I’ve just uploaded to the arXiv the D.H.J. Polymath paper “Variants of the Selberg sieve, and bounded intervals containing many primes“, which is the second paper to be produced from the Polymath8 project (the first one being discussed here). We’ll refer to this latter paper here as the Polymath8b paper, and the former as the Polymath8a paper. As with Polymath8a, the Polymath8b paper is concerned with the smallest asymptotic prime gap

$\displaystyle H_1 := \liminf_{n \rightarrow \infty}(p_{n+1}-p_n),$

where ${p_n}$ denotes the ${n^{th}}$ prime, as well as the more general quantities

$\displaystyle H_m := \liminf_{n \rightarrow \infty}(p_{n+m}-p_n).$

In the breakthrough paper of Goldston, Pintz, and Yildirim, the bound ${H_1 \leq 16}$ was obtained under the strong hypothesis of the Elliott-Halberstam conjecture. An unconditional bound on ${H_1}$, however, remained elusive until the celebrated work of Zhang last year, who showed that

$\displaystyle H_1 \leq 70{,}000{,}000.$

The Polymath8a paper then improved this to ${H_1 \leq 4{,}680}$. After that, Maynard introduced a new multidimensional Selberg sieve argument that gave the substantial improvement

$\displaystyle H_1 \leq 600$

unconditionally, and ${H_1 \leq 12}$ on the Elliott-Halberstam conjecture; furthermore, bounds on ${H_m}$ for higher ${m}$ were obtained for the first time, and specifically that ${H_m \ll m^3 e^{4m}}$ for all ${m \geq 1}$, with the improvements ${H_2 \leq 600}$ and ${H_m \ll m^3 e^{2m}}$ on the Elliott-Halberstam conjecture. (I had independently discovered the multidimensional sieve idea, although I did not obtain Maynard’s specific numerical results, and my asymptotic bounds were a bit weaker.)

In Polymath8b, we obtain some further improvements. Unconditionally, we have ${H_1 \leq 246}$ and ${H_m \ll m e^{(4 - \frac{28}{157}) m}}$, together with some explicit bounds on ${H_2,H_3,H_4,H_5}$; on the Elliott-Halberstam conjecture we have ${H_m \ll m e^{2m}}$ and some numerical improvements to the ${H_2,H_3,H_4,H_5}$ bounds; and assuming the generalised Elliott-Halberstam conjecture we have the bound ${H_1 \leq 6}$, which is best possible from sieve-theoretic methods thanks to the parity problem obstruction.

There were a variety of methods used to establish these results. Maynard’s paper obtained a criterion for bounding ${H_m}$ which reduced to finding a good solution to a certain multidimensional variational problem. When the dimension parameter ${k}$ was relatively small (e.g. ${k \leq 100}$), we were able to obtain good numerical solutions both by continuing the method of Maynard (using a basis of symmetric polynomials), or by using a Krylov iteration scheme. For large ${k}$, we refined the asymptotics and obtained near-optimal solutions of the variational problem. For the ${H_1}$ bounds, we extended the reach of the multidimensional Selberg sieve (particularly under the assumption of the generalised Elliott-Halberstam conjecture) by allowing the function ${F}$ in the multidimensional variational problem to extend to a larger region of space than was previously admissible, albeit with some tricky new constraints on ${F}$ (and penalties in the variational problem). This required some unusual sieve-theoretic manipulations, notably an “epsilon trick”, ultimately relying on the elementary inequality ${(a+b)^2 \geq a^2 + 2ab}$, that allowed one to get non-trivial lower bounds for sums such as ${\sum_n (a(n)+b(n))^2}$ even if the sum ${\sum_n b(n)^2}$ had no non-trivial estimates available; and a way to estimate divisor sums such as ${\sum_{n\leq x} \sum_{d|n} \lambda_d}$ even if ${d}$ was permitted to be comparable to or even exceed ${x}$, by using the fundamental theorem of arithmetic to factorise ${n}$ (after restricting to the case when ${n}$ is almost prime). I hope that these sieve-theoretic tricks will be useful in future work in the subject.

With this paper, the Polymath8 project is almost complete; there is still a little bit of scope to push our methods further and get some modest improvement for instance to the ${H_1 \leq 246}$ bound, but this would require a substantial amount of effort, and it is probably best to instead wait for some new breakthrough in the subject to come along. One final task we are performing is to write up a retrospective article on both the 8a and 8b experiences, an incomplete writeup of which can be found here. If anyone wishes to contribute some commentary on these projects (whether you were an active contributor, an occasional contributor, or a silent “lurker” in the online discussion), please feel free to do so in the comments to this post.