I just found out via Terry Tao’s blog (http://terrytao.wordpress.com/) that a new improvement for bounded gaps between primes have been announced, due to James Maynard. In particular, one can now show that for infinitely many primes the interval contains at least two primes. Further, Maynard achieved a fundamental advance in the form that no assumptions on higher level of distribution of primes is necessary; specifically, that the Bombieri-Vinogradov theorem suffices to establish the existence of bounded gaps between primes.

Perhaps more importantly, Maynard proved that for any , we have

which is the first finite bound for the limit inferior of gaps of this type for .

Hopefully by the time the talks begin or even by the organizational meeting the paper of Maynard would appear on arXiv at least, and we will add it to the list of talks.

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