The main purpose of this post is to roll over the discussion from the previous Polymath8 thread, which has become rather full with comments. We are still writing the paper, but it appears to have stabilised in a near-final form (source files available here); the main remaining tasks are proofreading, checking the mathematics, and polishing the exposition. We also have a tentative consensus to submit the paper to Algebra and Number Theory when the proofreading is all complete.
The paper is quite large now (164 pages!) but it is fortunately rather modular, and thus hopefully somewhat readable (particularly regarding the first half of the paper, which does not need any of the advanced exponential sum estimates). The size should not be a major issue for the journal, so I would not seek to artificially shorten the paper at the expense of readability or content.
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15 October, 2013 at 11:41 am
Eytan Paldi
In lines 12, 15 below (3.5), the “
” should be the standard one.
15 October, 2013 at 2:37 pm
Andrew Sutherland
I am giving a colloquium talk on Zhang’s proof and the polymath8 project on Thursday. Here is a draft of my slides.
I would welcome any comments and (especially) corrections. I have tried to give a brief overview of Zhang/GPY and to summarize the main Polymath8 improvements. I necessarily had to surpress a lot of detail, but I hope I haven’t said anything that is actually false by doing so. The last part of the talk focuses on admissible tuples, not because I think this part is particularly important (it obviously isn’t), but it is the easiest piece to explain to a general audience and the area with which I am most familiar. But I would definitely welcome suggestions if anyone thinks I have left out (or underemphasized) something important.
I also plan to add some graphs and to demo sieving with the applet, but I am still working on these pieces.
Once I get the talk finalized I can post the latex source in case anyone wants to adapt it for their own use (or just copy pieces of it).
15 October, 2013 at 3:24 pm
Gergely Harcos
Nice slides! Some comments:
Page 3 of 27: (log t)^k should be (log t)^2
Page 9 of 27: (??) should be (1)
Page 12 of 27: I would remark that the implication MPZ[…]->DHL[k,2] was
published earlier by Motohashi and Pintz.
Page 17 of 27: I would perhaps mention the H=5414 result that was definite for a while.
Page 18 of 27: theta should be 1/2
15 October, 2013 at 3:27 pm
Terence Tao
Looks good! Some minor corrections:
Slide 4: the equals sign in the Hardy-Littlewood conjecture should be a tilde. Also one needs to specify that c_H is positive :)
Slide 9: LaTeX reference error.
Slide 11: The first part of this theorem was also essentially proven independently by Motohashi and Pintz (though they didn’t write it so explicitly).
Slide 17: Very nice compression of the enormous table!
Slide 18: The second part of this theorem was found shortly before us by Farkas, Pintz, and Revesz (although we discovered it independently).
Slide 22: A very minor technical point: Strictly speaking, the Brun-Titchmarsh inequality as stated only implies the lower bound on H(k) assuming the prime tuples conjecture. But the Brun-Titchmarsh inequality applies not only to primes in an interval, but any set formed by sieving out one residue class mod p for each p from an interval, and this does give the lower bound on H(k). [Not sure how you would fit this technicality onto one slide though!]
15 October, 2013 at 4:36 pm
Andrew Sutherland
Thanks for the quick feedback! I have posted an updated version of the slides at the link above.
I didn’t put the k0=720 H=5414 line into the table yet, but I may add it. I do have one question about this — at this point do we regard the 4680 bound in our abstract as any less solid than 5414? (I notice that 4680 still has a question mark on the wiki).
15 October, 2013 at 4:59 pm
Terence Tao
Well, the 4680 bound relies on the argument in Section 10 of the paper, which I have gone through line-by-line, and Phillipe has also made some corrections to that part of the paper, but as far as I know we’re the only ones to have seriously looked at that part of the argument. If Emmanuel or Gergely makes it all the way to that section and signs off on it, I’m happy to remove the question mark from the wiki :).
15 October, 2013 at 5:27 pm
Andrew Sutherland
Thanks for the clarification — for the moment I went ahead and put in a line with the 5414 bound as well.
15 October, 2013 at 6:41 pm
Polymath8: Writing the paper, IV | What's new
[…] again it is time to roll over the previous discussion thread, which has become rather full with comments. The paper is nearly finished (see also the working […]
15 October, 2013 at 6:43 pm
Terence Tao
Time to roll over the thread again! Of course, any direct responses to comments here can still be made on this thread, but it will probably be easier for readers to follow new comments if they are made on the fresh thread.
15 October, 2013 at 7:16 pm
Pace Nielsen
Things are settling down, now that I’ve written my grant proposal that I can’t submit. :-p So I started reading through the paper. Here are just some minor typos/comments/questions I found in the first two sections. Feel free to ignore any/all of the comments if you so choose.
1. In the statement of Theorem 1.2 the reference to Zhang’s paper [79] is given, even though it is listed directly above the theorem. Probably it isn’t needed.
2. In the first paragraph after Theorem 1.2, there is a list of papers (currently numbered [23, 24, 25, 32, 7, 8, 9]). I was wondering if there was a reason they are not in numerical order.
3. Sentence above equation 1.1, currently near the top of page 4. It says that GPY “obtained the partial result…”. Perhaps “preliminary result” would be a better way to put it.
4. Two paragraphs later it says “weakened from of the Elliot-Halberstam conjecture…” The word “from” should be “form”.
5. Five paragraphs later (currently near the top of page 5), it mentions the “Motohashi-Pintz-Zhang conjecture”. First, I would refer to Claim 2.8 here so the reader knows what this is. Second, I would replace the word “conjecture” with “estimate” (as is done in Claim 2.8). Have any of these authors made a formal conjecture of this form? [If this change is made, perhaps a search-and-replace for “conjecture” in conjunction with MPZ should be done.]
6. Three paragraphs later (currently near the bottom of page 5 after the offset equation) there is a space missing at the beginning of the sentence “We obtain significant improvements…”
7. Later in the same paragraph, change “were also obtained” to “are also obtained” to make tenses match.
8. In the statement of Lemma 1.5, the variable q is used in two ways. In part (i) as a polynomial size variable, and in part (iii) as a modulus with apparently no size restriction. I would recommend a different variable name be used in the first case. Also, does q need to still have polynomial size in part (iii)? [Lemma 1.6 seems to use that restriction, but I may be missing something here.]
9. In the caption to Figure 1, there is a space missing after the comma in “Specifically,the”
10. When defining the notation DHL[k_0,2] in Claim 2.1, why not define the fuller notation DHL[k_0,m], so as to make it clear to the reader why there is a 2 here?
11. Paragraph before Remark 2.5 (currently near the bottom of page 12). In the sentence “It was a crucial progress…” remove “a”.
12. Two paragraphs after Theorem 2.6, change “Assuming a claim EH[1/2+2\varpi] with \varpi>0” to “Assuming the claim E[1/2+2\varpi] for some \varpi>0”
13. Footnote 4, currently at the bottom of page 13: is there a reference to the Bombieri-Davenport result?
14. Paragraph after Claim 2.8 (currently top of page 14). Change “the residue class modulo all q…” to “the residue class modulo each q….”
15. Three paragraphs later, the word “barrier” is misspelled as “bareer”
16. Two paragraphs later it says “for any fixed $a$ such that $1\leq |x| \leq x$”. Isn’t the second condition automatically satisfied for a nonzero fixed $a$?
17. Page 15, offset equation after the ternary divisor function. What does \Delta(x;a (q)) mean? The first entry of \Delta should be an arithmetic function with finite support. Also, it isn’t clear what this bound has to do with the ternary divisor bounds.
18. Paragraph preceding Theorem 2.12 (currently near the top of page 16). Change J_{k_0 -2} to J_{\nu} in two places, and change the equality j_{k_0 -2} = j_{k_0 -2, 1} by also substituting in \nu’s.
19. Top of page 18 change “consider densely divisible integers with prime factors…” by adding the word “squarefree”(near the beginning).
20. In the statement of Claim 2.15, since 1\leq q\leq Q, why does $I$ need to be a bounded set? (Taking $I$ to be the closed interval [1,Q] is the same as making no restriction on $I$, is it not?)
21. Statement of Theorem 2.17(ii): change “We can prove…” to “We have…” just as in part (i).
22. Paragraph above Definition 2.20: “Estimating sufficiently precisely these sums…” should probably be reworded, something like “Estimating the sums in a sufficiently precise way…”
23. Throughout the paper, commas and periods are placed outside of quotation marks, but they should be inside of them. [And yes, I’m not obeying that rule in this post myself!] I recommend a search and replace.
24. Definition 2.20, part (iii). Don’t you need N\geq 1, so that the last sentence can hold true (that \alpha is smoot at scale N)? Also, replace
“and takes the form…” with “if it takes the form…”.
That’s it! Hopefully this is helpful.
15 October, 2013 at 8:14 pm
Emmanuel Kowalski
Thanks a lot, I’ll go through the file and correct these some time today…
16 October, 2013 at 12:18 am
Emmanuel Kowalski
I corrected these; here are comments on some of these points.
2. In the first paragraph after Theorem 1.2, there is a list of papers (currently numbered [23, 24, 25, 32, 7, 8, 9]). I was wondering if there was a reason they are not in numerical order.
These are (intended: 25 should have been be last) in chronological order.
3. Sentence above equation 1.1, currently near the top of page 4. It says that GPY “obtained the partial result…”. Perhaps “preliminary result” would be a better way to put it.
I think “partial” is better because it wasn’t clear that this was preliminary to the bounded gap property.
8. Also, does q need to still have polynomial size in part (iii)? [Lemma 1.6 seems to use that restriction, but I may be missing something here
As the proof explains, there is no restriction on q because we deal separately with q>y and q\leq y (and for q>y there is at most one term tau(d)^C in the sum, which is << y^{eps} << x^{eps}.)
13. Footnote 4, currently at the bottom of page 13: is there a reference to the Bombieri-Davenport result?
I will add the reference — I had forgotten I had inserted this remark.
16. Two paragraphs later it says “for any fixed $a$ such that $1\leq |x| \leq x$”. Isn’t the second condition automatically satisfied for a nonzero fixed $a$?
This was meant to mean uniformly for such a, so "fixed" shouldn't be there.
17. Page 15, offset equation after the ternary divisor function. What does \Delta(x;a (q)) mean? The first entry of \Delta should be an arithmetic function with finite support. Also, it isn’t clear what this bound has to do with the ternary divisor bounds.
This should have been \Delta(\tau_3; a\ (q))
20. In the statement of Claim 2.15, since 1\leq q\leq Q, why does $I$ need to be a bounded set? (Taking $I$ to be the closed interval [1,Q] is the same as making no restriction on $I$, is it not?)
This is basically unchanged from the first version, the restriction seems indeed unnecessary. Maybe Terry can comment on why it was put there?
16 October, 2013 at 6:08 am
Terence Tao
The only reason
is required to be a bounded set is that so one can interpret the product 
of all the primes in 
as a finite integer (as opposed to a profinite integer or adele or something more exotic). This restriction also appears in Definition 2.7 and Definition 2.21.
One could of course restrict to subsets of [1,Q] without loss of generality (and for our application one can take [w,Q] for instance); actually, now that I think about it, the Chinese remainder theorem shows that the I = [1,Q] case implies all the other cases. So maybe a remark after the claim on these points would be in order.
16 October, 2013 at 6:59 am
Pace Nielsen
Emmanuel, thanks for answering my questions.