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

where denotes the prime, as well as the more general quantities

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

The Polymath8a paper then improved this to . After that, Maynard introduced a new multidimensional Selberg sieve argument that gave the substantial improvement

unconditionally, and on the Elliott-Halberstam conjecture; furthermore, bounds on for higher were obtained for the first time, and specifically that for all , with the improvements and 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 and , together with some explicit bounds on ; on the Elliott-Halberstam conjecture we have and some numerical improvements to the bounds; and assuming the generalised Elliott-Halberstam conjecture we have the bound , 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 which reduced to finding a good solution to a certain multidimensional variational problem. When the dimension parameter was relatively small (e.g. ), 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 , we refined the asymptotics and obtained near-optimal solutions of the variational problem. For the bounds, we extended the reach of the multidimensional Selberg sieve (particularly under the assumption of the generalised Elliott-Halberstam conjecture) by allowing the function in the multidimensional variational problem to extend to a larger region of space than was previously admissible, albeit with some tricky new constraints on (and penalties in the variational problem). This required some unusual sieve-theoretic manipulations, notably an “epsilon trick”, ultimately relying on the elementary inequality , that allowed one to get non-trivial lower bounds for sums such as even if the sum had no non-trivial estimates available; and a way to estimate divisor sums such as even if was permitted to be comparable to or even exceed , by using the fundamental theorem of arithmetic to factorise (after restricting to the case when 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 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.

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1 September, 2014 at 8:16 pm

arch1Polymath 8b paper:

Page 13.3: “establishing certain variational problems” -> “establishing certain variational results” ?

Page 13.7: “square integrable” -> “square-integrable”

Page 14.4: “be the defined” -> “be defined”

Page 14.4: “square-integrable F supported in” -> “square-integrable functions F that are supported in”

Page 15.3: “square integrable functions” -> “square-integrable functions”

Page 15.5: “one of the following..hold” -> “one of the following..holds”

Page 15.9, “there is also a version corresponding to part (i) also”: strike an “also”

Page 19.9: “…replace [dj , d’j] with…]” -> “…replace the denominators [dj , d’j] with…]”

[Corrected, thanks – T.]2 September, 2014 at 2:08 am

Eytan PaldiIn Polymathe8b paper, after the definition of in the abstract, it seems clearer to add (as in the introduction) the definition of .

Also, in the first line of page 46, “[52]” should be “[53]”.

[Corrected, thanks – T.]2 September, 2014 at 10:19 am

Terence TaoThanks to all for the many corrections on the Polymath8b paper! The RIMS production staff at Springer are now asking for the LaTeX source, so I’ll wait until this evening for any last remaining corrections and then submit the source to them (and also update the arXiv copy).

2 September, 2014 at 6:37 pm

Terence TaoOK, I have sent back the source files and updated the arXiv copy.

3 September, 2014 at 1:49 pm

Eytan PaldiIn the main page of Polymath8, the links for the arXiv copies of Polymath8a and Polymath8b papers are still for older versions.

[It may take about 24 hours or so for the arXiv to update. -T.]2 September, 2014 at 3:17 pm

arch1Polymath 8b paper:

Page 21.5: Is the meaning of “this contribution” sufficiently clear?

Page 25.3: In the inequality following “We see that”, and in the RHS expansion 2 lines further on, are the ‘3-bar-equals’ signs intended?

Page 29.0: “In particular, the…tuples..contributes…” -> “In particular, the…tuples…contribute…”

Page 30.2: In the LHS of the 2nd standalone equation, shouldn’t all inequalities between the pi (& thus the 5th inequality symbol) be strict?

Page 32.6, “Now that we have proven…we can now establish…”: strike the 2nd “now”

Page 34.8: “..convolves…by…” -> “…convolves…with…” ?

Page 36.2, 2nd equation: Is it allowed to have a top-heavy partial derivative like that?

Page 38.0: “…holds, then…” -> “…holds; then…” (ditto in the next sentence)

Page 38.4: “for some i0 = 1,…,k” -> “for some 1 lte i0 lte k”??

Page 38.6: “…this and Theorem 3.5 is not strong enough…” -> “…this combined with Theorem 3.5 is not strong enough…”?

Page 41.2 “supported of intervals” -> “supported on intervals”

Page 41.2 “with the support of each component…supported in” -> “with each component…supported in”

(end of casual “naive scan”)

[Corrected, thanks – T.]5 September, 2014 at 2:26 pm

AnonymousPolymath 8b paper, v3 on http://arxiv.org/pdf/1407.4897v3.pdf.

The following are for the bibliography:

– Ref. 7: $156$ –> $\mathbf{156}$

– Ref. 10: $70$ –> $\mathbf{70}$

– Ref. 11: $52$ –> $\mathbf{52}$

– Ref. 12: $33$ –> $\mathbf{33}$

– Ref. 13: $4$ –> $\mathbf{4}$

– Ref. 14: to appear in: –> to appear in

– Ref. 16: $152$ –> $\mathbf{152}$ + E. –> É [two things]

– Ref. 20: $4$ –> $\mathbf{4}$

– Ref. 21: 945—1040 –> 945–1040

– Ref. 22: $42$ –> $\mathbf{42}$

– Ref. 25: $170$ –> $\mathbf{170}$

– Ref. 26: $361$ –> $\mathbf{361}$

– Ref. 30: $44$ –> $\mathbf{44}$

– Ref. 31: $34$ –> $\mathbf{34}$

– Ref. 32: pp. 123–127 –> 123–127

– Ref. 33: $25$ –> $\mathbf{25}$

– Ref. 34: Publications Vol. $53$ –> Publications, vol. $53$ [two things]

– Ref. 36: $34$ –> $\mathbf{34}$

– Ref. 40: volume $227$ of Lecture Notes in Math. Springer –> Lecture Notes in Math. Springer, vol. 227 [two things]

– Ref. 42: $52$ –> $\mathbf{52}$

– Ref. 43: $40$ –> $\mathbf{40}$

– Ref. 44: pp. 525–559 –> 525–559

– Ref. 54: $80$ –> $\mathbf{80}$

– Ref. 55: $7$ (1961/1962) 1–8 –> $\mathbf{7}$ (1961/1962), 1–8 [two things]

– Ref. 56: $7$ –> $\mathbf{7}$

– Ref. 57: Vol. I –> vol. I

– Ref. 58: in: –> in

– Ref. 59: $44$ –> $\mathbf{44}$

– Ref. 22 should in between ref. 17 and 18

7 September, 2014 at 2:55 am

Eytan PaldiIn Polymath8 home page, the table of current records (“without EH”) should be updated to the bounds in Polymath8b paper (theorem 1.4(ii)-(vi)).

[Actually, the records are slightly better than the 8b paper, because they use a provisional equidistribution estimate whose proof has not been fully written up due to the need to check some tricky algebraic geometry. There are some people looking to flesh out the details of that estimate, though this would be done by a more traditional collaboration, i.e. outside of the Polymath8 project scope. -T.]18 September, 2014 at 10:50 am

Eytan PaldiIn Polymath8b paper (page 18), the factorization of the integral kernel into the factors (given by (39)) may be verified via the following very general (and simple!) factorization principle:

Let be a bijection from the product of finite nonempty sets onto a set . Let be a complex-valued function on with the (factorization) property:

For every , where are certain complex-valued functions on , correspondingly. Therefore

Remark: This factorization principle is for finite sums and products (but under certain regularity conditions it may be extended for countable sums and products.)

25 September, 2014 at 1:48 pm

Terence TaoJust a note that all of the people who had agreed to contribute to the retrospective article at https://www.dropbox.com/sh/koxbhwvw1ysybk9/AADTJ4w3yegvgTut_Tsv0Sana/retrospective.pdf have now done so. I’ll wait until Monday in case there are any last minute edits or suggestions, and then submit to the EMS newsletter.

26 September, 2014 at 12:56 pm

Eytan PaldiIn the definition of (in both the abstract and introduction), should be inside parentheses.

[Corrected, thanks – T.]28 September, 2014 at 1:13 am

Eytan PaldiIt should also be corrected in the introduction (page2, line 14 from below.)

[Corrected, thanks – T.]28 September, 2014 at 7:40 am

Eytan PaldiThe closing parenthesis is misplaced.

[Corrected, thanks – T.]29 September, 2014 at 4:40 am

Eytan PaldiIn the retrospective article, after “” in the introduction (line10 above the introduction end), “generalized” should be added before “Elliott-Halberstam conjecture”.

[Corrected, thanks – T.]25 September, 2014 at 3:03 pm

AnonymousFootnote 5: “Nelson and” –> “Nelson, and”

Footnote 6: “the” –> “The” + “2004” –> “2004.”

[Corrected, thanks – T.]27 September, 2014 at 5:36 am

Eytan PaldiSome typos:

1. In Polymath8a paper (definition 1.4 in the old unabridged version, definition 1.2 in the new version) and in Polymath8b paper (definition 1.6), it seems that “” should be ““.

2. In Polymath8a paper (page 22), “” should be “” in line 4, and “” in line 6.

[Corrected, thanks – T.]27 September, 2014 at 7:14 am

AnonymousRetrospective article, Terrence Tao’s input:

– P. 3, l. -4: “The first that” –> “The first time that”?

– P. 3, l. -1: “skeptical – the” –> “skeptical—the” or “skeptical\,—\,the”

– P. 7, 2nd paragraph, l. 7: “size – 163” –> “size—163” or “size\,—\,163”

[Corrected, thanks – T.]29 September, 2014 at 5:59 pm

Terence TaoI’ve submitted the retrospective to the newsletter of the EMS and uploaded it to the arXiv. I’ll make a blog post about it when the arXiv paper comes out (and it’s about time to roll over the polymath8 thread again anyway, though presumably this will be the final time to do so, as the project now seems to be more or less concluded).

30 September, 2014 at 12:20 pm

Eytan PaldiIn polymath8b paper, in the line below the estimate (2), “[52, Lemma 1.5]” should be “[53, Lemma 1.5]” (or “[52, Lemma 1.3]”), but it seems that (2) is not explicitly stated in these lemmas.

[Corrected, thanks – T.]1 October, 2014 at 2:22 am

Eytan PaldiIn Polymath8b paper (page 5), “[52, Lemma 1.5]” (in the line below the estimate (2)) should still be corrected to “[52, Lemma 1.3]” – since lemma 1.5 of the older unabridge version [53] is now lemma 1.3 of the new version [52] of Polymath8a paper.

Although the estimate (2) is not explicitly stated in [52, lemma 1.3], it easily follows (using summation by parts) from [52, lemma 1.3(ii)].

[Oops, the correction did not update properly before; fixed now. -T.]30 September, 2014 at 8:58 pm

The “bounded gaps between primes” Polymath project – a retrospective | What's new[…] 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 […]

6 October, 2014 at 5:09 pm

AnonThe “work of Zhang” anchor is broken.

[Corrected, thanks – T.]19 March, 2016 at 9:11 am

IlaI do believe all the ideas you have introducerd for our post.

They’rereally convincing and can definitely work.

Nonetheless, the posts are very brief for novices. May just you please extend

them a litttle from neext time? Thanks for the post.

17 July, 2016 at 8:54 am

Notes on the Bombieri asymptotic sieve | What's new[…] of the generalised Elliott-Halberstam (GEH) conjecture (as formulated for instance in Claim 2.6 of the Polymath8b paper), one can also swap the factor in the above asymptotics with other parity-violating weights and […]

1 March, 2017 at 6:32 pm

Terence TaoNot sure how many of the contributors to this paper will see this comment three years after the fact, but: the journal that this paper was published in (Research in the Mathematical Sciences) has offered to give our paper a video abstract to feature on their journal (created by their affiliate, Research Square). Two examples of the abstracts they have made can be found at https://www.youtube.com/watch?v=GMhZyYeBap4 and https://www.youtube.com/watch?v=PsGfuSGgHyk . I’m willing to work with them on this, but wondered if any of the other former participants had any thoughts on the matter.

2 March, 2017 at 1:57 am

Andrew SutherlandI’d be happy to help.

16 May, 2019 at 8:28 am

Hoa HuynhRecently I made some discovery in Twin Prime Conjecture, and posted the work on my Facebook page:

https://www.facebook.com/TwinPrimeConjecture/

I made one Conjecture and one Thesis from this discovery.

Conjecture:

On average, the number of Twin Prime Pairs grows as the interval (L1, L2) ascends the number line; where L1 = (6k-1)^2 and L2 = (6k+1)^2, and k is natural number.

Thesis:

The discovery of formulas for L0, L1, and L2 helps to predict the growth of number of Twin Prime Pairs (#TPPs) in a controlled manner, even the Twin Prime Density (TPD) is thinning out, up the number line.

( L0 = previous L2 = (6(k – 1) + 1)^2 = (6k – 5) ^2 )

This means Twin Prime Conjecture is TRUE!

Please visit, like, share and comment.

Thank you,

Hoa

16 May, 2019 at 1:26 pm

Hoa HuynhTest with k = 5, 50, 500, 5000

k = 5 50 500 5000

L1 = 841 89401 8994001 899940001

L2 = 961 90601 9006001 900060001

Pair = 2 14 54 376

k = natural number

length of (L1, L2) = 24k (grows linearly); 24 is the size of the sieve of 4 rows x 6 columns

(L1, L2) needs to grow to compensate for the thinning of Twin Prime Density.

Number of Twin Prime Pairs grows (on average, or from big picture.)

17 May, 2019 at 8:26 am

Hoa HuynhFacebook posts:

May 13, 2019. Report of data and graphs. 6 pages.

May 13, 2019: Appendix 4 and conclusion. 1 page.

22 May, 2019 at 9:08 pm

Hoa HuynhJust posted, May 22th:

a) Python program to count TPPs in (L1, L2.)

b) TPC: Graphical data tells better story than I do.

24 May, 2019 at 6:48 pm

Hoa HuynhJust posted: Appendix 5: Twin Prime Pairs are not as scarce as we thought.3 pages, 2 tables, 4 graphs.

31 May, 2019 at 9:41 am

Hoa HuynhJust posted, Appendix 6 Retrospect: My journey with Twin Prime Conjecture. 5 pages.

25 March, 2020 at 4:11 am

sylvainjulienMaybe this link can be of interest:

http://www.les-mathematiques.net/phorum/read.php?5,1758830

A proof that the set of integers n such that r0(n)=1 has positive density among the set of integers m such that r0(m) divides 6 would entail that the GEH conjecture implies the twin prime conjecture.