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

- 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** - Enrico Bombieri,
*Le grand crible dans la théorie analytique des nombres*, Astérisque**18**(1987), 103 (French, with English summary). MR**891718** - E. Bombieri and H. Davenport,
*Small differences between prime numbers*, Proc. Roy. Soc. London Ser. A**293**(1966), 1–18. MR**199165**, DOI https://doi.org/10.1098/rspa.1966.0155 - 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**, DOI https://doi.org/10.1007/BF02399204 - 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**17**(1966), 385–386. MR**207668** - 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** - Noam D. Elkies and Curtis T. McMullen,
*Gaps in ${\sqrt n}\bmod 1$ and ergodic theory*, Duke Math. J.**123**(2004), no. 1, 95–139. MR**2060024**, DOI https://doi.org/10.1215/S0012-7094-04-12314-0
[8]8 P. Erdős, - P. Erdös,
*The difference of consecutive primes*, Duke Math. J.**6**(1940), 438–441. MR**1759** - John Friedlander and Henryk Iwaniec,
*The polynomial $X^2+Y^4$ captures its primes*, Ann. of Math. (2)**148**(1998), no. 3, 945–1040. MR**1670065**, DOI https://doi.org/10.2307/121034 - P. X. Gallagher,
*On the distribution of primes in short intervals*, Mathematika**23**(1976), no. 1, 4–9. MR**409385**, DOI https://doi.org/10.1112/S0025579300016442
[12]12 D. Goldston, J. Pintz and C. Yıldırım, - 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]16 G.H. Hardy and J.E. Littlewood, - D. R. Heath-Brown,
*Prime twins and Siegel zeros*, Proc. London Math. Soc. (3)**47**(1983), no. 2, 193–224. MR**703977**, DOI https://doi.org/10.1112/plms/s3-47.2.193 - D. R. Heath-Brown,
*Differences between consecutive primes*, Jahresber. Deutsch. Math.-Verein.**90**(1988), no. 2, 71–89. MR**939754** - D. R. Heath-Brown,
*Primes represented by $x^3+2y^3$*, Acta Math.**186**(2001), no. 1, 1–84. MR**1828372**, DOI https://doi.org/10.1007/BF02392715 - M. N. Huxley,
*Small differences between consecutive primes. II*, Mathematika**24**(1977), no. 2, 142–152. MR**466042**, DOI https://doi.org/10.1112/S0025579300009037 - 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**, DOI https://doi.org/10.1090/S0273-0979-99-00766-1 - Helmut Maier,
*Small differences between prime numbers*, Michigan Math. J.**35**(1988), no. 3, 323–344. MR**978303**, DOI https://doi.org/10.1307/mmj/1029003814 - 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]24 H. Montgomery and R.C. Vaughan, - 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]27 E. Westzynthius,

*On the difference of consecutive primes*, Quart. J. Math. Oxford

**6**(1935), 124–128.

*Primes in tuples, I*, preprint, available at www.arxiv.org. [13]13 D. Goldston, S. Graham, J. Pintz and C. Yıldırım,

*Small gaps between primes and almost primes*, preprint, available at www.arxiv.org. [14]14 D. Goldston, Y. Motohashi, J. Pintz and C. Yıldırım,

*Small gaps between primes exist*, preprint, available at www.arxiv.org.

*Some problems of Parititio Numerorum (III): On the expression of a number as a sum of primes*, Acta Math.

**44**(1922), 1–70.

*Multiplicative number theory I: Classical theory*, Cambridge University Press, 2006. [25]25 R. Rankin,

*The difference between consecutive primes*, J. London Math. Soc.

**13**, 242–244.

*Über die Verteilung der Zahlen, die zu der $n$ ersten Primzahlen teilerfremd sind*, Comm. Phys. Math. Helsingfors

**25**(1931), 1–37.

Retrieve articles in *Bulletin of the American Mathematical Society*
with MSC (2000):
11N05

Retrieve articles in all journals with MSC (2000): 11N05

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

MR Author ID:
319775

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

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.