Primes in tuples I

Annals of Mathematics

Volumn 170, Issue 2, 2009, Pages 819-862

  Goldston, Daniel A ^a   Pintz, János ^b   Yildirim, Cem Y ^c  

^a SAN JOSE STATE UNIVERSITY   (United States)

^b ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS   (Hungary)

^c BOGAZICI UNIVERSITY   (Turkey)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 71449097991     PISSN: 0003486X     EISSN: None     Source Type: Journal    
DOI: 10.4007/annals.2009.170.819     Document Type: Article

References (33)

1. P. T. Bateman and R. A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363-367.
2. E. Bombieri and H. Davenport, Small differences between prime numbers, Proc. Roy. Soc. Ser. A 293 (1966), 1-18.
3. E. Bombieri, J. B. Friedlander, and H. Iwaniec, Primes in arithmetic progressions to large moduli. III, J. Amer. Math. Soc. 2 (1989), 215-224.
4. H. Davenport, Multiplicative Number Theory, second ed., Grad. Texts Math. 74, Springer-Verlag, New York, 1980, Revised by Hugh L. Montgomery.
5. P. D. T. A. Elliott and H. Halberstam, A conjecture in prime number theory, in Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), Academic Press, London, 1970, pp. 59-72.
6. T. J. Engelsma, k-tuple permissible patterns, 2005. Available at http://www.opertech.com/primes/k-tuples.html.
7. P. Erdös, The difference of consecutive primes, Duke Math. J. 6 (1940), 438-441.
8. K. Ford, Zero-free regions for the Riemann zeta function, in Number Theory for the Millennium, II (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, pp. 25-56.
9. É. Fouvry and F. Grupp, On the switching principle in sieve theory, J. Reine Angew. Math. 370 (1986), 101-126.
10. P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976), 4-9.
11. D. A. Goldston, On Bombieri and Davenport's theorem concerning small gaps between primes, Mathematika 39 (1992), 10-17.
12. D. A. Goldston and C. Y. Yildirim, Higher correlations of divisor sums related to primes. I. Triple correlations, Integers 3 (2003), A5, 66 pp.
13. D. A. Goldston and C. Y. Yildirim, Higher correlations of divisor sums related to primes III: Small gaps between primes, Proc. London Math. Soc. 95 (2007), 653-686.
14. D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yildirim, Small gaps between primes and almost primes, Trans. Amer. Math. Soc. 361 (2009), 5285-5330.
15. D. A. Goldston, Y. Motohashi, J. Pintz, and C. Y. Yildirim, Small gaps between primes exist, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), 61-65.
16. H. Halberstam and H.-E. Richert, Sieve Methods, London Math. Soc. Monogr., No. 4, Academic Press, New York, 1974.
17. 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.
18. G. H. Hardy and J. E. Littlewood, unpublished manuscript, see [26].
19. D. R. Heath-Brown, Almost-prime k-tuples, Mathematika 44 (1997), 245-266.
20. M. N. Huxley, On the differences of primes in arithmetical progressions, Acta Arith. 15 (1968/1969), 367-392.
21. M. N. Huxley, Small differences between consecutive primes. II, Mathematika 24 (1977), 142-152.
22. M. Huxley, An application of the Fouvry-Iwaniec theorem, Acta Arith. 43 (1984), 441-443.
23. H. Maier, Small differences between prime numbers, Michigan Math. J. 35 (1988), 323-344.
24. H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Math. 227, Springer-Verlag, New York, 1971.
25. G. Z. Pilt'ai, On the size of the difference between consecutive primes, Issledovania po teorii chisel 4 (1972), 73-79.
26. R. A. Rankin, The difference between consecutive prime numbers. II, Proc. Cambridge Philos. Soc. 36 (1940), 255-266.
27. G. Ricci, Sull'andamento della differenza di numeri primi consecutivi, Riv. Mat. Univ. Parma 5 (1954), 3-54.
28. A. Schinzel and W. Sierpiń Ski, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4, 185-208; Erratum 5 (1958), 259.
29. A. Selberg, Collected Papers. Vol. II, Springer-Verlag, New York, 1991.
30. J. Sivak, Méthodes de crible appliquées aux sommes de Kloosterman et aux petits écarts entre nombres premiers, 2005, Thèse de Doctorat de l'Université Paris Sud (Paris XI). Available at http://www.math.u-psud.fr/~sivak/these.pdf.
31. K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yi{dotless}ldi{dotless}ri{dotless}m, Bull. Amer. Math. Soc. 44 (2007), 1-18.
32. E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, second ed., The Clarendon Press, Oxford University Press, New York, 1986.
33. S. Uchiyama, On the difference between consecutive prime numbers, Acta Arith. 27 (1975), 153-157.