Primes in tuples IV: Density of small gaps between consecutive primes

Volume 160 / 2013

Daniel Alan Goldston, János Pintz, Cem Yalçın Yıldırım Acta Arithmetica 160 (2013), 37-53 MSC: Primary 11N05; Secondary 11N36. DOI: 10.4064/aa160-1-3


We prove that given any small but fixed $\eta > 0$, a positive proportion of all gaps between consecutive primes are smaller than $\eta $ times the average gap. We show some unconditional and conditional quantitative results in this vein. In the results the dependence on $\eta $ is given explicitly, providing a new quantitative way, in addition to that of the first paper in this series, of measuring the effect of the knowledge on the level of distribution of primes.


  • Daniel Alan GoldstonDepartment of Mathematics
    San Jose State University
    San Jose, CA 95192, U.S.A.
  • János PintzRényi Mathematical Institute
    of the Hungarian Academy of Sciences
    H-1364 Budapest, Pf 127, Hungary
  • Cem Yalçın YıldırımDepartment of Mathematics
    Boğaziçi University
    Bebek, İstanbul 34342, Turkey

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