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 Free Access

References | Similar Articles | Additional Information

**[1]**E. B. Bogomolny and J. P. Keating,*Random matrix theory and the Riemann zeros. I. Three- and four-point correlations*, Nonlinearity**8**(1995), no. 6, 1115–1131. MR**1363402****[2]**Enrico Bombieri,*Le grand crible dans la théorie analytique des nombres*, Astérisque**18**(1987), 103 (French, with English summary). MR**891718**

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. London Ser. A**293**(1966), 1–18. MR**199165**, https://doi.org/10.1098/rspa.1966.0155**[4]**E. Bombieri, J. B. Friedlander, and H. Iwaniec,*Primes in arithmetic progressions to large moduli*, Acta Math.**156**(1986), no. 3-4, 203–251. MR**834613**, https://doi.org/10.1007/BF02399204**[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**1759****[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**409385**, 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. (3)**47**(1983), no. 2, 193–224. MR**703977**, https://doi.org/10.1112/plms/s3-47.2.193**[18]**D. R. Heath-Brown,*Differences between consecutive primes*, Jahresber. Deutsch. Math.-Verein.**90**(1988), no. 2, 71–89. MR**939754****[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**466042**, 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]**Helmut Maier,*Small differences between prime numbers*, Michigan Math. J.**35**(1988), no. 3, 323–344. MR**978303**, https://doi.org/10.1307/mmj/1029003814**[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.

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Additional Information

**K. Soundararajan**

Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Address at time of publication:
Department of Mathematics, Stanford University, 450 Serra Mall, Building 380, Stanford, California 94305-2125

Email:
ksound@math.stanford.edu, ksound@umich.edu

DOI:
https://doi.org/10.1090/S0273-0979-06-01142-6

Received by editor(s):
July 18, 2006

Published electronically:
September 25, 2006

Additional Notes:
This article is based on a lecture presented January 14, 2006, at the AMS Special Session on Current Events, Joint Mathematics Meetings, San Antonio, TX

The author is partially supported by the National Science Foundation

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.