Bounded gaps between primes: Difference between revisions

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* [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. [http://www.ams.org/mathscinet-getitem?mr=976723 MathSciNet] [http://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0976723-6/ 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. [http://www.ams.org/mathscinet-getitem?mr=976723 MathSciNet] [http://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0976723-6/ Article]
* [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 [http://www.ams.org/mathscinet-getitem?mr=1836929 MathSciNet] [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Article]
* [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 [http://www.ams.org/mathscinet-getitem?mr=1836929 MathSciNet] [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Article]
* [FPR2013] B. Farkas, J. Pintz and Sz. Gy. R´ev´esz, On the optimal weight function in the Goldston–Pintz–Yıldırım method for finding small gaps between consecutive primes, submitted. [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf Article]
* [FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). [http://www.ams.org/mathscinet-getitem?mr=610700 MathSciNet] [http://www.math.ethz.ch/~kowalski/fouvry-iwaniec-on-a-theorem.pdf Article]  
* [FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). [http://www.ams.org/mathscinet-getitem?mr=610700 MathSciNet] [http://www.math.ethz.ch/~kowalski/fouvry-iwaniec-on-a-theorem.pdf Article]  
* [FI1983] Fouvry, E.; Iwaniec, H. Primes in arithmetic progressions. Acta Arith. 42 (1983), no. 2, 197–218. [http://www.ams.org/mathscinet-getitem?mr=719249 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4226.pdf Article]
* [FI1983] Fouvry, E.; Iwaniec, H. Primes in arithmetic progressions. Acta Arith. 42 (1983), no. 2, 197–218. [http://www.ams.org/mathscinet-getitem?mr=719249 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4226.pdf Article]

Revision as of 08:09, 5 June 2013

World records

Date [math]\displaystyle{ \varpi }[/math] [math]\displaystyle{ k_0 }[/math] [math]\displaystyle{ H }[/math] Comments
14 May 1/1168 (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)

Optimising Zhang's condition [math]\displaystyle{ \omega\gt 0 }[/math], and then using 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 1/1040? (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 34429? (Paldi/v08ltu) Uses the optimal Bessel function cutoff. Only provisional so far because the kappa error is not yet analysed

? - unconfirmed or conditional

?? - theoretical limit of an analysis, rather than a claimed record

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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
  • [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
  • [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 MathSciNet Article
  • [FPR2013] B. Farkas, J. Pintz and Sz. Gy. R´ev´esz, On the optimal weight function in the Goldston–Pintz–Yıldırım method for finding small gaps between consecutive primes, submitted. 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. 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
  • [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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