Primes in intervals of bounded length

TLDR: In 2014, Zhang as mentioned in this paper proved the existence of a finite bound B such that there are infinitely many pairs of distinct primes which differ by no more than B. This is a massive breakthrough, makes the twin prime conjecture look highly plausible (which can be reinterpreted as the conjecture that one can take B 2) and his work helps us to better understand other delicate questions about prime numbers that had previously seemed intractable.

Abstract: In April 2013, Yitang Zhang proved the existence of a finite bound B such that there are infinitely many pairs of distinct primes which differ by no more than B. This is a massive breakthrough, makes the twin prime conjecture look highly plausible (which can be re-interpreted as the conjecture that one can take B 2) and his work helps us to better understand other delicate questions about prime numbers that had previously seemed intractable. The original purpose of this talk was to discuss Zhang’s extraordinary work, putting it in its context in analytic number theory, and to sketch a proof of his theorem. Zhang had even proved the result with B 70 000 000. Moreover, a co-operative team, polymath8, collaborating only on-line, had been able to lower the value of B to 4680. Not only had they been more careful in several difficult arguments in Zhang’s original paper, they had also developed Zhang’s techniques to be both more powerful and to allow a much simpler proof. Indeed the proof of Zhang’s Theorem, that will be given in the write-up of this talk, is based on these developments. In November, inspired by Zhang’s extraordinary breakthrough, James Maynard dramatically slashed this bound to 600, by a substantially easier method. Both Maynard, and Terry Tao who had independently developed the same idea, were able to extend their proofs to show that for any given integer m ¥ 1 there exists a bound Bm such that there are infinitely many intervals of length Bm containing at least m distinct primes. We will also prove this much stronger result herein, even showing that one can take Bm e 8m . If Zhang’s method is combined with the Maynard-Tao set up then it appears that the bound can be further reduced to 576. If all of these techniques could be pushed to their limit then we would obtain B( B2) 12, so new ideas are still needed to have a feasible plan for proving the twin prime conjecture. The article will be split into two parts. The first half, which appears here, we will introduce the work of Zhang, Polymath8, Maynard and Tao, and explain their arguments that allow them to prove their spectacular results. As we will discuss, Zhang’s main novel contribution is an estimate for primes in relatively short arithmetic progressions. The second half of this article sketches a proof of this result; the Bulletin article will contain full details of this extraordinary work. Part 1. Primes in short intervals