Bounded prime gaps in short intervals

Abstract

We generalise Zhang's and Pintz recent results on bounded prime gaps to give a lower bound for the the number of prime pairs bounded by 6*10^7 in the short interval $[x,x+x(\log x{)}^{-A}]$. Our result follows only by analysing Zhang's proof of Theorem 1, but we also explain how a sharper variant of Zhang's Theorem 2 would imply the same result for shorter intervals.

v2: 6 pages. Fixed minor errors and added reference to polymath project

References in corpus (4)

• A Smoothed GPY Sieve
• Primes in Tuples II
• Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim
• A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million

Cited by in corpus (2)

• The "bounded gaps between primes" Polymath project - a retrospective
• Variants of the Selberg sieve, and bounded intervals containing many primes