Restriction theory of the Selberg sieve, with applications
Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 1, pp. 147-182.

Le crible de Selberg fournit des majorants pour certaines suites arithmétiques, comme les nombres premiers et les nombres premiers jumeaux. Nous démontrons un théorème de restriction L 2 -L p pour les majorants de ce type. Comme application immédiate, nous considérons l’estimation des sommes d’exponentielles sur les k-uplets premiers. Soient a 1 ,,a k et b 1 ,,b k les entiers positifs. On pose h(θ):= nX e(nθ), où X est l’ensemble des nN tels que tous les nombres a 1 n+b 1 ,,a k n+b k sont premiers. Nous obtenons des bornes supérieures pour h L p (𝕋) , p>2, qui sont (en supposant la vérité de la conjecture de Hardy et Littlewood sur les k-uplets premiers) d’ordre de magnitude correct. Une autre application est la suivante. En utilisant les théorèmes de Chen et de Roth et un « principe de transférence », nous démontrons qu’il existe une infinité de suites arithmétiques p 1 <p 2 <p 3 de nombres premiers, telles que chacun p i +2 est premier ou un produit de deux nombres premier.

The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L 2 L p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime k-tuples. Let a 1 ,,a k and b 1 ,,b k be positive integers. Write h(θ):= nX e(nθ), where X is the set of all nN such that the numbers a 1 n+b 1 ,,a k n+b k are all prime. We obtain upper bounds for h L p (𝕋) , p>2, which are (conditionally on the Hardy-Littlewood prime tuple conjecture) of the correct order of magnitude. As a second application we deduce from Chen’s theorem, Roth’s theorem, and a transference principle that there are infinitely many arithmetic progressions p 1 <p 2 <p 3 of primes, such that p i +2 is either a prime or a product of two primes for each i=1,2,3.

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DOI : https://doi.org/10.5802/jtnb.538
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     author = {Ben Green and Terence Tao},
     title = {Restriction theory of the {Selberg} sieve, with applications},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {147--182},
     publisher = {Universit\'e Bordeaux 1},
     volume = {18},
     number = {1},
     year = {2006},
     doi = {10.5802/jtnb.538},
     zbl = {05070452},
     mrnumber = {2245880},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.538/}
}
Ben Green; Terence Tao. Restriction theory of the Selberg sieve, with applications. Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 1, pp. 147-182. doi : 10.5802/jtnb.538. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.538/

[1] G.F. Bachelis, On the upper and lower majorant properties in L p (G). Quart. J. Math. (Oxford) (2) 24 (1973), 119–128. | MR 320636 | Zbl 0268.43003

[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), Birkhäuser, 1990, 47–75. | Zbl 0719.11066

[3] J. Bourgain, On Λ(p)-subsets of squares. Israel J. Math. 67 (1989), 291–311. | MR 1029904 | Zbl 0692.43005

[4] , Fourier restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I. Schrödinger equations. GAFA 3 (1993), 107–156. | Zbl 0787.35097

[5] , On triples in arithmetic progression. GAFA 9 (1999), no. 5, 968–984. | Zbl 0959.11004

[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. | MR 434997 | Zbl 0319.10056

[7] F. Dress, H. Iwaniec, G. Tenenbaum, Sur une somme liée à la fonction Möbius. J. reine angew. Math. 340 (1983), 53–58. | MR 691960 | Zbl 0497.10003

[8] J.B. Friedlander, D.A. Goldston, Variance of distribution of primes in residue classes. Quart. J. Math. (Oxford) (2) 47 (1996), 313–336. | MR 1412558 | Zbl 0859.11054

[9] J.B. Friedlander, H. Iwaniec, The polynomial X 2 +Y 4 captures its primes. Ann. Math. 148 (1998), no. 3, 965–1040. | Zbl 0926.11068

[10] D.A. Goldston, A lower bound for the second moment of primes in short intervals. Exposition Math. 13 (1995), no. 4, 366–376. | MR 1358214 | Zbl 0854.11044

[11] D.A. Goldston, C.Y. Yıldırım, Higher correlations of divisor sums related to primes, I: Triple correlations. Integers 3 (2003) A5, 66pp. | Zbl 01989994

[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. | MR 1631259 | Zbl 0907.11005

[15] S.W. Graham, An asymptotic estimate related to Selberg’s sieve. J. Num. Th. 10 (1978), 83–94. | MR 484449 | Zbl 0382.10031

[16] B.J. Green, Roth’s theorem in the primes. Ann. of Math. (2) 161 (2005), no. 3, 1609–1636. | MR 2180408 | Zbl 05004660

[17] B.J. Green, I.Z. Ruzsa, On the Hardy-Littlewood majorant problem. Math. Proc. Camb. Phil. Soc. 137 (2004), no. 3, 511–517. | MR 2103913 | Zbl 1066.42009

[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. | MR 424730 | Zbl 0298.10026

[20] C. Hooley, Applications of sieve methods in number theory. Cambridge Tracts in Math. 70, CUP 1976. | Zbl 0327.10044

[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. | MR 1689551 | Zbl 0943.11042

[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 harmonic analysis. CBMS Regional Conference Series in Mathematics 84, AMS, Providence, RI, 1994. | MR 1297543 | Zbl 0814.11001

[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). | Zbl 0671.10001

[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. | MR 2123862 | Zbl 1098.11049

[27] O. Ramaré, On Snirel’man’s constant. Ann. Scu. Norm. Pisa 22 (1995), 645–706. | Numdam | MR 1375315 | Zbl 0851.11057

[28] O. Ramaré, I.Z. Ruzsa, Additive properties of dense subsets of sifted sequences. J. Th. Nombres de Bordeaux 13 (2001), 559–581. | Numdam | MR 1879673 | Zbl 0996.11057

[29] K.F. Roth, On certain sets of integers. J. London Math. Soc. 28 (1953), 104–109. | MR 51853 | Zbl 0050.04002

[30] I.Z. Ruzsa, On an additive property of squares and primes. Acta Arithmetica 49 (1988), 281–289. | MR 932527 | Zbl 0636.10042

[31] A. Selberg, On an elementary method in the theory of primes. Kong. Norske Vid. Selsk. Forh. B 19 (18), 64–67. | MR 22871 | Zbl 0041.01903

[32] T.C. Tao, Recent progress on the restriction conjecture. In Fourier analysis and convexity (Milan, 2001), 217–243, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2004. | Zbl 1083.42008

[33] D.I. Tolev, Arithmetic progressions of prime-almost-prime twins. Acta. Arith. 88 (1999), no. 1, 67–98. | MR 1698353 | Zbl 0929.11043

[34] P. Tomas, A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc. 81 (1975), 477–478. | MR 358216 | Zbl 0298.42011

[35] P. Varnavides, On certain sets of positive density. J. London Math. Soc. 34 (1959), 358–360. | MR 106865 | Zbl 0088.25702

[36] R. Vaughan, The Hardy-Littlewood method, 2nd ed. Cambridge Tracts in Math. 125, CUP 1997. | MR 1435742 | Zbl 0868.11046

[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. | MR 107776