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

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.

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.

DOI: 10.5802/jtnb.538
Green, Ben 1; Tao, Terence 2

1 School of Mathematics University of Bristol Bristol BS8 1TW, England
2 Department of Mathematics University of California at Los Angeles Los Angeles CA 90095, USA
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Green, Ben; Tao, Terence. Restriction theory of the Selberg sieve, with applications. Journal de théorie des nombres de Bordeaux, Volume 18 (2006) no. 1, pp. 147-182. doi : 10.5802/jtnb.538. http://www.numdam.org/articles/10.5802/jtnb.538/

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