Small gaps between primes

@article{Maynard2013SmallGB,
  title={Small gaps between primes},
  author={James Maynard},
  journal={Annals of Mathematics},
  year={2013},
  volume={181},
  pages={383-413}
}
  • J. Maynard
  • Published 19 November 2013
  • Mathematics
  • Annals of Mathematics
We introduce a renement of the GPY sieve method for studying prime k-tuples and small gaps between primes. This renement 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, lim infn(pn+m pn) <1 for every integer m. We also show that lim inf(pn+1 pn) 600 and, if we assume the Elliott-Halberstam conjecture, that lim infn(pn+1 pn) 12 and lim infn(pn+2 pn) 600. 
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173 Citations
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References

SHOWING 1-10 OF 13 REFERENCES
Primes in tuples I
We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the…
Bounded gaps between primes
It is proved that lim inf n?8 (p n+1 -p n )<7×10 7 , where p n is the n -th prime. Our method is a refinement of the recent work of Goldston, Pintz and Yildirim on the small gaps between…
Small gaps between products of two primes
Let qn denote the nth number that is a product of exactly two distinct primes. We prove that qn+1 − qn ⩽ 6 infinitely often. This sharpens an earlier result of the authors, which had 26 in place of…
Higher correlations of divisor sums related to primes III: small gaps between primes
We use divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η > 0, a…
Limitations to the equi-distribution of primes I
In an earlier paper FG] we showed that the expected asymptotic formula (x; q; a) (x)==(q) does not hold uniformly in the range q < x= log N x, for any xed N > 0. There are several reasons to suspect…
Higher correlations of divisor sums related to primes II: variations of the error term in the prime number theorem
We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We…
Collected Papers
THIS volume is the first to be produced of the projected nine volumes of the collected papers of the late Prof. H. A. Lorentz. It contains a number of papersnineteen in all, mainly printed…
Primes in tup les
  • I.Ann. of Math. (2) , 170(2):819–862
  • 2009
A conjecture in prime number theory
  • Symposia Mathematica, Vol. IV
  • 1968
A conjecture in prim e number theory
  • InSymposia Mathematica, Vol. IV
  • 1968
...
...