Restriction theory of the Selberg sieve, with applications

Green, Ben; Tao, Terence

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 $a_1,\cdots ,a_k$ 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.

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 $i=1,2,3$.

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