There’s been some progress on the bounded gaps between primes front since we last checked in.

The Polymath8 project has got the gap down to $4,680$. But that’s small beans: James Maynard, a postgrad student at Oxford, announced at a meeting in Oberwolfach that he has got the gap down to $700$. Emmanuel Kowalski has written an effusive post on his blog singing the praises of Maynard’s achievement.

Meanwhile, Andrew Granville is writing a survey article on “gaps between primes”. It explains everything involved in getting to Zhang’s result, starting at the prime number theorem that $\pi(x) \sim \frac{x}{\ln x}$. Granville has uploaded a draft of that survey, mainly for the benefit of the people writing the Polymath8 paper, to his website.

More information

Polymath8 project page with the table of best upper bounds for gaps.

James Maynard’s arXiv preprints

“James Maynard, auteur du théorème de l’année” at E. Kowalski’s blog, explaining Maynard’s result. (via David Roberts on Google+)

“Bounded gaps between primes” – Andrew Granville’s survey paper (draft).

“A new bound for gaps between primes” by ‘D.H.J. Polymath” – the Polymath8 project’s write-up of their results (draft).

One of the many very long comment threads on Terry Tao’s blog discussing the Polymath8 paper.