Bounded gaps between primes
Contents
World records
 [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].
 [math]\varpi[/math] is a technical parameter related to a specialized form of the ElliottHalberstam 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].)
Date  [math]\varpi[/math]  [math]k_0[/math]  [math]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]\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](p_{m+1},\ldots,p_{m+k_0})[/math] with [math]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](p_{m+1},\ldots,p_{m+k_0})[/math] and then [math](\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/21})[/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]\omega\gt0[/math], and then uses an improved bound on [math]\delta_2[/math]  
1 Jun  42,342,924 (Tao)  Tiny improvement using the parity of [math]k_0[/math]  
2 Jun  866,605 (Morrison)  13,008,612 (Morrison)  Uses a further improvement on the quantity [math]\Sigma_2[/math] in Zhang's analysis (replacing the previous bounds on [math]\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]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 HensleyRichards 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) 
k_0 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.
H bound obtained by a hybrid Schinzel/greedy (or "greedygreedy") sieve  
6 Jun  387,960 (Angelveit)

Experimentation with different residue classes and different intervals 
?  unconfirmed or conditional
??  theoretical limit of an analysis, rather than a claimed record
Benchmarks
Let H be the minimal diameter of an admissible tuple of cardinality [math]k_0 = 34,429[/math]. For benchmark upper bounds:
 The Zhang sieve (as optimized by Trudgian and later Morrison) gives [math]H \leq 411,932[/math].
 The HensleyRichards sieve gives [math]H \leq 402,790[/math] after optimization.
 The shifted HensleyRichards sieve gives [math]H \leq 401,700[/math] after optimization.
For benchmark lower bounds:
 The easier version of the MontgomeryVaughan large sieve inequality [math]k_0 \sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H+1+Q^2[/math] gives [math]H \geq 196,729[/math].
 The MontgomeryVaughan BrunTitchmarsh bound [math]k_0 \leq \frac{2(H+1)}{\log(H+1)}[/math] gives [math]H \geq 211,046[/math].
 The MontgomeryVaughan large sieve inequality [MV1973, Corollary 1] gives [math]H \geq 226,987[/math] after optimization.
 Using a refinement of this inequality [B1995, p.162] improves this to [math]H \geq 227,078[/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 GoldstonPintzYildirim, MotohashiPintz, and Zhang, Terence Tao, 3 June 2013. Active
 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. Active
 More narrow admissible sets, Scott Morrison, 5 June 2013. Active
Code and data
 HenselyRichards sequences, Scott Morrison
 A mathematica notebook for finding k_0, Scott Morrison
 python implementation, Avi Levy
 ktuple 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 "greedygreedy" algorithm, Andrew Sutherland
Other relevant blog posts
 Marker lecture III: “Small gaps between primes”, Terence Tao, 19 Nov 2008.
 The GoldstonPintzYildirim 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.
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]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
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 GoldstonPintzYildirim 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.
 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 FriedlanderIwaniec 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 almostprime II, James Maynard, 5 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. 34, 203–251. MathSciNet
 [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 BombieriVinogradov 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. JSTOR
 [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
 [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
 [MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. 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 GoldstonPintzYıldırım. Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 1–18. MathSciNet Article arXiv
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