Small gaps between primes

2014/10/16 by James Maynard · 7 citations
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #History and Theory of Mathematics

paper · pdf · doi:10.4007/annals.2015.181.1.7

openalex publication_date 2014/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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.

Cited by

• Large gaps between consecutive prime numbers
• Could Euler Have Conjectured the Prime Number Theorem?
• Klaus Friedrich Roth. 29 October 1925 — 10 November 2015
• Obstructions to Uniformity and Arithmetic Patterns in the Primes
• The structure factor of primes
• Limit points of normalized prime gaps
• List of unsolved problems in mathematics [wikipedia]
• Sieve theory [wikipedia]

Related

• Small gaps between primes
• Gaps between primes
• Primes in Tuples IV: Density of small gaps between consecutive primes
• On the frequency of small gaps between the primes
• Sieve Method and Prime Gaps via Probabilistic Method