# 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}
}```
• D. Polymath
• Published 2014
• Mathematics
• Research in the Mathematical Sciences
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… Expand
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#### References

SHOWING 1-10 OF 74 REFERENCES
A note on the theorem of Maynard and Tao
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 interestingExpand
On the distribution of gaps between consecutive primes
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 anExpand
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. WeExpand
On the Ratio of Consecutive Gaps Between Primes
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 consecutiveExpand
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.Expand
• Mathematics, Computer Science
• ANTS
• 1998
The behavior of ρ *(x), in particular, the point at which ρ (x) first exceeds π (x), and its asymptotic growth is examined. Expand
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 someExpand
The existence of small prime gaps in subsets of the integers
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 inExpand
Primes in tuples I
• Mathematics
• 2009
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 theExpand
Bounded gaps between primes with a given primitive root
Fix an integer \$g \neq -1\$ that is not a perfect square. In 1927, Artin conjectured that there are infinitely many primes for which \$g\$ is a primitive root. Forty years later, Hooley showed thatExpand