Primes in tuples I

@article{Goldston2009PrimesIT,
  title={Primes in tuples I},
  author={D. A. Goldston and Janos Pintz and C. Y. Yildirim},
  journal={Annals of Mathematics},
  year={2009},
  volume={170},
  pages={819-862}
}
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 Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any… 

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References

SHOWING 1-10 OF 40 REFERENCES

SMALL GAPS BETWEEN PRIMES OR ALMOST PRIMES

Let p n denote the n th prime. Goldston, Pintz, and Yildirim recently proved that li m in f (pn+1 ― p n ) n→∞ log p n = 0. We give an alternative proof of this result. We also prove some

The difference of consecutive primes

is greater than (c2/2)n. A simple calculation now shows that the primes satisfying (4) also satisfy the first inequality of (3) i΀ = e(ci) is chosen small enough. The second inequality of (3) is

ON THE DISTRIBUTION OF PRIMES IN SHORT INTERVALS

One of the formulations of the prime number theorem is the statement that the number of primes in an interval (n, n + ft], averaged over n ^ JV, tends to the limit A, when JV and h tend to infinity

Small gaps between primes exist

In the recent preprint (3), Goldston, Pintz, and Yoldorom established, among other things, (0) liminf n→∞ pn+1 − pn log pn = 0, with pn the nth prime. In the present article, which is essentially

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

A heuristic asymptotic formula concerning the distribution of prime numbers

Suppose fi, f2, -*, fk are polynomials in one variable with all coefficients integral and leading coefficients positive, their degrees being hi, h2, **. , hk respectively. Suppose each of these

On Bombieri and Davenport's theorem concerning small gaps between primes

§1. Introduction . In this paper we give a new proof of a theorem of Bombieri and Davenport [2, Theorem 1]. Let t (– k ) = t ( k ) be real, where e ( u ) = e 2 πiu . Let p and p ' denote primes, k an

The difference between consecutive prime numbers. II

  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1940
In a previous paper, under the same title, I considered the problem of how far apart two consecutive primes can be. The present paper is concerned with the opposite question. How near together can

Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim

In early 2005, Dan Goldston, Janos Pintz, and Cem Yildirim [12] made a spectacular breakthrough in the study of prime numbers. Resolving a long-standing open problem, they proved that there are

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