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The twin prime conjecture

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Japanese Journal of Mathematics Aims and scope

Abstract

The Twin Prime Conjecture asserts that there should be infinitely many pairs of primes which differ by 2. Unfortunately this long-standing conjecture remains open, but recently there has been several dramatic developments making partial progress. We survey the key ideas behind proofs of bounded gaps between primes (due to Zhang, Tao and the author) and developments on Chowla's conjecture (due to Matomäki, Radziwiłł and Tao).

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References

  1. R.C. Baker and T. Freiberg, Limit points and long gaps between primes, Q. J. Math., 67 (2016), 233–260.

    Article  MathSciNet  Google Scholar 

  2. R.C. Baker and P. Pollack, Bounded gaps between primes with a given primitive root. II, Forum Math., 28 (2016), 675–687.

    MathSciNet  MATH  Google Scholar 

  3. R.C. Baker and L. Zhao, Gaps between primes in Beatty sequences, Acta Arith., 172 (2016), 207–242.

    MathSciNet  MATH  Google Scholar 

  4. R.C. Baker and L. Zhao, Gaps of smallest possible order between primes in an arithmetic progression, Int. Math. Res. Not. IMRN, 2016 (2016), 7341–7368.

    MathSciNet  MATH  Google Scholar 

  5. W.D. Banks, T. Freiberg and J. Maynard, On limit points of the sequence of normalized prime gaps, Proc. Lond. Math. Soc. (3), 113 (2016), 515–539.

    Article  MathSciNet  Google Scholar 

  6. W.D. Banks, T. Freiberg and C.L. Turnage-Butterbaugh, Consecutive primes in tuples}, Acta Arith., 167 (2015), 261–266.

    Article  MathSciNet  Google Scholar 

  7. E. Bombieri, On the large sieve, Mathematika, 12 (1965), 201–225.

    Article  MathSciNet  Google Scholar 

  8. E. Bombieri and H. Davenport, Small differences between prime numbers, Proc. Roy. Soc. Ser. A, 293 (1966), 1–18.

    Article  MathSciNet  Google Scholar 

  9. E. Bombieri, J.B. Friedlander and H. Iwaniec, Primes in arithmetic progressions to large moduli, Acta Math., 156 (1986), 203–251.

    Article  MathSciNet  Google Scholar 

  10. E. Bombieri, J.B. Friedlander and H. Iwaniec, Primes in arithmetic progressions to large moduli. II, Math. Ann., 277 (1987), 361–393.

    Article  MathSciNet  Google Scholar 

  11. E. Bombieri, J.B. Friedlander and H. Iwaniec, Primes in arithmetic progressions to large moduli. III, J. Amer. Math. Soc., 2 (1989), 215–224.

    Article  MathSciNet  Google Scholar 

  12. A. Castillo, C. Hall, R.J. Lemke Oliver, P. Pollack and L. Thompson, Bounded gaps between primes in number fields and function fields, Proc. Amer. Math. Soc., 143 (2015), 2841–2856.

    Article  MathSciNet  Google Scholar 

  13. J.R. Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica, 16 (1973), 157–176.

    MathSciNet  MATH  Google Scholar 

  14. J.R. Chen, On the Goldbach's problem and the sieve methods, Sci. Sinica, 21 (1978), 701–739.

    MathSciNet  MATH  Google Scholar 

  15. S. Chowla, The Riemann Hypothesis and Hilbert's Tenth Problem, Mathematics and Its Applications. Vol. 4, Gordon and Breach Science Publishers, New York-London-Paris, 1965.

  16. L. Chua, S. Park and G.D. Smith, Bounded gaps between primes in special sequences, Proc. Amer. Math. Soc., 143 (2015), 4597–4611.

    Article  MathSciNet  Google Scholar 

  17. A. de Polignac, Recherches nouvelles sur les nombres premiers, C. R. Acad. Sci. Paris Math., 29 (1849), 397–401.

    Google Scholar 

  18. P.D.T.A. Elliott and H. Halberstam, A conjecture in prime number theory, In: 1970 Symposia Mathematica. Vol. IV, INDAM, Rome, 1968/69, Academic Press, London, 1970, pp. 59–72.

    Google Scholar 

  19. P. Erdős, The difference of consecutive primes, Duke Math. J., 6 (1940), 438–441.

    Article  MathSciNet  Google Scholar 

  20. K. Ford, B. Green, S. Konyagin, J. Maynard and T. Tao, Long gaps between primes, J. Amer. Math. Soc., 31 (2018), 65–105.

    Article  MathSciNet  Google Scholar 

  21. K. Ford, B. Green, S. Konyagin and T. Tao, Large gaps between consecutive prime numbers, Ann. of Math. (2), 183 (2016), 935–974.

    Article  MathSciNet  Google Scholar 

  22. É. Fouvry, Autour du théorème de Bombieri–Vinogradov, Acta Math., 152 (1984), 219–244.

    Article  MathSciNet  Google Scholar 

  23. É. Fouvry and F. Grupp, J. Reine Angew. Math., 370 (1986), 101–126.

  24. É. Fouvry and H. Iwaniec, On a theorem of Bombieri–Vinogradov type, Mathematika, 27 (1980), 135–152.

    Article  MathSciNet  Google Scholar 

  25. É. Fouvry and H. Iwaniec, Primes in arithmetic progressions, Acta Arith., 42 (1983), 197–218.

    Article  MathSciNet  Google Scholar 

  26. N. Frantzikinakis, An averaged Chowla and Elliott conjecture along independent polynomials, Int. Math. Res. Not. IMRN, 2018 (2018), 3721–3743.

    MathSciNet  MATH  Google Scholar 

  27. N. Frantzikinakis and B. Host, The logarithmic Sarnak conjecture for ergodic weights, Ann. of Math. (2), 187 (2018), 869–931.

    Article  MathSciNet  Google Scholar 

  28. T. Freiberg, Short intervals with a given number of primes, J. Number Theory, 163 (2016), 159–171.

    Article  MathSciNet  Google Scholar 

  29. D.A. Goldston, J. Pintz and C.Y. Yıldırım, Primes in tuples. I, Ann. of Math. (2), 170 (2009), 819–862.

    Article  MathSciNet  Google Scholar 

  30. É. Goudout, Théorème d'Erdős–Kac dans presque tous les petits intervalles, Acta Arith., 182 (2018), 101–116.

    Article  MathSciNet  Google Scholar 

  31. A. Granville, Primes in intervals of bounded length, Bull. Amer. Math. Soc. (N.S.), 52 (2015), 171–222.

    Article  MathSciNet  Google Scholar 

  32. G.H. Hardy and J.E. Littlewood, Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes, Acta Math., 44 (1923), 1–70.

    Article  MathSciNet  Google Scholar 

  33. W. Huang and X. Wu, On the set of the difference of primes, Proc. Amer. Math. Soc., 145 (2017), 3787–3793.

    Article  MathSciNet  Google Scholar 

  34. M.N. Huxley, Small differences between consecutive primes, Mathematika, 20 (1973), 229–232.

    Article  MathSciNet  Google Scholar 

  35. M.N. Huxley, Small differences between consecutive primes. II, Mathematika, 24 (1977), 142–152.

    Article  MathSciNet  Google Scholar 

  36. M.N. Huxley, An application of the Fouvry–Iwaniec theorem, Acta Arith., 43 (1984), 441–443.

    Article  MathSciNet  Google Scholar 

  37. D.A. Kaptan, A note on small gaps between primes in arithmetic progressions, Acta Arith., 172 (2016), 351–375.

    MathSciNet  MATH  Google Scholar 

  38. O. Klurman, Correlations of multiplicative functions and applications, Compos. Math., 153 (2017), 1622–1657.

    Article  MathSciNet  Google Scholar 

  39. O. Klurman and A.P. Mangerel, Rigidity theorems for multiplicative functions, Math. Ann., 372 (2018), 651–697.

    Article  MathSciNet  Google Scholar 

  40. E. Kowalski, Gaps between prime numbers and primes in arithmetic progressions [after Y. Zhang and J. Maynard], Astérisque, 367-368 (2015), 327–366.

    MathSciNet  MATH  Google Scholar 

  41. S. Lester, K. Matomäki and M. Radziwiłł, Small scale distribution of zeros and mass of modular forms, J. Eur. Math. Soc. (JEMS)}, 20 (2018), 1595–1627.

    Article  MathSciNet  Google Scholar 

  42. H. Li and H. Pan, Bounded gaps between primes of a special form, Int. Math. Res. Not. IMRN, 2015 (2015), 12345–12365.

    Article  MathSciNet  Google Scholar 

  43. J. Li, K. Pratt and G. Shakan, Q. J. Math., 68 (2017), 729–758.

  44. H. Maier, Small differences between prime numbers, Michigan Math. J., 35 (1988), 323–344.

    Article  MathSciNet  Google Scholar 

  45. H. Maier and M.Th. Rassias, Large gaps between consecutive prime numbers containing perfect k-th powers of prime numbers, J. Funct. Anal., 272 (2017), 2659–2696.

    Article  MathSciNet  Google Scholar 

  46. K. Matomäki and M. Radziwiłł, Multiplicative functions in short intervals, Ann. of Math. (2), 183 (2016), 1015–1056.

    Article  MathSciNet  Google Scholar 

  47. K. Matomäki, M. Radziwiłł and T. Tao, An averaged form of Chowla's conjecture, Algebra Number Theory, 9 (2015), 2167–2196.

    Article  MathSciNet  Google Scholar 

  48. K. Matomäki, M. Radziwiłł and }T. Tao, Sign patterns of the Liouville and Möbius functions, Forum Math. Sigma, 4 (2016), e14.

    Article  Google Scholar 

  49. K. Matomäki and X. Shao, Vinogradov's three primes theorem with almost twin primes, Compos. Math., 153 (2017), 1220–1256.

    Article  MathSciNet  Google Scholar 

  50. J. Maynard, Small gaps between primes, Ann. of Math. (2), 181 (2015), 383–413.

    Article  MathSciNet  Google Scholar 

  51. J. Maynard, Dense clusters of primes in subsets, Compos. Math., 152 (2016), 1517–1554.

    Article  MathSciNet  Google Scholar 

  52. J. Maynard, Large gaps between primes, Ann. of Math. (2), 183 (2016), 915–933.

    Article  MathSciNet  Google Scholar 

  53. H. Parshall, Small gaps between configurations of prime polynomials, J. Number Theory, 162 (2016), 35–53.

    Article  MathSciNet  Google Scholar 

  54. G.Z. Pil'tjaĭ, The magnitude of the difference between consecutive primes, In: Studies in Number Theory. No. 4, Izdat. Saratov. Univ., Saratov, 1972, pp. 73–79.

    MathSciNet  Google Scholar 

  55. J. Pintz, On the ratio of consecutive gaps between primes, In: Analytic Number Theory, Springer-Verlag, 2015, pp. 285–304.

    Chapter  Google Scholar 

  56. J. Pintz, Patterns of primes in arithmetic progressions, In: Number Theory—Diophantine Problems, Uniform Distribution and Applications, Springer-Verlag, 2017, pp. 369–379.

    Chapter  Google Scholar 

  57. P. Pollack, Bounded gaps between primes with a given primitive root, Algebra Number Theory, 8 (2014), 1769–1786.

    Article  MathSciNet  Google Scholar 

  58. P. Pollack and L. Thompson, Arithmetic functions at consecutive shifted primes, Int. J. Number Theory, 11 (2015), 1477–1498.

    Article  MathSciNet  Google Scholar 

  59. D.H.J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes, Res. Math. Sci., 1 (2014), Art. 12.

  60. D.H.J. Polymath, New equidistribution estimates of Zhang type, Algebra Number Theory, 8 (2014), 2067–2199.

    Article  MathSciNet  Google Scholar 

  61. R.A. Rankin, The difference between consecutive prime numbers. IV, Proc. Amer. Math. Soc., 1 (1950), 143–150.

    Article  MathSciNet  Google Scholar 

  62. G. Ricci, Sull'andamento della differenza di numeri primi consecutivi, Riv. Mat. Univ. Parma, 5 (1954), 3–54.

    MathSciNet  MATH  Google Scholar 

  63. K. Soundararajan, Small gaps between prime numbers: the work of Goldston–Pintz–Yıldırım, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 1–18.

    Article  MathSciNet  Google Scholar 

  64. K. Soundararajan, The Liouville function in short intervals, Astérisque, 390 (2017), 453–479; Séminaire Bourbaki. Vol. 2015/2016, 1104–1119.

    MathSciNet  MATH  Google Scholar 

  65. T. Tao, The Erdős discrepancy problem, Discrete Anal., 1 (2016).

  66. T. Tao, The logarithmically averaged Chowla and Elliott conjectures for two-point correlations, Forum Math. Pi, 4 (2016), e8.

  67. J. Teräväinen, Almost primes in almost all short intervals, Math. Proc. Cambridge Philos. Soc., 161 (2016), 247–281.

    Article  MathSciNet  Google Scholar 

  68. J. Teräväinen, On binary correlations of multiplicative functions, Forum Math. Sigma, 6 (2018), e10.

    Article  MathSciNet  Google Scholar 

  69. J. Thorner, Bounded gaps between primes in Chebotarev sets, Res. Math. Sci., 1 (2014), Art. 4.

  70. L. Troupe, Bounded gaps between prime polynomials with a given primitive root, Finite Fields Appl., 37 (2016), 295–310.

    Article  MathSciNet  Google Scholar 

  71. S. Uchiyama, On the difference between consecutive prime numbers. Collection of articles in memory of Juriĭ Vladimirovič Linnik, Acta Arith., 27 (1975), 153–157.

    Article  MathSciNet  Google Scholar 

  72. A. Vatwani, Bounded gaps between Gaussian primes, J. Number Theory, 171 (2017), 449–473.

    Article  MathSciNet  Google Scholar 

  73. A.I. Vinogradov, The density hypothesis for Dirichet L-series, Izv. Akad. Nauk SSSR Ser. Mat., 29 (1965), 903–934.

    MathSciNet  Google Scholar 

  74. J. Wu, Chen's double sieve, Goldbach's conjecture and the twin prime problem, Acta Arith., 114 (2004), 215–273.

    Article  MathSciNet  Google Scholar 

  75. Y. Zhang, Bounded gaps between primes, Ann. of Math. (2), 179} (2014), 1121–1174.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    James Maynard

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  1. James Maynard
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Correspondence to James Maynard.

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Communicated by: Takeshi Saito

This article is based on the 22nd Takagi Lectures that the author delivered at The University of Tokyo on November 17–18, 2018.

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Maynard, J. The twin prime conjecture. Jpn. J. Math. 14, 175–206 (2019). https://doi.org/10.1007/s11537-019-1837-z

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  • DOI: https://doi.org/10.1007/s11537-019-1837-z

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