Can someone offer an overview of the idea of the proof of prime gaps bounds?

Question

I saw on a YouTube video that Yitang Zhang's original proof was too sophisticated but a British mathematician called James Maynard, later proposed a more elementary proof that sharpened the upper bound to 600. The person said that Maynard's proof can be explained to undergraduates as it was more elementary. Unfortunately, I couldn't find his paper.

Can someone offer a brief sketch of the ideas of Maynard's proof ? I'm more interested in Maynard's proof than Yitang's because I'll have a better chance of understanding Maynard's proof.

The paper is too long to provide a sketch so if you could just list the perquisites for understanding the proof, that would be fine too.

On an additional note, does anyone know what that upper bound has been sharpened too today ? How far are we from the twin prime conjecture? Do you think this method will lead to the conjecture or that we've reached limits of this method and the answer will come through another method. Thanks

@MorganRodgers Thanks for the paper. I wasn't able to find it. — Saikat, Jun 09 '16 at 13:47
Take a look at these lecture notes. They provide an introduction to the various sieving techniques involved, with Zhang's and Maynard's work being the ultimate goal. — Mathmo123, Jun 09 '16 at 13:49
For the last question: apparently, the current least known prime gap is $246$. According to the timeline of Polymath (column $H$), though, it still has a question mark. The last one without a question mark is $270$, dating 6th January. Be sure to read the legenda, because the table contains the results to similar problems, some of which under the assumption of some open conjectures. — , Jun 09 '16 at 14:27
The link to $246$ in my previous comment should have been this one. Also, the table stops at year 2014. — , Jun 09 '16 at 14:35

score 2 · Answer 1 · answered Jun 09 '16 at 20:54

Maynard's paper is on the arXiv:

Small gaps between primes, arXiv:1311.4600 [math.NT]

It obtained a bound of 600, as you mention; a collaborative effort improved this to 246 in this paper:

Variants of the Selberg sieve, and bounded intervals containing many primes, arXiv:1407.4897 [math.NT]

as one of the first papers in the open-access journal Research in the Mathematical Sciences. A retrospective of the project was written up as

The "bounded gaps between primes" Polymath project - a retrospective, arXiv:1409.8361 [math.HO]

and the project itself lives at

Bounded gaps between primes

As mentioned in the comments, the paper is complex and not easy to summarize. Tao explains Zhang's original proof here

Online reading seminar for Zhang’s “bounded gaps between primes”

and Maynard's proof here

Polymath8b: Bounded intervals with many primes, after Maynard

It's not an easy read for an undergraduate as it uses sieve theory -- the Bombieri–Vinogradov sieve and a modified Goldston–Yıldırım sieve.

If your goal is to get a general understanding of the proof then the Polymath page and Tao's blog posts do a good job. If you want a more in-depth understanding I recommend Andrew Granville's

Primes in intervals of bounded length, arXiv:1410.8400 [math.NT]

which, although longer than Maynard's paper, is substantially easier to understand. Another possibility, as recommended by Mathmo123 above, is to read Adam Harper's lecture notes on Elementary Methods in Analytic Number Theory.

Can someone offer an overview of the idea of the proof of prime gaps bounds?

1 Answers1