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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
Published electronically:
September 25, 2006
MathSciNet review:
2265008
Full-text PDF
References |
Similar Articles |
Additional Information
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- [1]
- E. Bogomolny and J. Keating, Random matrix theory and the Riemann zeros. II.
 -point correlations, Nonlinearity 9 (1996), 911-935. MR 1399479 (97d:11132b)
- [2]
- E. Bombieri, Le grand crible dans la théorie analytique des nombres, vol. 18, Astérisque, 1987/1974. MR 0891718 (88g:11064), MR 0371840 (51:8057)
- [3]
- E. Bombieri and H. Davenport, Small differences between prime numbers, Proc. Roy. Soc. Ser. A 293 (1966), 1-18. MR 0199165 (33:7314)
- [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]
- J.R. Chen, On the representation of a large even number 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 (34:7483)
- [6]
- H. Davenport, Multiplicative number theory, Springer Verlag, 2000. MR 1790423 (2001f:11001)
- [7]
- N. Elkies and C. McMullen, Gaps in
 and ergodic theory, Duke Math. J. 123 (2004), 95-139. MR 2060024 (2005f:11143)
- [8]
- P. Erdos, On the difference of consecutive primes, Quart. J. Math. Oxford 6 (1935), 124-128.
- [9]
- P. Erdos, The difference between consecutive primes, Duke Math. J. 6 (1940), 438-441. MR 0001759 (1:292h)
- [10]
- J. Friedlander and H. Iwaniec, The polynomial
 captures its primes, Ann. of Math. 148 (1998), 945-1040. MR 1670065 (2000c:11150a)
- [11]
- P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976), 4-9. MR 0409385 (53:13140)
- [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]
- A. Granville, Unexpected irregularities in the distribution of prime numbers, Proc. of the Int. Congr. of Math., Vols. 1, 2 (Zürich, 1994) (1995), 388-399. MR 1403939 (97d:11139)
- [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
 , Acta Math. 186 (2001), 1-84. MR 1828372 (2002b:11122)
- [20]
- M. Huxley, Small differences between consecutive primes. II, Mathematika 24 (1977), 142-152. MR 0466042 (57:5925)
- [21]
- N. Katz and P. Sarnak, Zeros of zeta functions and symmetry, Bull. Amer. Math. Soc. 36 (1999), 1-26. MR 1640151 (2000f:11114)
- [22]
- H. Maier, Small differences between prime numbers, Michigan Math. J. 35 (1988), 323-344. MR 0978303 (90e:11126)
- [23]
- H. Montgomery, The pair correlation of zeros of the zeta-function, Proc. Symp. Pure Math. 24 (St. Louis, MO, 1972), 181-193, Amer. Math. Soc., 1973. MR 0337821 (49:2590)
- [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. Szemeredi, On sets of integers containing no
 elements in arithmetic progression, Proc. International Congress of Math. (Vancouver) 2 (1975), 503-505. MR 0422191 (54:10183)
- [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:
http://dx.doi.org/10.1090/S0273-0979-06-01142-6
PII:
S 0273-0979(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.
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