz-{Y}\i ld\i r\i m},
      JOURNAL = {Bull. Amer. Math. Soc.},
      FJOURNAL = {American Mathematical Society. Bulletin. New Series},
      VOLUME = {44},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {1--18},
      ISSN = {0273-0979},
      CODEN = {BAMOAD},
      MRCLASS = {11N05 (11-02 11N36)},
      MRNUMBER = {2265008},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.1090/S0273-0979-06-01142-6},
      ZBLNUMBER = {1193.11086},
     }

Authors

Yitang Zhang

Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824