Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim

Soundararajan, K.

doi:10.1090/S0273-0979-06-01142-6

Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim

Author: K. Soundararajan
Journal: Bull. Amer. Math. Soc. 44 (2007), 1-18
MSC (2000): Primary 11N05
DOI: https://doi.org/10.1090/S0273-0979-06-01142-6
Published electronically: September 25, 2006
MathSciNet review: 2265008
Full-text PDF

[1] E. B. Bogomolny and J. P. Keating, Random matrix theory and the Riemann zeros. II. 𝑛-point correlations, Nonlinearity 9 (1996), no. 4, 911–935. MR 1399479, https://doi.org/10.1088/0951-7715/9/4/006
[2] Enrico Bombieri, Le grand crible dans la théorie analytique des nombres, Société Mathématique de France, Paris, 1974 (French). Avec une sommaire en anglais; Astérisque, No. 18. MR 0371840
[3] E. Bombieri and H. Davenport, Small differences between prime numbers, Proc. Roy. Soc. Ser. A 293 (1966), 1–18. MR 0199165
[4] E. Bombieri, J. Friedlander and H. Iwaniec, Primes in arithmetic progressions to large moduli, Acta Math. 156 (1986), 203-251. MR 0834613 (88b:11058)
[5] Chen Jing-run, On the representation of a large even integer as the sum of a prime and the product of at most two primes, Kexue Tongbao (Foreign Lang. Ed.) 17 (1966), 385–386. MR 0207668
[6] Harold Davenport, Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. Revised and with a preface by Hugh L. Montgomery. MR 1790423
[7] Noam D. Elkies and Curtis T. McMullen, Gaps in √𝑛\bmod1 and ergodic theory, Duke Math. J. 123 (2004), no. 1, 95–139. MR 2060024, https://doi.org/10.1215/S0012-7094-04-12314-0
[8] P. Erdos, On the difference of consecutive primes, Quart. J. Math. Oxford 6 (1935), 124-128.
[9] P. Erdös, The difference of consecutive primes, Duke Math. J. 6 (1940), 438–441. MR 0001759
[10] John Friedlander and Henryk Iwaniec, The polynomial 𝑋²+𝑌⁴ captures its primes, Ann. of Math. (2) 148 (1998), no. 3, 945–1040. MR 1670065, https://doi.org/10.2307/121034
[11] P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976), no. 1, 4–9. MR 0409385, https://doi.org/10.1112/S0025579300016442
[12] D. Goldston, J. Pintz and C. Yildirim, Primes in tuples, I, preprint, available at www.arxiv.org.
[13] D. Goldston, S. Graham, J. Pintz and C. Yildirim, Small gaps between primes and almost primes, preprint, available at www.arxiv.org.
[14] D. Goldston, Y. Motohashi, J. Pintz and C. Yildirim, Small gaps between primes exist, preprint, available at www.arxiv.org.
[15] Andrew Granville, Unexpected irregularities in the distribution of prime numbers, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 388–399. MR 1403939
[16] G.H. Hardy and J.E. Littlewood, Some problems of Parititio Numerorum (III): On the expression of a number as a sum of primes, Acta Math. 44 (1922), 1-70.
[17] D.R. Heath-Brown, Prime twins and Siegel zeros, Proc. London Math. Soc. 47 (1983), 193-224. MR 0703977 (84m:10029)
[18] D.R. Heath-Brown, Differences between consecutive primes, Jahresber. Deutsch. Math.-Verein. 90 (1988), 71-89. MR 0939754 (89i:11100)
[19] D. R. Heath-Brown, Primes represented by 𝑥³+2𝑦³, Acta Math. 186 (2001), no. 1, 1–84. MR 1828372, https://doi.org/10.1007/BF02392715
[20] M. N. Huxley, Small differences between consecutive primes. II, Mathematika 24 (1977), no. 2, 142–152. MR 0466042, https://doi.org/10.1112/S0025579300009037
[21] Nicholas M. Katz and Peter Sarnak, Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 1, 1–26. MR 1640151, https://doi.org/10.1090/S0273-0979-99-00766-1
[22] H. Maier, Small differences between prime numbers, Michigan Math. J. 35 (1988), 323-344. MR 0978303 (90e:11126)
[23] H. L. Montgomery, The pair correlation of zeros of the zeta function, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 181–193. MR 0337821
[24] H. Montgomery and R.C. Vaughan, Multiplicative number theory I: Classical theory, Cambridge University Press, 2006.
[25] R. Rankin, The difference between consecutive primes, J. London Math. Soc. 13, 242-244.
[26] E. Szemerédi, On sets of integers containing no 𝑘 elements in arithmetic progression, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 503–505. MR 0422191
[27] E. Westzynthius, Über die Verteilung der Zahlen, die zu der ersten Primzahlen teilerfremd sind, Comm. Phys. Math. Helsingfors 25 (1931), 1-37.