This is the eighth thread for the Polymath8b project to obtain new bounds for the quantity

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

either for small values of ${m}$ (in particular ${m=1,2}$) or asymptotically as ${m \rightarrow \infty}$. The previous thread may be found here. The currently best known bounds on ${H_m}$ can be found at the wiki page.

The big news since the last thread is that we have managed to obtain the (sieve-theoretically) optimal bound of ${H_1 \leq 6}$ assuming the generalised Elliott-Halberstam conjecture (GEH), which pretty much closes off that part of the story. Unconditionally, our bound on ${H_1}$ is still ${H_1 \leq 270}$. This bound was obtained using the “vanilla” Maynard sieve, in which the cutoff ${F}$ was supported in the original simplex ${\{ t_1+\dots+t_k \leq 1\}}$, and only Bombieri-Vinogradov was used. In principle, we can enlarge the sieve support a little bit further now; for instance, we can enlarge to ${\{ t_1+\dots+t_k \leq \frac{k}{k-1} \}}$, but then have to shrink the J integrals to ${\{t_1+\dots+t_{k-1} \leq 1-\epsilon\}}$, provided that the marginals vanish for ${\{ t_1+\dots+t_{k-1} \geq 1+\epsilon \}}$. However, we do not yet know how to numerically work with these expanded problems.

Given the substantial progress made so far, it looks like we are close to the point where we should declare victory and write up the results (though we should take one last look to see if there is any room to improve the ${H_1 \leq 270}$ bounds). There is actually a fair bit to write up:

• Improvements to the Maynard sieve (pushing beyond the simplex, the epsilon trick, and pushing beyond the cube);
• Asymptotic bounds for ${M_k}$ and hence ${H_m}$;
• Explicit bounds for ${H_m, m \geq 2}$ (using the Polymath8a results)
• ${H_1 \leq 270}$;
• ${H_1 \leq 6}$ on GEH (and parity obstructions to any further improvement).

I will try to create a skeleton outline of such a paper in the Polymath8 Dropbox folder soon. It shouldn’t be nearly as big as the Polymath8a paper, but it will still be quite sizeable.