Bounded gaps between primes: Difference between revisions

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| [http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-268732 6] (on GEH)
| [http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-268732 6] (on GEH)
[http://arxiv.org/abs/1311.4600 12] (on EH only)
[http://arxiv.org/abs/1311.4600 12] [M] (on EH only)
| <B>[http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-297456 246]</B>
| <B>[http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-297456 246]</B>
| [http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-297456 246]
| [http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-297456 246]
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|<math>\displaystyle (1+o(1)) m \log m</math>
|<math>\displaystyle (1+o(1)) m \log m</math>
|<math>\displaystyle O( m e^{2m} )</math>
|<math>\displaystyle O( m e^{2m} )</math>
|<math>O( m \exp((4 - \frac{52}{283}) m) )</math>
|<math>O( \exp( 3.815 m) ) [BI]</math>
|<math>O( m \exp((4 - \frac{4}{43}) m) )</math>
|<math>O( m \exp((4 - \frac{4}{43}) m) )</math>
|}
|}
Unless listed below, all the above bounds were produced by the Polymath8 project.
* [BI]: R. C. Baker, A. J. Irving, [http://arxiv.org/abs/1505.01815 Bounded intervals containing many primes]
* [M]: J. Maynard, [http://annals.math.princeton.edu/articles/8772 Small gaps between primes]


We have been working on improving a number of other quantities, including the quantity <math>H_m</math> mentioned above:
We have been working on improving a number of other quantities, including the quantity <math>H_m</math> mentioned above:
Line 256: Line 261:
* [http://arxiv.org/abs/1411.2989 Gaps between Primes in Beatty Sequences], Roger Baker, Liangyi Zhao, 11 Nov 2014.
* [http://arxiv.org/abs/1411.2989 Gaps between Primes in Beatty Sequences], Roger Baker, Liangyi Zhao, 11 Nov 2014.
* [http://arxiv.org/abs/1501.06690 On the Density of Weak Polignac Numbers], Stijn Hanson, 27 Jan 2015.
* [http://arxiv.org/abs/1501.06690 On the Density of Weak Polignac Numbers], Stijn Hanson, 27 Jan 2015.
* [http://arxiv.org/abs/1504.06860 On a conjecture of ErdŐs, Pólya and Turán on consecutive gaps between primes], Janos Pintz, 26 Apr 2015.
* [http://arxiv.org/abs/1504.06860 On a conjecture of Erdős, Pólya and Turán on consecutive gaps between primes], Janos Pintz, 26 Apr 2015.
* [http://arxiv.org/abs/1505.01815 Bounded intervals containing many primes], R. C. Baker, A. J. Irving, 7 May 2015.
* [http://arxiv.org/abs/1505.03104 Goldbach versus de Polignac numbers], Jacques Benatar, 12 May 2015.
* [http://www.ams.org/notices/201506/rnoti-p660.pdf Prime numbers: A much needed gap is finally found], John Friedlander, June 2015.
* [http://math.mit.edu/~drew/PrimeGapsOberwolfach1.pdf Sieve theory and small gaps between primes: Introduction], [http://math.mit.edu/~drew/PrimeGapsOberwolfach2.pdf A variational problem], [http://math.mit.edu/~drew/PrimeGapsOberwolfach3.pdf Narrow admissible tuples]  Andrew V. Sutherland, July 2015.
* "Small gaps between the set of primes and products of two primes", Keiju Sono, August 2015.
* [http://arxiv.org/abs/1509.01564 Patterns of primes in arithmetic progressions], Janos Pintz, 4 Sep 2015.
* [http://arxiv.org/abs/1510.04577 A note on the distribution of normalized prime gaps], Janos Pintz, 15 Oct 2015.
* [http://arxiv.org/abs/1510.08054 Limit points and long gaps between primes], Roger Baker, Tristan Freiberg, 27 Oct 2015.
* [http://www.mast.queensu.ca/~akshaa/gaussian.pdf Bounded gaps between Gaussian primes], Akshaa Vatwani, 3 Nov 2015.
* [http://arxiv.org/abs/1512.01470 General divisor functions in arithmetic progressions to large moduli], Fei Wei, Boqing Xue, Yitang Zhang, Dec 2015.
* [http://arxiv.org/abs/1512.03936 Large gaps between consecutive primes containing perfect k-th powers of prime numbers], Helmut Maier, Michael Rassias, 12 Dec 2015.
* [http://arxiv.org/abs/1604.01761 Increasing and decreasing prime gaps], Daniel Shiu, 6 Apr 2016.
* [http://arxiv.org/abs/1604.06903 Golomb's conjecture on prime gaps], Christian Elsholtz, 23 Apr 2016.
* Factors of Carmichael numbers and a weak k-tuples conjecture, Thomas Wright, J. Aust. Math. Soc. 100 (2016), 421-429.
* [https://arxiv.org/abs/1605.02920 Small gaps between the set of products of at most two primes], Keiju Sono, May 10 2016.
* [http://arxiv.org/abs/1607.03887 Bounded Gaps Between Products of Distinct Primes], Yang Liu, Peter S. Park, Zhuo Qun Song, Jul 14 2016.
* [https://arxiv.org/abs/1705.08034 Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds], Benjamin Linowitz, D. B. McReynolds, Paul Pollack, Lola Thompson, May 24, 2017.
* [https://arxiv.org/abs/1707.05437 Bounded gaps between primes in short intervals], Ryan Alweiss, Sammy Luo, Jul 18 2017.
* [https://arxiv.org/abs/1711.01949 Bounded gaps between product of two primes in number fields], Pranendu Darbar, Anirban Mukhopadhyay, Nov 6 2017.
* [http://arxiv.org/abs/1802.10327 Short intervals containing a prescribed number of primes], Daniele Mastrostefano, Mar 1 2018.
* [http://arxiv.org/abs/1806.09034 Almost primes in various settings], Paweł Lewulis, Jun 23 2018.
* [https://arxiv.org/abs/1907.02246 On the largest square divisor of shifted primes], Jori Merikoski, Jul 4 2019.


== Media ==
== Media ==
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* [http://podacademy.org/podcasts/maths-isnt-standing-still/ Maths isn’t standing still], Adam Smith and Vicky Neale, Pod Academy, March 3, 2014.
* [http://podacademy.org/podcasts/maths-isnt-standing-still/ Maths isn’t standing still], Adam Smith and Vicky Neale, Pod Academy, March 3, 2014.
* [https://www.quantamagazine.org/20141210-prime-gap-grows-after-decades-long-lull/ Prime Gap Grows After Decades-Long Lull], Erica Klarreich, Quanta, Dec 10, 2014.
* [https://www.quantamagazine.org/20141210-prime-gap-grows-after-decades-long-lull/ Prime Gap Grows After Decades-Long Lull], Erica Klarreich, Quanta, Dec 10, 2014.
* [http://www.zalafilms.com/films/countingindex.html Counting from infinity: Yitang Zhang and the twin prime conjecture] (Documentary), George Csicsery, released Jan 2015.
* [http://www.newyorker.com/magazine/2015/02/02/pursuit-beauty The Pursuit of Beauty], Alec Wilkinson, New Yorker, Feb 2 2015.
* [http://www.newyorker.com/magazine/2015/02/02/pursuit-beauty The Pursuit of Beauty], Alec Wilkinson, New Yorker, Feb 2 2015.
** [http://video.newyorker.com/watch/annals-of-ideas-yitang-zhang-s-discovery-2015-01-28 Yitang Zhang's discovery] (Video), Alec Wilkinson, New Yorker, Jan 28, 2015.
* [http://digitaleditions.walsworthprintgroup.com/publication/?i=247647&p=16 Prime Progress Invigorates Math Minds], Katherine Merow, MAA Focus, February/March 2015.
* [http://digitaleditions.walsworthprintgroup.com/publication/?i=247647&p=16 Prime Progress Invigorates Math Minds], Katherine Merow, MAA Focus, February/March 2015.
* [http://www.ams.org/notices/201506/rnoti-p660.pdf Prime Numbers: A Much Needed Gap Is Finally Found], John Friedlander, Notices of the AMS, June/July 2015.


== Bibliography ==
== Bibliography ==

Latest revision as of 07:51, 11 July 2019

This is the home page for the Polymath8 project, which has two components:

  • Polymath8a, "Bounded gaps between primes", was a project to improve the bound H=H_1 on the least gap between consecutive primes that was attained infinitely often, by developing the techniques of Zhang. This project concluded with a bound of H = 4,680.
  • Polymath8b, "Bounded intervals with many primes", was project to improve the value of H_1 further, as well as H_m (the least gap between primes with m-1 primes between them that is attained infinitely often), by combining the Polymath8a results with the techniques of Maynard. This project concluded with a bound of H=246, as well as additional bounds on H_m (see below).

World records

Current records

This table lists the current best upper bounds on [math]\displaystyle{ H_m }[/math] - the least quantity for which it is the case that there are infinitely many intervals [math]\displaystyle{ n, n+1, \ldots, n+H_m }[/math] which contain [math]\displaystyle{ m+1 }[/math] consecutive primes - both on the assumption of the Elliott-Halberstam conjecture (or more precisely, a generalization of this conjecture, formulated as Conjecture 1 in [BFI1986]), without this assumption, and without EH or the use of Deligne's theorems. The boldface entry - the bound on [math]\displaystyle{ H_1 }[/math] without assuming Elliott-Halberstam, but assuming the use of Deligne's theorems - is the quantity that has attracted the most attention. The conjectured value [math]\displaystyle{ H_1=2 }[/math] for [math]\displaystyle{ H_1 }[/math] is the twin prime conjecture.

[math]\displaystyle{ m }[/math] Conjectural Assuming EH Without EH Without EH or Deligne
1 2 6 (on GEH)

12 [M] (on EH only)

246 246
2 6 252 (on GEH)

270 (on EH only)

395,106 474,266
3 8 52,116 24,462,654 32,285,928
4 12 474,266 1,404,556,152 2,031,558,336
5 16 4,137,854 78,602,310,160 124,840,189,042
[math]\displaystyle{ m }[/math] [math]\displaystyle{ \displaystyle (1+o(1)) m \log m }[/math] [math]\displaystyle{ \displaystyle O( m e^{2m} ) }[/math] [math]\displaystyle{ O( \exp( 3.815 m) ) [BI] }[/math] [math]\displaystyle{ O( m \exp((4 - \frac{4}{43}) m) ) }[/math]

Unless listed below, all the above bounds were produced by the Polymath8 project.

We have been working on improving a number of other quantities, including the quantity [math]\displaystyle{ H_m }[/math] mentioned above:

  • [math]\displaystyle{ H = H_1 }[/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.) More recent improvements on [math]\displaystyle{ k_0 }[/math] have come from solving a Selberg sieve variational problem.
  • [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.

Timeline of bounds

A table of bounds as a function of time may be found at timeline of prime gap bounds. In this table, infinitesimal losses in [math]\displaystyle{ \delta,\varpi }[/math] are ignored.

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. Inactive
  21. Polymath8: writing the paper, IV, Terence Tao, 15 October 2013. Inactive
  22. Polymath8: Writing the first paper, V, and a look ahead, Terence Tao, 17 November 2013. Inactive
  23. Polymath8b: Bounded intervals with many primes, after Maynard, Terence Tao, 19 November 2013. Inactive
  24. Polymath8b, II: Optimising the variational problem and the sieve Terence Tao, 22 November 2013. Inactive
  25. Polymath8b, III: Numerical optimisation of the variational problem, and a search for new sieves, Terence Tao, 8 December 2013. Inactive
  26. Polymath8b, IV: Enlarging the sieve support, more efficient numerics, and explicit asymptotics, Terence Tao, 20 December 2013. Inactive
  27. Polymath8b, V: Stretching the sieve support further, Terence Tao, 8 January 2014. Inactive
  28. Polymath8b, VI: A low-dimensional variational problem, Terence Tao, 17 January 2014. Inactive
  29. Polymath8b, VII: Using the generalised Elliott-Halberstam hypothesis to enlarge the sieve support yet further, Terence Tao, 28 January 2014. Inactive
  30. “New equidistribution estimates of Zhang type, and bounded gaps between primes” – and a retrospective, Terence Tao, 7 February 2013. Inactive
  31. Polymath8b, VIII: Time to start writing up the results?, Terence Tao, 9 February 2014. Inactive
  32. Polymath8b, IX: Large quadratic programs, Terence Tao, 21 February 2014. Inactive
  33. Polymath8b, X: Writing the paper, and chasing down loose ends, Terence Tao, 14 April 2014. Inactive
  34. Polymath 8b, XI: Finishing up the paper, Terence Tao, 17 May 2014.Inactive
  35. Polymath8: wrapping up, Terence Tao, 19 June 2014. Inactive
  36. Variants of the Selberg sieve, and bounded intervals containing many primes, Terence Tao, 21 July 2014. Inactive
  37. The "bounded gaps between primes" Polymath project - a retrospective, Terence Tao, 30 September 2014. Active

Writeup

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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