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Ben Green; Terence Tao
Restriction theory of the Selberg sieve, with applications
Journal de théorie des nombres de Bordeaux, 18 no. 1 (2006), p. 147-182, doi: 10.5802/jtnb.538
Article PDF | Reviews MR 2245880 | Zbl 05070452

Résumé - Abstract

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 $a_1,\dots ,a_k$ and $b_1,\dots ,b_k$ be positive integers. Write $h(\theta ) := \sum _{n \in X} e(n\theta )$, where $X$ is the set of all $n \le N$ such that the numbers $a_1n + b_1,\dots , a_kn + b_k$ are all prime. We obtain upper bounds for $\Vert h \Vert _{L^p(\mathbb{T})}$, $p > 2$, 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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