Restriction theory of the Selberg sieve, with applications

Journal de Theorie des Nombres de Bordeaux

Volumn 18, Issue 1, 2006, Pages 147-182

  Green, Ben ^a   Tao, Terence ^b  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85009959882     PISSN: 12467405     EISSN: None     Source Type: Journal    
DOI: 10.5802/jtnb.538     Document Type: Article

References (38)

1. p(G). Quart. J. Math. (Oxford) (2) 24 (1973), 119-128.
2. A. Balog, The Hardy-Littlewood k-tuple conjecture on average. Analytic Number Theory (eds. B. Brendt, H.G. Diamond, H. Halberstam and A. Hildebrand), Birkhauser, 1990, 47-75.
3. J. Bourgain, On Λ(p)-subsets of squares. Israel J. Math. 67 (1989), 291-311.
4. J. Bourgain, Fourier restriction phenomena for certain lattice subsets and application to non- linear evolution equations I. Schrödinger equations. GAFA 3 (1993), 107-156.
5. J. Bourgain, On triples in arithmetic progression. GAFA 9 (1999), no. 5, 968-984.
6. J.-R. Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes. Sci. Sinica 16 (1973), 157-176.
7. F. Dress, H. Iwaniec, G. Tenenbaum, Sur une somme liée à la fonction Möbius. J. reine angew. Math. 340 (1983), 53-58.
8. J.B. Friedlander, D.A. Goldston, Variance of distribution of primes in residue classes. Quart. J. Math. (Oxford) (2) 47 (1996), 313-336.
9. 4captures its primes. Ann. Math. 148 (1998), no. 3, 965-1040.
10. D.A. Goldston, A lower bound for the second moment of primes in short intervals. Exposition Math. 13 (1995), no. 4, 366-376.
11. D.A. Goldston, C.Y. Yıldırım, Higher correlations of divisor sums related to primes, I: Triple correlations. Integers 3 (2003) A5, 66 pp.
12. D.A. Goldston, C.Y. Yıldırım, Higher correlations of divisor sums related to primes, III: k-correlations. Preprint. Available at: http://www.arxiv.org/pdf/math.NT/0209102.
13. D.A. Goldston, C.Y. Yıldırım, Small gaps between primes I. Preprint. Available at: http://www.arxiv.org/pdf/math.NT/0504336.
14. W.T. Gowers, A new proof of Szemerédi’s theorem for arithmetic progressions of length four. GAFA 8 (1998), 529-551.
15. S.W. Graham, An asymptotic estimate related to Selberg’s sieve. J. Num. Th. 10 (1978), 83-94.
16. B.J. Green, Roth’s theorem in the primes. Ann. of Math. (2) 161 (2005), no. 3, 1609-1636.
17. B.J. Green, I.Z. Ruzsa, On the Hardy-Littlewood majorant problem. Math. Proc. Camb. Phil. Soc. 137 (2004), no. 3, 511-517.
18. B.J. Green, T.C. Tao, The primes contain arbitrarily long arithmetic progressions. To appear in Ann. of Math.
19. H. Halberstam, H.E. Richert, Sieve Methods. London Math. Soc. Monographs 4, Academic Press 1974.
20. C. Hooley, Applications of sieve methods in number theory. Cambridge Tracts in Math. 70, CUP 1976.
21. C. Hooley, On the Barban-Davenport-Halberstam theorem, XII. Number Theory in Progress, Vol. 2 (Zakopan-Kościelisko 1997), 893-910, de Gruyter, Berlin 1999.
22. H. Iwaniec, Sieve methods. Graduate course, Rutgers 1996.
23. G. Mockenhaupt, W. Schlag, manuscript.
24. H.L. Montgomery, Ten lectures on the interface between analytic number theory and har- monic analysis. CBMS Regional Conference Series in Mathematics 84, AMS, Providence, RI, 1994.
25. H.L. Montgomery, Selberg’s work on the zeta-function. In Number Theory, Trace Formulas and Discrete Groups (a Symposium in Honor of Atle Selberg, Oslo, Norway, July 14-21, 1987).
26. Y. Motohashi, A multiple sum involving the Möbius function. Publ. Inst. Math. (Beograd) (N.S.) 76(90) (2004), 31-39. Also available at: http://www.arxiv.org/pdf/math.NT/0310064.
27. O. Ramaré, On Snirel’man’s constant. Ann. Scu. Norm. Pisa 22 (1995), 645-706.
28. O. Ramaré, I.Z. Ruzsa, Additive properties of dense subsets of sifted sequences. J. Th. Nombres de Bordeaux 13 (2001), 559-581.
29. K.F. Roth, On certain sets of integers. J. London Math. Soc. 28 (1953), 104-109.
30. I.Z. Ruzsa, On an additive property of squares and primes. Acta Arithmetica 49 (1988), 281-289.
31. A. Selberg, On an elementary method in the theory of primes. Kong. Norske Vid. Selsk. Forh. B 19 (18), 64-67.
32. T.C. Tao, Recent progress on the restriction conjecture. In Fourier analysis and convexity (Milan, 2001), 217-243, Appl. Numer. Harmon. Anal., Birkhauser Boston, Boston, MA, 2004.
33. D.I. Tolev, Arithmetic progressions of prime-almost-prime twins. Acta. Arith. 88 (1999), no. 1, 67-98.
34. P. Tomas, A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc. 81 (1975), 477-478.
35. P. Varnavides, On certain sets of positive density. J. London Math. Soc. 34 (1959), 358-360.
36. R. Vaughan, The Hardy-Littlewood method, 2nd ed. Cambridge Tracts in Math. 125, CUP 1997.
37. D.R. Ward, Some series involving Euler’s function. J. London Math. Soc. 2 (1927), 210-214.
38. A. Zygmund, Trigonometric series, 2nd ed. Vols I, II. CUP 1959.