Annals of Mathematics· 2014Q1
Small gaps between primes
- 309citations
- Q1SCImago
- 2014year
Short summary
A refined sieve method demonstrates that the prime k-tuples conjecture holds for a positive proportion of admissible k-tuples, proving that the infimum of gaps between consecutive primes is bounded.
AI-generated from the title and abstract; the full text is not read.
Key points
- A refined GPY sieve method is introduced for prime k-tuple analysis.
- The prime k-tuples conjecture is shown to hold for a positive proportion of admissible k-tuples.
- The infimum of the difference between consecutive primes (lim inf(pn+m - pn)) is proven to be bounded for every integer m.
- Under the Elliott-Halberstam conjecture, the infimum of gaps between consecutive primes is shown to be 12, and the infimum of gaps between primes separated by one prime is 600.
AI-generated from the title and abstract; the full text is not read.
Abstract
We introduce a refinement of the GPY sieve method for studying prime k-tuples and small gaps between primes. This refinement 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) < 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.
The authors' abstract, as published at the source. Annals of Mathematics, 2014 · DOI ↗
Continue with a free account
Ask the paper: 3 free questions a day about this paper; save it, get its citation, new summaries every day for your field. Takeaways are Premium.
Continue free on the webSign in with Google or Apple; no card needed. You come back to this paper.
On your phone:
Field: Algebra and Number Theory
Algebra and Number TheoryMathematics