Primes in tuples I

• 成果类型: Article
• 署名作者: Goldston, Daniel A.; Pintz, Janos; Yildirim, Cem Y.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2009.170.819
• 发表日期: 2009
• 页码: 819-862
• 关键词: small gaps; numbers

摘要

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 arbitrarily small multiple of the average spacing, that is, lim inf(n -> infinity) p(n+1) - p(n)/log p(n) = 0. We will quantify this result further in a later paper.