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The (presumably) final article arising from the Polymath8 project has now been uploaded to the arXiv as “The “bounded gaps between primes” Polymath project – a retrospective“.  This article, submitted to the Newsletter of the European Mathematical Society, consists of personal contributions from ten different participants (at varying levels of stage of career, and intensity of participation) on their own experiences with the project, and some thoughts as to what lessons to draw for any subsequent Polymath projects.  (At present, I do not know of any such projects being proposed, but from recent experience I would imagine that some opportunity suitable for a Polymath approach will present itself at some point in the near future.)

This post will also serve as the latest (and probably last) of the Polymath8 threads (rolling over this previous post), to wrap up any remaining discussion about any aspect of this project.

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.

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

H_m := \liminf_{n \to\infty} p_{n+m} - p_n;

the previous thread may be found here.

The main focus is now on writing up the results, with a draft paper close to completion here (with the directory of source files here).    Most of the sections are now written up more or less completely, with the exception of the appendix on narrow admissible tuples, which was awaiting the bounds on such tuples to stabilise.  There is now also an acknowledgments section (linking to the corresponding page on the wiki, which participants should check to see if their affiliations etc. are posted correctly), and in the final remarks section there is now also some discussion about potential improvements to the H_m bounds.  I’ve also added some mention of a recent paper of Banks, Freiberg and Maynard which makes use of some of our results (in particular, that M_{50,1/25} > 4).  On the other hand, the portions of the writeup relating to potential improvements to the MPZ estimates have been commented out, as it appears that one cannot easily obtain the exponential sum estimates required to make those go through.  (Perhaps, if there are significant new developments, one could incorporate them into a putative Polymath8c project, although at present I think there’s not much urgency to start over once again.)

Regarding the numerics in Section 7 of the paper, one thing which is missing at present is some links to code in case future readers wish to verify the results; alternatively one could include such code and data into the arXiv submission.

It’s about time to discuss possible journals to submit the paper to.  Ken Ono has invited us to submit to his new journal, “Research in the Mathematical Sciences“.  Another option would be to submit to the same journal “Algebra & Number Theory” that is currently handling our Polymath8a paper (no news on the submission there, but it is a very long paper), although I think the papers are independent enough that it is not absolutely necessary to place them in the same journal.  A third natural choice is “Mathematics of Computation“, though I should note that when the Polymath4 paper was submitted there, the editors required us to use our real names instead of the D.H.J. Polymath pseudonym as it would have messed up their metadata system otherwise.  (But I can check with the editor there before submitting to see if there is some workaround now, perhaps their policies have changed.)  At present I have no strong preferences regarding journal selection, and would welcome further thoughts and proposals.  (It is perhaps best to avoid the journals that I am editor or associate editor of, namely Amer. J. Math, Forum of Mathematics, Analysis & PDE, and Dynamics and PDE, due to conflict of interest (and in the latter two cases, specialisation to a different area of mathematics)).

 

 

 

 

This is the ninth 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 focus is now on bounding {H_1} unconditionally (in particular, without resorting to the Elliott-Halberstam conjecture or its generalisations). We can bound {H_1 \leq H(k)} whenever one can find a symmetric square-integrable function {F} supported on the simplex {{\cal R}_k := \{ (t_1,\dots,t_k) \in [0,+\infty)^k: t_1+\dots+t_k \leq 1 \}} such that

\displaystyle  k \int_{{\cal R}_{k-1}} (\int_0^\infty F(t_1,\dots,t_k)\ dt_k)^2\ dt_1 \dots dt_{k-1} \ \ \ \ \ (1)

\displaystyle > 4 \int_{{\cal R}_{k}} F(t_1,\dots,t_k)^2\ dt_1 \dots dt_{k-1} dt_k.

Our strategy for establishing this has been to restrict {F} to be a linear combination of symmetrised monomials {[t_1^{a_1} \dots t_k^{a_k}]_{sym}} (restricted of course to {{\cal R}_k}), where the degree {a_1+\dots+a_k} is small; actually, it seems convenient to work with the slightly different basis {(1-t_1-\dots-t_k)^i [t_1^{a_1} \dots t_k^{a_k}]_{sym}} where the {a_i} are restricted to be even. The criterion (1) then becomes a large quadratic program with explicit but complicated rational coefficients. This approach has lowered {k} down to {54}, which led to the bound {H_1 \leq 270}.

Actually, we know that the more general criterion

\displaystyle  k \int_{(1-\epsilon) \cdot {\cal R}_{k-1}} (\int_0^\infty F(t_1,\dots,t_k)\ dt_k)^2\ dt_1 \dots dt_{k-1} \ \ \ \ \ (2)

\displaystyle  > 4 \int F(t_1,\dots,t_k)^2\ dt_1 \dots dt_{k-1} dt_k

will suffice, whenever {0 \leq \epsilon < 1} and {F} is supported now on {2 \cdot {\cal R}_k} and obeys the vanishing marginal condition {\int_0^\infty F(t_1,\dots,t_k)\ dt_k = 0} whenever {t_1+\dots+t_k > 1+\epsilon}. The latter is in particular obeyed when {F} is supported on {(1+\epsilon) \cdot {\cal R}_k}. A modification of the preceding strategy has lowered {k} slightly to {53}, giving the bound {H_1 \leq 264} which is currently our best record.

However, the quadratic programs here have become extremely large and slow to run, and more efficient algorithms (or possibly more computer power) may be required to advance further.

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.

There are multiple purposes to this blog post.

The first purpose is to announce the uploading of the paper “New equidistribution estimates of Zhang type, and bounded gaps between primes” by D.H.J. Polymath, which is the main output of the Polymath8a project on bounded gaps between primes, to the arXiv, and to describe the main results of this paper below the fold.

The second purpose is to roll over the previous thread on all remaining Polymath8a-related matters (e.g. updates on the submission status of the paper) to a fresh thread. (Discussion of the ongoing Polymath8b project is however being kept on a separate thread, to try to reduce confusion.)

The final purpose of this post is to coordinate the writing of a retrospective article on the Polymath8 experience, which has been solicited for the Newsletter of the European Mathematical Society. I suppose that this could encompass both the Polymath8a and Polymath8b projects, even though the second one is still ongoing (but I think we will soon be entering the endgame there). I think there would be two main purposes of such a retrospective article. The first one would be to tell a story about the process of conducting mathematical research, rather than just describe the outcome of such research; this is an important aspect of the subject which is given almost no attention in most mathematical writing, and it would be good to be able to capture some sense of this process while memories are still relatively fresh. The other would be to draw some tentative conclusions with regards to what the strengths and weaknesses of a Polymath project are, and how appropriate such a format would be for other mathematical problems than bounded gaps between primes. In my opinion, the bounded gaps problem had some fairly unique features that made it particularly amenable to a Polymath project, such as (a) a high level of interest amongst the mathematical community in the problem; (b) a very focused objective (“improve {H}!”), which naturally provided an obvious metric to measure progress; (c) the modular nature of the project, which allowed for people to focus on one aspect of the problem only, and still make contributions to the final goal; and (d) a very reasonable level of ambition (for instance, we did not attempt to prove the twin prime conjecture, which in my opinion would make a terrible Polymath project at our current level of mathematical technology). This is not an exhaustive list of helpful features of the problem; I would welcome other diagnoses of the project by other participants.

With these two objectives in mind, I propose a format for the retrospective article consisting of a brief introduction to the polymath concept in general and the polymath8 project in particular, followed by a collection of essentially independent contributions by different participants on their own experiences and thoughts. Finally we could have a conclusion section in which we make some general remarks on the polymath project (such as the remarks above). I’ve started a dropbox subfolder for this article (currently in a very skeletal outline form only), and will begin writing a section on my own experiences; other participants are of course encouraged to add their own sections (it is probably best to create separate files for these, and then input them into the main file retrospective.tex, to reduce edit conflicts. If there are participants who wish to contribute but do not currently have access to the Dropbox folder, please email me and I will try to have you added (or else you can supply your thoughts by email, or in the comments to this post; we may have a section for shorter miscellaneous comments from more casual participants, for people who don’t wish to write a lengthy essay on the subject).

As for deadlines, the EMS Newsletter would like a submitted article by mid-April in order to make the June issue, but in the worst case, it will just be held over until the issue after that.

Read the rest of this entry »

This is the seventh 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 current focus is on improving the upper bound on {H_1} under the assumption of the generalised Elliott-Halberstam conjecture (GEH) from {H_1 \leq 8} to {H_1 \leq 6}. Very recently, we have been able to exploit GEH more fully, leading to a promising new expansion of the sieve support region. The problem now reduces to the following:

Problem 1 Does there exist a (not necessarily convex) polytope {R \subset [0,2]^3} with quantities {0 \leq \varepsilon_1,\varepsilon_2,\varepsilon_3 \leq 1}, and a non-trivial square-integrable function {F: {\bf R}^3 \rightarrow {\bf R}} supported on {R} such that

  • {R + R \subset \{ (x,y,z) \in [0,4]^3: \min(x+y,y+z,z+x) \leq 2 \},}
  • {\int_0^\infty F(x,y,z)\ dx = 0} when {y+z \geq 1+\varepsilon_1};
  • {\int_0^\infty F(x,y,z)\ dy = 0} when {x+z \geq 1+\varepsilon_2};
  • {\int_0^\infty F(x,y,z)\ dz = 0} when {x+y \geq 1+\varepsilon_3};

and such that we have the inequality

\displaystyle  \int_{y+z \leq 1-\varepsilon_1} (\int_{\bf R} F(x,y,z)\ dx)^2\ dy dz

\displaystyle + \int_{z+x \leq 1-\varepsilon_2} (\int_{\bf R} F(x,y,z)\ dy)^2\ dz dx

\displaystyle + \int_{x+y \leq 1-\varepsilon_3} (\int_{\bf R} F(x,y,z)\ dz)^2\ dx dy

\displaystyle  > 2 \int_R F(x,y,z)^2\ dx dy dz?

An affirmative answer to this question will imply {H_1 \leq 6} on GEH. We are “within two percent” of this claim; we cannot quite reach {2} yet, but have got as far as {1.962998}. However, we have not yet fully optimised {F} in the above problem. In particular, the simplex

\displaystyle  R = \{ (x,y,z) \in [0,2]^3: x+y+z \leq 3/2 \}

is now available, and should lead to some noticeable improvement in the numerology.

There is also a very slim chance that the twin prime conjecture is now provable on GEH. It would require an affirmative solution to the following problem:

Problem 2 Does there exist a (not necessarily convex) polytope {R \subset [0,2]^2} with quantities {0 \leq \varepsilon_1,\varepsilon_2 \leq 1}, and a non-trivial square-integrable function {F: {\bf R}^2 \rightarrow {\bf R}} supported on {R} such that

  • {R + R \subset \{ (x,y) \in [0,4]^2: \min(x,y) \leq 2 \}}

    \displaystyle  = [0,2] \times [0,4] \cup [0,4] \times [0,2],

  • {\int_0^\infty F(x,y)\ dx = 0} when {y \geq 1+\varepsilon_1};
  • {\int_0^\infty F(x,y)\ dy = 0} when {x \geq 1+\varepsilon_2};

and such that we have the inequality

\displaystyle  \int_{y \leq 1-\varepsilon_1} (\int_{\bf R} F(x,y)\ dx)^2\ dy

\displaystyle + \int_{x \leq 1-\varepsilon_2} (\int_{\bf R} F(x,y)\ dy)^2\ dx

\displaystyle  > 2 \int_R F(x,y)^2\ dx dy?

We suspect that the answer to this question is negative, but have not formally ruled it out yet.

For the rest of this post, I will justify why positive answers to these sorts of variational problems are sufficient to get bounds on {H_1} (or more generally {H_m}).

Read the rest of this entry »

This is the sixth 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 (which has recently returned to full functionality, after a partial outage).

The current focus is on improving the upper bound on {H_1} under the assumption of the generalised Elliott-Halberstam conjecture (GEH) from {H_1 \leq 8} to {H_1 \leq 6}, which looks to be the limit of the method (see this previous comment for a semi-rigorous reason as to why {H_1 \leq 4} is not possible with this method). With the most general Selberg sieve available, the problem reduces to the following three-dimensional variational one:

Problem 1 Does there exist a (not necessarily convex) polytope {R \subset [0,1]^3} with quantities {0 \leq \varepsilon_1,\varepsilon_2,\varepsilon_3 \leq 1}, and a non-trivial square-integrable function {F: {\bf R}^3 \rightarrow {\bf R}} supported on {R} such that

  • {R + R \subset \{ (x,y,z) \in [0,2]^3: \min(x+y,y+z,z+x) \leq 2 \},}
  • {\int_0^\infty F(x,y,z)\ dx = 0} when {y+z \geq 1+\varepsilon_1};
  • {\int_0^\infty F(x,y,z)\ dy = 0} when {x+z \geq 1+\varepsilon_2};
  • {\int_0^\infty F(x,y,z)\ dz = 0} when {x+y \geq 1+\varepsilon_3};

and such that we have the inequality

\displaystyle  \int_{y+z \leq 1-\varepsilon_1} (\int_{\bf R} F(x,y,z)\ dx)^2\ dy dz

\displaystyle + \int_{z+x \leq 1-\varepsilon_2} (\int_{\bf R} F(x,y,z)\ dy)^2\ dz dx

\displaystyle + \int_{x+y \leq 1-\varepsilon_3} (\int_{\bf R} F(x,y,z)\ dz)^2\ dx dy

\displaystyle  > 2 \int_R F(x,y,z)^2\ dx dy dz?

(Initially it was assumed that {R} was convex, but we have now realised that this is not necessary.)

An affirmative answer to this question will imply {H_1 \leq 6} on GEH. We are “within almost two percent” of this claim; we cannot quite reach {2} yet, but have got as far as {1.959633}. However, we have not yet fully optimised {F} in the above problem.

The most promising route so far is to take the symmetric polytope

\displaystyle  R = \{ (x,y,z) \in [0,1]^3: x+y+z \leq 3/2 \}

with {F} symmetric as well, and {\varepsilon_1=\varepsilon_2=\varepsilon_3=\varepsilon} (we suspect that the optimal {\varepsilon} will be roughly {1/6}). (However, it is certainly worth also taking a look at easier model problems, such as the polytope {{\cal R}'_3 := \{ (x,y,z) \in [0,1]^3: x+y,y+z,z+x \leq 1\}}, which has no vanishing marginal conditions to contend with; more recently we have been looking at the non-convex polytope {R = \{x+y,x+z \leq 1 \} \cup \{ x+y,y+z \leq 1 \} \cup \{ x+z,y+z \leq 1\}}.) Some further details of this particular case are given below the fold.

There should still be some progress to be made in the other regimes of interest – the unconditional bound on {H_1} (currently at {270}), and on any further progress in asymptotic bounds for {H_m} for larger {m} – but the current focus is certainly on the bound on {H_1} on GEH, as we seem to be tantalisingly close to an optimal result here.

Read the rest of this entry »

This is the fifth 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 (which has recently returned to full functionality, after a partial outage). In particular, the upper bound for {H_1} has been shaved a little from {272} to {270}, and we have very recently achieved the bound {H_1 \leq 8} on the generalised Elliott-Halberstam conjecture GEH, formulated as Conjecture 1 of this paper of Bombieri, Friedlander, and Iwaniec. We also have explicit bounds for {H_m} for {m \leq 5}, both with and without the assumption of the Elliott-Halberstam conjecture, as well as slightly sharper asymptotics for the upper bound for {H_m} as {m \rightarrow \infty}.

The basic strategy for bounding {H_m} still follows the general paradigm first laid out by Goldston, Pintz, Yildirim: given an admissible {k}-tuple {(h_1,\dots,h_k)}, one needs to locate a non-negative sieve weight {\nu: {\bf Z} \rightarrow {\bf R}^+}, supported on an interval {[x,2x]} for a large {x}, such that the ratio

\displaystyle  \frac{\sum_{i=1}^k \sum_n \nu(n) 1_{n+h_i \hbox{ prime}}}{\sum_n \nu(n)} \ \ \ \ \ (1)

is asymptotically larger than {m} as {x \rightarrow \infty}; this will show that {H_m \leq h_k-h_1}. Thus one wants to locate a sieve weight {\nu} for which one has good lower bounds on the numerator and good upper bounds on the denominator.

One can modify this paradigm slightly, for instance by adding the additional term {\sum_n \nu(n) 1_{n+h_1,\dots,n+h_k \hbox{ composite}}} to the numerator, or by subtracting the term {\sum_n \nu(n) 1_{n+h_1,n+h_k \hbox{ prime}}} from the numerator (which allows one to reduce the bound {h_k-h_1} to {\max(h_k-h_2,h_{k-1}-h_1)}); however, the numerical impact of these tweaks have proven to be negligible thus far.

Despite a number of experiments with other sieves, we are still relying primarily on the Selberg sieve

\displaystyle  \nu(n) := 1_{n=b\ (W)} 1_{[x,2x]}(n) \lambda(n)^2

where {\lambda(n)} is the divisor sum

\displaystyle  \lambda(n) := \sum_{d_1|n+h_1, \dots, d_k|n+h_k} \mu(d_1) \dots \mu(d_k) f( \frac{\log d_1}{\log R}, \dots, \frac{\log d_k}{\log R})

with {R = x^{\theta/2}}, {\theta} is the level of distribution ({\theta=1/2-} if relying on Bombieri-Vinogradov, {\theta=1-} if assuming Elliott-Halberstam, and (in principle) {\theta = \frac{1}{2} + \frac{13}{540}-} if using Polymath8a technology), and {f: [0,+\infty)^k \rightarrow {\bf R}} is a smooth, compactly supported function. Most of the progress has come by enlarging the class of cutoff functions {f} one is permitted to use.

The baseline bounds for the numerator and denominator in (1) (as established for instance in this previous post) are as follows. If {f} is supported on the simplex

\displaystyle  {\cal R}_k := \{ (t_1,\dots,t_k) \in [0,+\infty)^k: t_1+\dots+t_k < 1 \},

and we define the mixed partial derivative {F: [0,+\infty)^k \rightarrow {\bf R}} by

\displaystyle  F(t_1,\dots,t_k) = \frac{\partial^k}{\partial t_1 \dots \partial t_k} f(t_1,\dots,t_k)

then the denominator in (1) is

\displaystyle  \frac{Bx}{W} (I_k(F) + o(1)) \ \ \ \ \ (2)

where

\displaystyle  B := (\frac{W}{\phi(W) \log R})^k

and

\displaystyle  I_k(F) := \int_{[0,+\infty)^k} F(t_1,\dots,t_k)^2\ dt_1 \dots dt_k.

Similarly, the numerator of (1) is

\displaystyle  \frac{Bx}{W} \frac{2}{\theta} (\sum_{j=1}^m J^{(m)}_k(F) + o(1)) \ \ \ \ \ (3)

where

\displaystyle  J_k^{(m)}(F) := \int_{[0,+\infty)^{k-1}} (\int_0^\infty F(t_1,\ldots,t_k)\ dt_m)^2\ dt_1 \dots dt_{m-1} dt_{m+1} \dots dt_k.

Thus, if we let {M_k} be the supremum of the ratio

\displaystyle  \frac{\sum_{m=1}^k J_k^{(m)}(F)}{I_k(F)}

whenever {F} is supported on {{\cal R}_k} and is non-vanishing, then one can prove {H_m \leq h_k - h_1} whenever

\displaystyle  M_k > \frac{2m}{\theta}.

We can improve this baseline in a number of ways. Firstly, with regards to the denominator in (1), if one upgrades the Elliott-Halberstam hypothesis {EH[\theta]} to the generalised Elliott-Halberstam hypothesis {GEH[\theta]} (currently known for {\theta < 1/2}, thanks to Motohashi, but conjectured for {\theta < 1}), the asymptotic (2) holds under the more general hypothesis that {F} is supported in a polytope {R}, as long as {R} obeys the inclusion

\displaystyle  R + R \subset \bigcup_{m=1}^k \{ (t_1,\ldots,t_k) \in [0,+\infty)^k: \ \ \ \ \ (4)

\displaystyle  t_1+\dots+t_{m-1}+t_{m+1}+\dots+t_k < 2; t_m < 2/\theta \} \cup \frac{2}{\theta} \cdot {\cal R}_k;

examples of polytopes {R} obeying this constraint include the modified simplex

\displaystyle  {\cal R}'_k := \{ (t_1,\ldots,t_k) \in [0,+\infty)^k: t_1+\dots+t_{m-1}+t_{m+1}+\dots+t_k < 1

\displaystyle \hbox{ for all } 1 \leq m \leq k \},

the prism

\displaystyle  {\cal R}_{k-1} \times [0, 1/\theta)

the dilated simplex

\displaystyle  \frac{1}{\theta} \cdot {\cal R}_k

and the truncated simplex

\displaystyle  \frac{k}{k-1} \cdot {\cal R}_k \cap [0,1/\theta)^k.

See this previous post for a proof of these claims.

With regards to the numerator, the asymptotic (3) is valid whenever, for each {1 \leq m \leq k}, the marginals {\int_0^\infty F(t_1,\ldots,t_k)\ dt_m} vanish outside of {{\cal R}_{k-1}}. This is automatic if {F} is supported on {{\cal R}_k}, or on the slightly larger region {{\cal R}'_k}, but is an additional constraint when {F} is supported on one of the other polytopes {R} mentioned above.

More recently, we have obtained a more flexible version of the above asymptotic: if the marginals {\int_0^\infty F(t_1,\ldots,t_k)\ dt_m} vanish outside of {(1+\varepsilon) \cdot {\cal R}_{k-1}} for some {0 < \varepsilon < 1}, then the numerator of (1) has a lower bound of

\displaystyle  \frac{Bx}{W} \frac{2}{\theta} (\sum_{j=1}^m J^{(m)}_{k,\varepsilon}(F) + o(1))

where

\displaystyle  J_{k,\varepsilon}^{(m)}(F) := \int_{(1-\varepsilon) \cdot {\cal R}_{k-1}} (\int_0^\infty F(t_1,\ldots,t_k)\ dt_m)^2\ dt_1 \dots dt_{m-1} dt_{m+1} \dots dt_k.

A proof is given here. Putting all this together, we can conclude

Theorem 1 Suppose we can find {0 \leq \varepsilon < 1} and a function {F} supported on a polytope {R} obeying (4), not identically zero and with all marginals {\int_0^\infty F(t_1,\ldots,t_k)\ dt_m} vanishing outside of {(1+\varepsilon) \cdot {\cal R}_{k-1}}, and with

\displaystyle  \frac{\sum_{m=1}^k J_{k,\varepsilon}^{(m)}(F)}{I_k(F)} > \frac{2m}{\theta}.

Then {GEH[\theta]} implies {H_m \leq h_k-h_1}.

In principle, this very flexible criterion for upper bounding {H_m} should lead to better bounds than before, and in particular we have now established {H_1 \leq 8} on GEH.

Another promising direction is to try to improve the analysis at medium {k} (more specifically, in the regime {k \sim 50}), which is where we are currently at without EH or GEH through numerical quadratic programming. Right now we are only using {\theta=1/2} and using the baseline {M_k} analysis, basically for two reasons:

  • We do not have good numerical formulae for integrating polynomials on any region more complicated than the simplex {{\cal R}_k} in medium dimension.
  • The estimates {MPZ^{(i)}[\varpi,\delta]} produced by Polymath8a involve a {\delta} parameter, which introduces additional restrictions on the support of {F} (conservatively, it restricts {F} to {[0,\delta']^k} where {\delta' := \frac{\delta}{1/4+\varpi}} and {\theta = 1/2 + 2 \varpi}; it should be possible to be looser than this (as was done in Polymath8a) but this has not been fully explored yet). This then triggers the previous obstacle of having to integrate on something other than a simplex.

However, these look like solvable problems, and so I would expect that further unconditional improvement for {H_1} should be possible.

This is the fourth 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} are:

  • (Maynard) Assuming the Elliott-Halberstam conjecture, {H_1 \leq 12}.
  • (Polymath8b, tentative) {H_1 \leq 272}. Assuming Elliott-Halberstam, {H_2 \leq 272}.
  • (Polymath8b, tentative) {H_2 \leq 429{,}822}. Assuming Elliott-Halberstam, {H_4 \leq 493{,}408}.
  • (Polymath8b, tentative) {H_3 \leq 26{,}682{,}014}. (Presumably a comparable bound also holds for {H_6} on Elliott-Halberstam, but this has not been computed.)
  • (Polymath8b) {H_m \leq \exp( 3.817 m )} for sufficiently large {m}. Assuming Elliott-Halberstam, {H_m \ll m e^{2m}} for sufficiently large {m}.

While the {H_1} bound on the Elliott-Halberstam conjecture has not improved since the start of the Polymath8b project, there is reason to hope that it will soon fall, hopefully to {8}. This is because we have begun to exploit more fully the fact that when using “multidimensional Selberg-GPY” sieves of the form

\displaystyle  \nu(n) := \sigma_{f,k}(n)^2

with

\displaystyle  \sigma_{f,k}(n) := \sum_{d_1|n+h_1,\dots,d_k|n+h_k} \mu(d_1) \dots \mu(d_k) f( \frac{\log d_1}{\log R},\dots,\frac{\log d_k}{\log R}),

where {R := x^{\theta/2}}, it is not necessary for the smooth function {f: [0,+\infty)^k \rightarrow {\bf R}} to be supported on the simplex

\displaystyle {\cal R}_k := \{ (t_1,\dots,t_k)\in [0,1]^k: t_1+\dots+t_k \leq 1\},

but can in fact be allowed to range on larger sets. First of all, {f} may instead be supported on the slightly larger polytope

\displaystyle {\cal R}'_k := \{ (t_1,\dots,t_k)\in [0,1]^k: t_1+\dots+t_{j-1}+t_{j+1}+\dots+t_k \leq 1

\displaystyle  \hbox{ for all } j=1,\dots,k\}.

However, it turns out that more is true: given a sufficiently general version of the Elliott-Halberstam conjecture {EH[\theta]} at the given value of {\theta}, one may work with functions {f} supported on more general domains {R}, so long as the sumset {R+R := \{ t+t': t,t'\in R\}} is contained in the non-convex region

\displaystyle  \bigcup_{j=1}^k \{ (t_1,\dots,t_k)\in [0,\frac{2}{\theta}]^k: t_1+\dots+t_{j-1}+t_{j+1}+\dots+t_k \leq 2 \} \cup \frac{2}{\theta} \cdot {\cal R}_k, \ \ \ \ \ (1)

and also provided that the restriction

\displaystyle  (t_1,\dots,t_{j-1},t_{j+1},\dots,t_k) \mapsto f(t_1,\dots,t_{j-1},0,t_{j+1},\dots,t_k) \ \ \ \ \ (2)

is supported on the simplex

\displaystyle {\cal R}_{k-1} := \{ (t_1,\dots,t_{j-1},t_{j+1},\dots,t_k)\in [0,1]^{k-1}:

\displaystyle t_1+\dots+t_{j-1}+t_{j+1}+\dots t_k \leq 1\}.

More precisely, if {f} is a smooth function, not identically zero, with the above properties for some {R}, and the ratio

\displaystyle  \sum_{j=1}^k \int_{{\cal R}_{k-1}} f_{1,\dots,j-1,j+1,\dots,k}(t_1,\dots,t_{j-1},0,t_{j+1},\dots,t_k)^2 \ \ \ \ \ (3)

\displaystyle dt_1 \dots dt_{j-1} dt_{j+1} \dots dt_k

\displaystyle  / \int_R f_{1,\dots,k}^2(t_1,\dots,t_k)\ dt_1 \dots dt_k

is larger than {\frac{2m}{\theta}}, then the claim {DHL[k,m+1]} holds (assuming {EH[\theta]}), and in particular {H_m \leq H(k)}.

I’ll explain why one can do this below the fold. Taking this for granted, we can rewrite this criterion in terms of the mixed derivative {F := f_{1,\dots,k}}, the upshot being that if one can find a smooth function {F} supported on {R} that obeys the vanishing marginal conditions

\displaystyle  \int F( t_1,\dots,t_k )\ dt_j = 0

whenever {1 \leq j \leq k} and {t_1+\dots+t_{j-1}+t_{j+1}+\dots+t_k > 1}, and the ratio

\displaystyle  \frac{\sum_{j=1}^k J_k^{(j)}(F)}{I_k(F)} \ \ \ \ \ (4)

is larger than {\frac{2m}{\theta}}, where

\displaystyle  I_k(F) := \int_R F(t_1,\dots,t_k)^2\ dt_1 \dots dt_k

and

\displaystyle  J_k^{(j)}(F) := \int_{{\cal R}_{k-1}} (\int_0^{1/\theta} F(t_1,\dots,t_k)\ dt_j)^2 dt_1 \dots dt_{j-1} dt_{j+1} \dots dt_k

then {DHL[k,m+1]} holds. (To equate these two formulations, it is convenient to assume that {R} is a downset, in the sense that whenever {(t_1,\dots,t_k) \in R}, the entire box {[0,t_1] \times \dots \times [0,t_k]} lie in {R}, but one can easily enlarge {R} to be a downset without destroying the containment of {R+R} in the non-convex region (1).) One initially requires {F} to be smooth, but a limiting argument allows one to relax to bounded measurable {F}. (To approximate a rough {F} by a smooth {F} while retaining the required moment conditions, one can first apply a slight dilation and translation so that the marginals of {F} are supported on a slightly smaller version of the simplex {{\cal R}_{k-1}}, and then convolve by a smooth approximation to the identity to make {F} smooth, while keeping the marginals supported on {{\cal R}_{k-1}}.)

We are now exploring various choices of {R} to work with, including the prism

\displaystyle  \{ (t_1,\dots,t_k) \in [0,1/\theta]^k: t_1+\dots+t_{k-1} \leq 1 \}

and the symmetric region

\displaystyle  \{ (t_1,\dots,t_k) \in [0,1/\theta]^k: t_1+\dots+t_k \leq \frac{k}{k-1} \}.

By suitably subdividing these regions into polytopes, and working with piecewise polynomial functions {F} that are polynomial of a specified degree on each subpolytope, one can phrase the problem of optimising (4) as a quadratic program, which we have managed to work with for {k=3}. Extending this program to {k=4}, there is a decent chance that we will be able to obtain {DHL[4,2]} on EH.

We have also been able to numerically optimise {M_k} quite accurately for medium values of {k} (e.g. {k \sim 50}), which has led to improved values of {H_1} without EH. For large {k}, we now also have the asymptotic {M_k=\log k - O(1)} with explicit error terms (details here) which have allowed us to slightly improve the {m=2} numerology, and also to get explicit {m=3} numerology for the first time.

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