# Variants of the Selberg sieve, and bounded intervals containing many primes

@article{Polymath2014VariantsOT, title={Variants of the Selberg sieve, and bounded intervals containing many primes}, author={D. H. J. Polymath}, journal={Research in the Mathematical Sciences}, year={2014}, volume={1}, pages={1-83} }

For any m≥1, let Hm denote the quantity liminfn→∞(pn+m−pn). A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1≤70,000,000. This was then improved by us (the Polymath8 project) to H1≤4680, and then by Maynard to H1≤600, who also established for the first time a finiteness result for Hm for m≥2, and specifically that Hm≪m3e4m. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound H1≤12, improving upon the previous bound H1≤16 of…

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## References

SHOWING 1-10 OF 74 REFERENCES

A note on the theorem of Maynard and Tao

- Mathematics
- 2015

In a recent and much celebrated breakthrough, Maynard and Tao have independently proved a certain approximation to the prime k-tuple conjecture. We have subsequently seen numerous interesting…

On the distribution of gaps between consecutive primes

- Mathematics
- 2014

Erd\"os conjectured that the set J of limit points of d_n/logn contains all nonnegative numbers, where d_n denotes the nth primegap. The author proved a year ago (arXiv: 1305.6289) that J contains an…

Higher correlations of divisor sums related to primes II: variations of the error term in the prime number theorem

- Mathematics
- 2004

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…

On the Ratio of Consecutive Gaps Between Primes

- Mathematics
- 2015

In the present work we prove a common generalization of Maynard–Tao’s recent result about consecutive bounded gaps between primes and of the Erdős–Rankin bound about large gaps between consecutive…

A smoothed GPY sieve

- Mathematics
- 2008

Combining the arguments developed in the works of D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yildirim [Preprint, 2005, arXiv: math.NT/506067] and Y. Motohashi [Number theory in progress – A.…

Prime Numbers in Short Intervals and a Generalized Vaughan Identity

- MathematicsCanadian Journal of Mathematics
- 1982

1. Introduction. Many problems involving prime numbers depend on estimating sums of the form ΣΛ(n)f(n), for appropriate functions f(n), (here, as usual, Λ(n) is the von Mangoldt function). Three…

Bounded length intervals containing two primes and an almost‐prime

- Mathematics
- 2013

Goldston, Pintz and Yıldırım have shown that if the primes have ‘level of distribution’ θ for some θ > ½, then there exists a constant C(θ), such that there are infinitely many integers n for which…

Dense Admissible Sets

- MathematicsANTS
- 1998

The behavior of ρ *(x), in particular, the point at which ρ (x) first exceeds π (x), and its asymptotic growth is examined.

SMALL GAPS BETWEEN PRIMES OR ALMOST PRIMES

- Mathematics
- 2005

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 existence of small prime gaps in subsets of the integers

- Mathematics
- 2013

We consider the problem of finding small prime gaps in various sets . Following the work of Goldston–Pintz–Yildirim, we will consider collections of natural numbers that are well-controlled in…