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 ${a}_{1},\cdots ,{a}_{k}$ and ${b}_{1},\cdots ,{b}_{k}$ be positive integers. Write $h\left(\theta \right):={\sum }_{n\in X}e\left(n\theta \right)$, where $X$ is the set of all $n\le N$ such that the numbers ${a}_{1}n+{b}_{1},\cdots ,{a}_{k}n+{b}_{k}$ are all prime. We obtain upper bounds for ${\parallel h\parallel }_{{L}^{p}\left(𝕋\right)}$, $p>2$, 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 ${a}_{1},\cdots ,{a}_{k}$ et ${b}_{1},\cdots ,{b}_{k}$ les entiers positifs. On pose $h\left(\theta \right):={\sum }_{n\in X}e\left(n\theta \right)$, où $X$ est l’ensemble des $n\le N$ tels que tous les nombres ${a}_{1}n+{b}_{1},\cdots ,{a}_{k}n+{b}_{k}$ sont premiers. Nous obtenons des bornes supérieures pour ${\parallel h\parallel }_{{L}^{p}\left(𝕋\right)}$, $p>2$, 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
Ben Green 1; Terence Tao 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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Ben Green; Terence Tao. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.538/

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