Bounded gaps between primes: Difference between revisions
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* <math>H</math> is a quantity such that there are infinitely many pairs of consecutive primes of distance at most <math>H</math> apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project). | * <math>H</math> is a quantity such that there are infinitely many pairs of consecutive primes of distance at most <math>H</math> apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project). | ||
* <math>k_0</math> is a quantity such that every admissible <math>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>k_0</math> lead to improvements in <math>H</math>. (The relationship is roughly of the form <math>H \sim k_0 \log k_0</math>; see the page on [[finding narrow admissible tuples]].) | * <math>k_0</math> is a quantity such that every admissible <math>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>k_0</math> lead to improvements in <math>H</math>. (The relationship is roughly of the form <math>H \sim k_0 \log k_0</math>; see the page on [[finding narrow admissible tuples]].) | ||
* <math>\varpi</math> is a technical parameter related to a specialized form of the [ | * <math>\varpi</math> is a technical parameter related to a specialized form of the [http://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture Elliott-Halberstam conjecture]. Would like to be as large as possible. Improvements in <math>\varpi</math> lead to improvements in <math>k_0</math>. (The relationship is roughly of the form <math>k_0 \sim \varpi^{-3/2}</math>; there is an active discussion on optimising these improvements [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ here].) In more recent work, the single parameter <math>\varpi</math> is replaced by a pair <math>(\varpi,\delta)</math> (in previous work we had <math>\delta=\varpi</math>). Discussion on improving the values of <math>(\varpi,\delta)</math> is currently being held [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ here]. In this table, infinitesimal losses in <math>\delta,\varpi</math> are ignored. | ||
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| 5,937? (Pintz/[ | | 5,937? (Pintz/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz Tao]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235124 v08ltu]) | ||
5,672? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235135 v08ltu]) | 5,672? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235135 v08ltu]) | ||
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[http://math.mit.edu/~drew/admissible_6329_60732 60,732] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23999 Sutherland]) | [http://math.mit.edu/~drew/admissible_6329_60732 60,732] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23999 Sutherland]) | ||
[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726] ([ | [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24002 xfxie]-Sutherland) | ||
58,866? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun]) | 58,866? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun]) | ||
Revision as of 17:46, 24 June 2013
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]. (The relationship is roughly of the form [math]\displaystyle{ k_0 \sim \varpi^{-3/2} }[/math]; there is an active discussion on optimising these improvements here.) 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]). Discussion on improving the values of [math]\displaystyle{ (\varpi,\delta) }[/math] is currently being held here. 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) | 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) 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 |
|
387,960 (Angelveit)
387,904 (Angeltveit)
|
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 |
[math]\displaystyle{ 207 \varpi + 43\delta \lt \frac{1}{4} }[/math] (v08ltu/Green) |
26,024? (vo8ltu) |
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)
285,752 (pedant-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) | New bounds on [math]\displaystyle{ H }[/math] obtained by interleaving iterated merging with local optimizations. | |
| Jun 10 | 23,283? (Harcos/v08ltu) | 285,210 (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) | 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) | Improved bound on [math]\displaystyle{ k_0 }[/math] avoids the technical issue in previous computations. | |
| Jun 13 | ||||
| Jun 14 | 248,898 (Sutherland) | |||
| Jun 15 | [math]\displaystyle{ 87\varpi+17\delta \lt \frac{1}{4} }[/math]? (Tao) | 6,330? (v08ltu)
6,329? (Harcos) 6,329 (v08ltu) |
60,830? (Sutherland) | Taking more advantage of the [math]\displaystyle{ \alpha }[/math] convolution in the Type III sums |
| Jun 16 | [math]\displaystyle{ 87\varpi+17\delta \lt \frac{1}{4} }[/math] (v08ltu)
|
60,760* (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)
58,866? (Sun) 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)
53,672*? (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) | Systematic tables of tuples of small length have been set up here and here |
| Jun 22 | 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{ 101\varpi + (21+O(1))\delta \lt 3/4 }[/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 |
Legend:
- ? - unconfirmed or conditional
- ?? - theoretical limit of an analysis, rather than a claimed record
- * - is majorized by an earlier but independent result
- 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
- I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart, Scott Morrison, 30 May 2013. Inactive
- The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang, Terence Tao, 3 June 2013. Inactive
- Polymath proposal: bounded gaps between primes, Terence Tao, 4 June 2013. Active
- Online reading seminar for Zhang’s “bounded gaps between primes, Terence Tao, 4 June 2013. Inactive
- More narrow admissible sets, Scott Morrison, 5 June 2013. Active
- The elementary Selberg sieve and bounded prime gaps, Terence Tao, 8 June 2013. Inactive
- A combinatorial subset sum problem associated with bounded prime gaps, Terence Tao, 10 June 2013. Active
- Further analysis of the truncated GPY sieve, Terence Tao, 11 June 2013. Inactive
- Estimation of the Type I and Type II sums, Terence Tao, 12 June 2013. Inactive
- Estimation of the Type III sums, Terence Tao, 14 June 2013. Inactive
- A truncated elementary Selberg sieve of Pintz, Terence Tao, 18 June, 2013. Active
- Bounding short exponential sums on smooth moduli via Weyl differencing, Terence Tao, 22 June, 2013. Active
- The distribution of primes in densely divisible moduli, Terence Tao, 23 June, 2013. Active
Code and data
- Hensely-Richards sequences, Scott Morrison
- A mathematica notebook for finding k_0, Scott Morrison
- python implementation, Avi Levy
- k-tuple pattern data, Thomas J Engelsma
- Sifted sequences, Vit Tucek
- Other sifted sequences, Christian Elsholtz
- Size of largest admissible tuples in intervals of length up to 1050, Clark and Jarvis
- C implementation of the "greedy-greedy" algorithm, Andrew Sutherland
- Dropbox for sequences, pedant
- Spreadsheet for admissible sequences, Vit Tucek
- Java code for optimising a given tuple V1.1, xfxie
Errata
Page numbers refer to the file linked to for the relevant paper.
- Errata for Zhang's "Bounded gaps between primes"
- 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).
- Page 14: In the final display, the constraint [math]\displaystyle{ (n,d_1=1 }[/math] should be [math]\displaystyle{ (n,d_1)=1 }[/math].
- Page 35: In the display after (10.5), the subscript on [math]\displaystyle{ {\mathcal J}_i }[/math] should be deleted.
- 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).
- 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].
- Page 42: In (12.3), [math]\displaystyle{ B }[/math] should probably be 2.
- 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].
- Page 49: In the top line, a comma in [math]\displaystyle{ (h_1,h_2;,n_1,n_2) }[/math] should be deleted.
- Page 51: In the penultimate display, one of the two consecutive commas should be deleted.
- 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].
- Errata for Motohashi-Pintz's "A smoothed GPY sieve"
- 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).
- 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.
- 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.
- 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
- Marker lecture III: “Small gaps between primes”, Terence Tao, 19 Nov 2008.
- The Goldston-Pintz-Yildirim result, and how far do we have to walk to twin primes ?, Emmanuel Kowalski, 22 Jan 2009.
- Number Theory News, Peter Woit, 12 May 2013.
- Bounded Gaps Between Primes, Emily Riehl, 14 May 2013.
- Bounded gaps between primes!, Emmanuel Kowalski, 21 May 2013.
- Bounded gaps between primes: some grittier details, Emmanuel Kowalski, 4 June 2013.
- Bound on prime gaps bound decreasing by leaps and bounds, Christian Perfect, 8 June 2013.
- Narrowing the Gap, Brie Finegold, AMS Blog on Math Blogs, 14 June 2013.
- Bounded gaps between primes: the dawn of (some) enlightenment, Emmanuel Kowalski, 22 June 2013.
MathOverflow
- Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture, 20 May 2013.
- A technical question related to Zhang’s result of bounded prime gaps, 25 May 2013.
- How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the [math]\displaystyle{ S_2 }[/math] sum, 31 May 2013.
- Tightening Zhang’s bound, 3 June 2013.
- Does Zhang’s theorem generalize to 3 or more primes in an interval of fixed length?, 3 June 2013.
Wikipedia and other references
- Bessel function
- Bombieri-Vinogradov theorem
- Brun-Titchmarsh theorem
- Dispersion method
- Prime gap
- Second Hardy-Littlewood conjecture
- Twin prime conjecture
Recent papers and notes
- On the exponent of distribution of the ternary divisor function, Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, 11 Apr 2013.
- On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes, Bálint Farkas, János Pintz and Szilárd Gy. Révész, To appear in: Paul Turán Memorial Volume: Number Theory, Analysis and Combinatorics, de Gruyter, Berlin, 2013. 23 pages. arXiv
- Bounded gaps between primes, Yitang Zhang, to appear, Annals of Mathematics. Released 21 May, 2013.
- Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture, Janos Pintz, 27 May 2013.
- A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million, T. S. Trudgian, 28 May 2013.
- The Friedlander-Iwaniec sum, É. Fouvry, E. Kowalski, Ph. Michel., May 2013.
- Notes on Zhang's prime gaps paper, Terence Tao, 1 June 2013.
- Bounded prime gaps in short intervals, Johan Andersson, 3 June 2013.
- Bounded length intervals containing two primes and an almost-prime II, James Maynard, 5 June 2013.
- A note on bounded gaps between primes, Janos Pintz, 6 June 2013.
- Lower Bounds for Admissible k-tuples, Avishay Tal, 15 June 2013.
- Notes in a truncated elementary Selberg sieve (Section 1, Section 2, Section 3), Janos Pintz, 18 June 2013.
Media
- First proof that infinitely many prime numbers come in pairs, Maggie McKee, Nature, 14 May 2013.
- Proof that an infinite number of primes are paired, Lisa Grossman, New Scientist, 14 May 2013.
- Unheralded Mathematician Bridges the Prime Gap, Erica Klarreich, Simons science news, 20 May 2013.
- The article also appeared on Wired as "Unknown Mathematician Proves Elusive Property of Prime Numbers".
- The Beauty of Bounded Gaps, Jordan Ellenberg, Slate, 22 May 2013.
- Game of proofs boosts prime pair result by millions, Jacob Aron, New Scientist, 4 June 2013.
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
