There’s a new paper out on the arXiv last night, Small gaps between primes, by James Maynard, which brings the bound on the size of gaps between primes down to 600. This uses some new methods, beating out the Polymath8 project, which has been improving Zhang’s original bound of 70,000,000, getting it down to 4680.
To follow the Polymath8 project, the place to look is Terence Tao’s blog, here. They’re working on a paper, with the current draft version available here. This is a remarkable collaborative project bringing together a sizable group of mathematicians in an unusual way.
For more about this, see this expository article by Andrew Granville, which is pre-Maynard. At Quanta magazine, Erica Klarreich has an excellent long popular article telling the story to date, including that of Maynard’s new result.