Bounded gaps between primes

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This is the home page for the Polymath8 project "bounded gaps between primes".

World records

  • [math]\displaystyle{ H }[/math] is a quantity such that there are infinitely many pairs of consecutive primes of distance at most [math]\displaystyle{ H }[/math] apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).
  • [math]\displaystyle{ k_0 }[/math] is a quantity such that every admissible [math]\displaystyle{ k_0 }[/math]-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in [math]\displaystyle{ k_0 }[/math] lead to improvements in [math]\displaystyle{ H }[/math]. (The relationship is roughly of the form [math]\displaystyle{ H \sim k_0 \log k_0 }[/math]; see the page on finding narrow admissible tuples.)
  • [math]\displaystyle{ \varpi }[/math] is a technical parameter related to a specialized form of the Elliott-Halberstam conjecture. Would like to be as large as possible. Improvements in [math]\displaystyle{ \varpi }[/math] lead to improvements in [math]\displaystyle{ k_0 }[/math], as described in the page on Dickson-Hardy-Littlewood theorems. In more recent work, the single parameter [math]\displaystyle{ \varpi }[/math] is replaced by a pair [math]\displaystyle{ (\varpi,\delta) }[/math] (in previous work we had [math]\displaystyle{ \delta=\varpi }[/math]). These estimates are obtained in turn from Type I, Type II, and Type III estimates, as described at the page on distribution of primes in smooth moduli.

In this table, infinitesimal losses in [math]\displaystyle{ \delta,\varpi }[/math] are ignored.

Date [math]\displaystyle{ \varpi }[/math] or [math]\displaystyle{ (\varpi,\delta) }[/math] [math]\displaystyle{ k_0 }[/math] [math]\displaystyle{ H }[/math] Comments
14 May 1/1,168 (Zhang) 3,500,000 (Zhang) 70,000,000 (Zhang) All subsequent work is based on Zhang's breakthrough paper.
21 May 63,374,611 (Lewko) Optimises Zhang's condition [math]\displaystyle{ \pi(H)-\pi(k_0) \gt k_0 }[/math]; can be reduced by 1 by parity considerations
28 May 59,874,594 (Trudgian) Uses [math]\displaystyle{ (p_{m+1},\ldots,p_{m+k_0}) }[/math] with [math]\displaystyle{ p_{m+1} \gt k_0 }[/math]
30 May 59,470,640 (Morrison)

58,885,998? (Tao)

59,093,364 (Morrison)

57,554,086 (Morrison)

Uses [math]\displaystyle{ (p_{m+1},\ldots,p_{m+k_0}) }[/math] and then [math]\displaystyle{ (\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1}) }[/math] following [HR1973], [HR1973b], [R1974] and optimises in m
31 May 2,947,442 (Morrison)

2,618,607 (Morrison)

48,112,378 (Morrison)

42,543,038 (Morrison)

42,342,946 (Morrison)

Optimizes Zhang's condition [math]\displaystyle{ \omega\gt 0 }[/math], and then uses an improved bound on [math]\displaystyle{ \delta_2 }[/math]
1 Jun 42,342,924 (Tao) Tiny improvement using the parity of [math]\displaystyle{ k_0 }[/math]
2 Jun 866,605 (Morrison) 13,008,612 (Morrison) Uses a further improvement on the quantity [math]\displaystyle{ \Sigma_2 }[/math] in Zhang's analysis (replacing the previous bounds on [math]\displaystyle{ \delta_2 }[/math])
3 Jun 1/1,040? (v08ltu) 341,640 (Morrison) 4,982,086 (Morrison)

4,802,222 (Morrison)

Uses a different method to establish [math]\displaystyle{ DHL[k_0,2] }[/math] that removes most of the inefficiency from Zhang's method.
4 Jun 1/224?? (v08ltu)

1/240?? (v08ltu)

4,801,744 (Sutherland)

4,788,240 (Sutherland)

Uses asymmetric version of the Hensley-Richards tuples
5 Jun 34,429? (Paldi/v08ltu)

34,429? (Tao/v08ltu/Harcos)

4,725,021 (Elsholtz)

4,717,560 (Sutherland)

397,110? (Sutherland)

4,656,298 (Sutherland)

389,922 (Sutherland)

388,310 (Sutherland)

388,284 (Castryck)

388,248 (Sutherland)

388,188 (Sutherland)

387,982 (Castryck)

387,974 (Castryck)

[math]\displaystyle{ k_0 }[/math] bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.

[math]\displaystyle{ H }[/math] bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

6 Jun (1/488,3/9272) (Pintz)

1/552 (Pintz, Tao)

60,000* (Pintz)

52,295* (Peake)

11,123 (Tao)

387,960 (Angelveit)

387,910 (Sutherland)

387,904 (Angeltveit)

387,814 (Sutherland)

387,766 (Sutherland)

387,754 (Sutherland)

387,620 (Sutherland)

768,534* (Pintz)

Improved [math]\displaystyle{ H }[/math]-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
7 Jun (1/538, 1/660) (v08ltu)

(1/538, 31/20444) (v08ltu)

(1/942, 19/27004) (v08ltu)

[math]\displaystyle{ 828 \varpi + 172\delta \lt 1 }[/math] (v08ltu/Green)

11,018 (Tao)

10,721 (v08ltu)

10,719 (v08ltu)

25,111 (v08ltu)

26,024? (vo8ltu)

113,520? (Angeltveit)

109,314? (Angeltveit/Sutherland)

707,328* (Sutherland)

108,990 (Sutherland)

113,462* (Sutherland)

112,302* (Sutherland)

112,272* (Sutherland)

116,386* (Sun)

108,978 (Sutherland)

108,634 (Sutherland)

108,632 (Castryck)

108,600 (Sutherland)

108,570 (Castryck)

108,556 (Sutherland)

108,550 (xfxie)

275,424 (Sutherland)

108,540 (Sutherland)

275,418 (Sutherland)

275,404 (Sutherland)

275,292 (Castryck-Sutherland)

275,262 (Castryck-pedant-Sutherland)

275,388* (xfxie-Sutherland)

275,126 (Castryck-pedant-Sutherland)

274,970 (Castryck-pedant-Sutherland)

275,208* (xfxie)

387,534 (pedant-Sutherland)

Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
Jun 8 286,224 (Sutherland)

285,810 (Sutherland)

286,216 (xfxie-Sutherland)

386,750* (Sutherland)

285,752 (pedant-Sutherland)

285,456 (Sutherland)

values of [math]\displaystyle{ \varpi,\delta,k_0 }[/math] now confirmed; most tuples available on dropbox. New bounds on [math]\displaystyle{ H }[/math] obtained via iterated merging using a randomized greedy sieve.
Jun 9 181,000*? (Pintz) 2,530,338*? (Pintz)

285,278 (Sutherland/xfxie)

285,272 (Sutherland)

285,248 (Sutherland)

285,246 (xfxie-Sutherland)

285,232 (Sutherland)

New bounds on [math]\displaystyle{ H }[/math] obtained by interleaving iterated merging with local optimizations.
Jun 10 23,283? (Harcos/v08ltu) 285,210 (Sutherland)

253,118 (xfxie)

386,532* (Sutherland)

253,048 (Sutherland)

252,990 (Sutherland)

252,976 (Sutherland)

More efficient control of the [math]\displaystyle{ \kappa }[/math] error using the fact that numbers with no small prime factor are usually coprime
Jun 11 252,804 (Sutherland)

2,345,896* (Sutherland)

More refined local "adjustment" optimizations, as detailed here.

An issue with the [math]\displaystyle{ k_0 }[/math] computation has been discovered, but is in the process of being repaired.

Jun 12 22,951 (Tao/v08ltu)

22,949 (Harcos)

249,180 (Castryck)

249,046 (Sutherland)

249,034 (Sutherland)

Improved bound on [math]\displaystyle{ k_0 }[/math] avoids the technical issue in previous computations.
Jun 13

248,970 (Sutherland)

248,910 (Sutherland)

Jun 14 248,898 (Sutherland)
Jun 15 [math]\displaystyle{ 348\varpi+68\delta \lt 1 }[/math]? (Tao) 6,330? (v08ltu)

6,329? (Harcos)

6,329 (v08ltu)

60,830? (Sutherland)

60,812? (Sutherland)

60,764 (xfxie)

60,772* (xfxie)

60,760 (xfxie)

Taking more advantage of the [math]\displaystyle{ \alpha }[/math] convolution in the Type III sums
Jun 16 [math]\displaystyle{ 348\varpi+68\delta \lt 1 }[/math] (v08ltu)

155\varpi+31\delta < 1 and 220\varpi + 60\delta < 1 (Tao)

3,405 (v08ltu) 60,760* (Sutherland)

60,756 (Sutherland)

60,754 (xfxie)

60,744 (Sutherland)

30,610* (Sutherland)

30,606 (Engelsma)

30,600 (Sutherland)

Attempting to make the Weyl differencing more efficient; unfortunately, it did not work
Jun 18 5,937? (Pintz/Tao/v08ltu)

5,672? (v08ltu)

5,459? (v08ltu)

5,454? (v08ltu)

5,453? (v08ltu)

60,740 (xfxie)

60,732 (Sutherland)

60,726 (xfxie-Sutherland)

58,866? (Sun)

56,660? (Sutherland)

56,640? (Sutherland)

53,898? (Sun)

53,842? (Sun)

A new truncated sieve of Pintz virtually eliminates the influence of [math]\displaystyle{ \delta }[/math]
Jun 19 5,455? (v08ltu)

5,453? (v08ltu)

5,452? (v08ltu)

53,774? (Sun)

51,544? (Sutherland)

51,540? (xfxie/Sutherland)

51,532? (Sutherland)

51,526? (Sutherland)

53,672*? (Sun)

51,520? (Sutherland/Hou-Sun)

Some typos in [math]\displaystyle{ \kappa_3 }[/math] estimation had placed the 5,454 and 5,453 values of [math]\displaystyle{ k_0 }[/math] into doubt; however other refinements have counteracted this
Jun 20 [math]\displaystyle{ 178\varpi + 52\delta \lt 1 }[/math]? (Tao)

[math]\displaystyle{ 148\varpi + 33\delta \lt 1 }[/math]? (Tao)

Replaced "completion of sums + Weil bounds" in estimation of incomplete Kloosterman-type sums by "Fourier transform + Weyl differencing + Weil bounds", taking advantage of factorability of moduli
Jun 21 [math]\displaystyle{ 148\varpi + 33\delta \lt 1 }[/math] (v08ltu) 1,470 (v08ltu)

1,467 (v08ltu)

12,042 (Engelsma)

12,012 (Engelsma)

Systematic tables of tuples of small length have been set up here and here (update: As of June 27 these tables have been merged and uploaded to an online database of current bounds on [math]\displaystyle{ H(k) }[/math] for [math]\displaystyle{ k }[/math] up to 5000).
Jun 22 1,466 (Harcos/v08ltu) 12,006 (Engelsma) Slight improvement in the [math]\displaystyle{ \tilde \theta }[/math] parameter in the Pintz sieve; unfortunately, it does not seem to currently give an actual improvement to the optimal value of [math]\displaystyle{ k_0 }[/math]
Jun 23 1,466 (Paldi/Harcos) 12,006 (Engelsma) An improved monotonicity formula for [math]\displaystyle{ G_{k_0-1,\tilde \theta} }[/math] reduces [math]\displaystyle{ \kappa_3 }[/math] somewhat
Jun 24 [math]\displaystyle{ (134 + \tfrac{2}{3}) \varpi + 28\delta \le 1 }[/math]? (v08ltu)

[math]\displaystyle{ 140\varpi + 32 \delta \lt 1 }[/math]? (Tao)

1/88?? (Tao)

1/74?? (Tao)

1,268? (v08ltu) 10,206? (Engelsma) A theoretical gain from rebalancing the exponents in the Type I exponential sum estimates
Jun 25 [math]\displaystyle{ 116\varpi+30\delta\lt 1 }[/math]? (Fouvry-Kowalski-Michel-Nelson/Tao) 1,346? (Hannes)

502?? (Trevino)

1,007? (Hannes)

10,876? (Engelsma)

3,612?? (Engelsma)

7,860? (Engelsma)

Optimistic projections arise from combining the Graham-Ringrose numerology with the announced Fouvry-Kowalski-Michel-Nelson results on d_3 distribution
Jun 26 [math]\displaystyle{ 116\varpi + 25.5 \delta \lt 1 }[/math]? (Nielsen)

[math]\displaystyle{ (112 + \tfrac{4}{7}) \varpi + (27 + \tfrac{6}{7}) \delta \lt 1 }[/math]? (Tao)

962? (Hannes) 7,470? (Engelsma) Beginning to flesh out various "levels" of Type I, Type II, and Type III estimates, see this page, in particular optimising van der Corput in the Type I sums. Integrated tuples page now online.
Jun 27 [math]\displaystyle{ 108\varpi + 30 \delta \lt 1 }[/math]? (Tao) 902? (Hannes) 6,966? (Engelsma) Improved the Type III estimates by averaging in [math]\displaystyle{ \alpha }[/math]; also some slight improvements to the Type II sums. Tuples page is now accepting submissions.
Jul 1 [math]\displaystyle{ (93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta \lt 1 }[/math]? (Tao)

873? (Hannes)

872? (xfxie)

6,712? (Sutherland)

6,696? (Engelsma)

Refactored the final Cauchy-Schwarz in the Type I sums to rebalance the off-diagonal and diagonal contributions
Jul 5 [math]\displaystyle{ (93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta \lt 1 }[/math] (Tao)

720 (xfxie/Harcos)

5,414 (Engelsma)

Weakened the assumption of [math]\displaystyle{ x^\delta }[/math]-smoothness of the original moduli to that of double [math]\displaystyle{ x^\delta }[/math]-dense divisibility

Jul 10 7/600? (Tao) An in principle refinement of the van der Corput estimate based on exploiting additional averaging
Jul 19 [math]\displaystyle{ (85 + \frac{5}{7})\varpi + (25 + \frac{5}{7}) \delta \lt 1 }[/math]? (Tao) A more detailed computation of the Jul 10 refinement
Jul 20 Jul 5 computations now confirmed
Jul 27 633? (Tao)

632? (Harcos)

4,686? (Engelsma)

4,680? (Engelsma)

Jul 30 [math]\displaystyle{ 168\varpi + 48\delta \lt 1 }[/math]**? (Tao) 1,788**? (Tao) 14,994**? (Sutherland) Bound obtained without using Deligne's theorems.
Aug 17 1,783**? (xfxie) 14,950**? (Sutherland)
Oct 3 13/1080?? (Nelson/Michel/Tao) 604?? (Tao) 4,428?? (Engelsma) Found an additional variable to apply van der Corput to
Oct 11 [math]\displaystyle{ \frac{1080}{13}\varpi + \frac{330}{13} \delta \lt 1 }[/math]? (Tao) 603? (xfxie) 4,422?(Engelsma) Worked out the dependence on [math]\displaystyle{ \delta }[/math] in the Oct 3 calculation


Legend:

  1. ? - unconfirmed or conditional
  2. ?? - theoretical limit of an analysis, rather than a claimed record
  3. * - is majorized by an earlier but independent result
  4. ** - bound does not rely on Deligne's theorems
  5. strikethrough - values relied on a computation that has now been retracted


See also the article on Finding narrow admissible tuples for benchmark values of [math]\displaystyle{ H }[/math] for various key values of [math]\displaystyle{ k_0 }[/math].

Polymath threads

  1. I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart, Scott Morrison, 30 May 2013. Inactive
  2. The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang, Terence Tao, 3 June 2013. Inactive
  3. Polymath proposal: bounded gaps between primes, Terence Tao, 4 June 2013. Inactive
  4. Online reading seminar for Zhang’s “bounded gaps between primes”, Terence Tao, 4 June 2013. Inactive
  5. More narrow admissible sets, Scott Morrison, 5 June 2013. Inactive
  6. The elementary Selberg sieve and bounded prime gaps, Terence Tao, 8 June 2013. Inactive
  7. A combinatorial subset sum problem associated with bounded prime gaps, Terence Tao, 10 June 2013. Inactive
  8. Further analysis of the truncated GPY sieve, Terence Tao, 11 June 2013. Inactive
  9. Estimation of the Type I and Type II sums, Terence Tao, 12 June 2013. Inactive
  10. Estimation of the Type III sums, Terence Tao, 14 June 2013. Inactive
  11. A truncated elementary Selberg sieve of Pintz, Terence Tao, 18 June, 2013. Inactive
  12. Bounding short exponential sums on smooth moduli via Weyl differencing, Terence Tao, 22 June, 2013. Inactive
  13. The distribution of primes in densely divisible moduli, Terence Tao, 23 June, 2013. Inactive
  14. Bounded gaps between primes (Polymath8) – a progress report, Terence Tao, 30 June 2013. Inactive
  15. The quest for narrow admissible tuples, Andrew Sutherland, 2 July 2013. Inactive
  16. The distribution of primes in doubly densely divisible moduli, Terence Tao, 7 July 2013. Inactive.
  17. An improved Type I estimate, Terence Tao, 27 July 2013. Inactive
  18. Polymath8: writing the paper, Terence Tao, 17 August 2013. Inactive
  19. Polymath8: writing the paper, II, Terence Tao, 2 September 2013. Inactive
  20. Polymath8: writing the paper, III, Terence Tao, 22 September 2013. Active

Writeup

Files for the draft paper for this project may be found in this directory. The compiled PDF is available here.

Here are the Polymath8 grant acknowledgments.

Code and data

Tuples applet

Here is a small javascript applet that illustrates the process of sieving for an admissible 632-tuple of diameter 4680 (the sieved residue classes match the example in the paper when translated to [0,4680]).

The same applet can also be used to interactively create new admissible tuples. The default sieve interval is [0,400], but you can change the diameter to any value you like by adding “?d=nnnn” to the URL (e.g. use https://math.mit.edu/~primegaps/sieve.html?d=4680 for diameter 4680). The applet will highlight a suggested residue class to sieve in green (corresponding to a greedy choice that doesn’t hit the end points), but you can sieve any classes you like.

You can also create sieving demos similar to the k=632 example above by specifying a list of residues to sieve. A longer but equivalent version of the “?ktuple=632″ URL is

https://math.mit.edu/~primegaps/sieve.html?d=4680&r=1,1,4,3,2,8,2,14,13,8,29,31,33,28,6,49,21,47,58,35,57,44,1,55,50,57,9,91,87,45,89,50,16,19,122,114,151,66

The numbers listed after “r=” are residue classes modulo increasing primes 2,3,5,7,…, omitting any classes that do not require sieving — the applet will automatically skip such primes (e.g. it skips 151 in the k=632 example).

A few caveats: You will want to run this on a PC with a reasonably wide display (say 1360 or greater), and it has only been tested on the current versions of Chrome and Firefox.

Errata

Page numbers refer to the file linked to for the relevant paper.

  1. Errata for Zhang's "Bounded gaps between primes"
    1. Page 5: In the first display, [math]\displaystyle{ \mathcal{E} }[/math] should be multiplied by [math]\displaystyle{ \mathcal{L}^{2k_0+2l_0} }[/math], because [math]\displaystyle{ \lambda(n)^2 }[/math] in (2.2) can be that large, cf. (2.4).
    2. Page 14: In the final display, the constraint [math]\displaystyle{ (n,d_1=1 }[/math] should be [math]\displaystyle{ (n,d_1)=1 }[/math].
    3. Page 35: In the display after (10.5), the subscript on [math]\displaystyle{ {\mathcal J}_i }[/math] should be deleted.
    4. Page 36: In the third display, a factor of [math]\displaystyle{ \tau(q_0r)^{O(1)} }[/math] may be needed on the right-hand side (but is ultimately harmless).
    5. Page 38: In the display after (10.14), [math]\displaystyle{ \xi(r,a;q_1,b_1;q_2,b_2;n,k) }[/math] should be [math]\displaystyle{ \xi(r,a;k;q_1,b_1;q_2,b_2;n) }[/math].
    6. Page 42: In (12.3), [math]\displaystyle{ B }[/math] should probably be 2.
    7. Page 47: In the third display after (13.13), the condition [math]\displaystyle{ l \in {\mathcal I}_i(h) }[/math] should be [math]\displaystyle{ l \in {\mathcal I}_i(sh) }[/math].
    8. Page 49: In the top line, a comma in [math]\displaystyle{ (h_1,h_2;,n_1,n_2) }[/math] should be deleted.
    9. Page 51: In the penultimate display, one of the two consecutive commas should be deleted.
    10. Page 54: Three displays before (14.17), [math]\displaystyle{ \bar{r_2}(m_1+m_2)q }[/math] should be [math]\displaystyle{ \bar{r_2}(m_1+m_2)/q }[/math].
  2. Errata for Motohashi-Pintz's "A smoothed GPY sieve", version 1. Update: the errata below have been corrected in the most recent arXiv version of the paper.
    1. Page 31: The estimation of (5.14) by (5.15) does not appear to be justified. In the text, it is claimed that the second summation in (5.14) can be treated by a variant of (4.15); however, whereas (5.14) contains a factor of [math]\displaystyle{ (\log \frac{R}{|D|})^{2\ell+1} }[/math], (4.15) contains instead a factor of [math]\displaystyle{ (\log \frac{R/w}{|K|})^{2\ell+1} }[/math] which is significantly smaller (K in (4.15) plays a similar role to D in (5.14)). As such, the crucial gain of [math]\displaystyle{ \exp(-k\omega/3) }[/math] in (4.15) does not seem to be available for estimating the second sum in (5.14).
  3. Errata for Pintz's "A note on bounded gaps between primes", version 1. Update: the errata below have been corrected in subsequent versions of Pintz's paper.
    1. Page 7: In (2.39), the exponent of [math]\displaystyle{ 3a/2 }[/math] should instead be [math]\displaystyle{ -5a/2 }[/math] (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper.
    2. Page 8: The "easy calculation" that the relative error caused by discarding all but the smooth moduli appears to be unjustified, as it relies on the treatment of (5.14) in Motohashi-Pintz which has the issue pointed out in 2.1 above.

Other relevant blog posts

MathOverflow

Wikipedia and other references

Recent papers and notes

Media

Bibliography

Additional links for some of these references (e.g. to arXiv versions) would be greatly appreciated.

  • [BFI1986] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. Acta Math. 156 (1986), no. 3-4, 203–251. MathSciNet Article
  • [BFI1987] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. II. Math. Ann. 277 (1987), no. 3, 361–393. MathSciNet Article
  • [BFI1989] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. III. J. Amer. Math. Soc. 2 (1989), no. 2, 215–224. MathSciNet Article
  • [B1995] Jörg Brüdern, Einführung in die analytische Zahlentheorie, Springer Verlag 1995
  • [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 MathSciNet Article
  • [FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). MathSciNet Article
  • [FI1983] Fouvry, E.; Iwaniec, H. Primes in arithmetic progressions. Acta Arith. 42 (1983), no. 2, 197–218. MathSciNet Article
  • [FI1985] Friedlander, John B.; Iwaniec, Henryk, Incomplete Kloosterman sums and a divisor problem. With an appendix by Bryan J. Birch and Enrico Bombieri. Ann. of Math. (2) 121 (1985), no. 2, 319–350. MathSciNet JSTOR Appendix
  • [GPY2009] Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862. arXiv MathSciNet
  • [GR1998] Gordon, Daniel M.; Rodemich, Gene Dense admissible sets. Algorithmic number theory (Portland, OR, 1998), 216–225, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998. MathSciNet Article
  • [GR1980] S. W. Graham, C. J. Ringrose, Lower bounds for least quadratic nonresidues. Analytic number theory (Allerton Park, IL, 1989), 269–309, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990. MathSciNet Article
  • [HB1978] D. R. Heath-Brown, Hybrid bounds for Dirichlet L-functions. Invent. Math. 47 (1978), no. 2, 149–170. MathSciNet Article
  • [HB1986] D. R. Heath-Brown, The divisor function d3(n) in arithmetic progressions. Acta Arith. 47 (1986), no. 1, 29–56. MathSciNet Article
  • [HR1973] Hensley, Douglas; Richards, Ian, On the incompatibility of two conjectures concerning primes. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 123–127. Amer. Math. Soc., Providence, R.I., 1973. MathSciNet Article
  • [HR1973b] Hensley, Douglas; Richards, Ian, Primes in intervals. Acta Arith. 25 (1973/74), 375–391. MathSciNet Article
  • [MP2008] Motohashi, Yoichi; Pintz, János A smoothed GPY sieve. Bull. Lond. Math. Soc. 40 (2008), no. 2, 298–310. arXiv MathSciNet Article
  • [MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. MathSciNet Article
  • [M1978] Hugh L. Montgomery, The analytic principle of the large sieve. Bull. Amer. Math. Soc. 84 (1978), no. 4, 547–567. MathSciNet Article
  • [R1974] Richards, Ian On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem. Bull. Amer. Math. Soc. 80 (1974), 419–438. MathSciNet Article
  • [S1961] Schinzel, A. Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers". Acta Arith. 7 1961/1962 1–8. MathSciNet Article
  • [S2007] K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım. Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 1–18. MathSciNet Article arXiv
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