Restriction theory of the Selberg sieve, with applications

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

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 1, page 147-182
  • ISSN: 1246-7405

Abstract

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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 ) : = ∑ n ∈ X e ( n θ ) , where X is the set of all n ≤ N 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 .

How to cite

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Green, Ben, and Tao, Terence. "Restriction theory of the Selberg sieve, with applications." Journal de Théorie des Nombres de Bordeaux 18.1 (2006): 147-182. <http://eudml.org/doc/249661>.

@article{Green2006,
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 $\Vert h \Vert _\{L^p(\mathbb\{T\})\}$, $p &gt; 2$, 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 $p_1 &lt; p_2 &lt; p_3$ of primes, such that $p_i + 2$ is either a prime or a product of two primes for each $i=1,2,3$.},
affiliation = {School of Mathematics University of Bristol Bristol BS8 1TW, England; Department of Mathematics University of California at Los Angeles Los Angeles CA 90095, USA},
author = {Green, Ben, Tao, Terence},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {primes; arithmetic progressions; Selberg sieve},
language = {eng},
number = {1},
pages = {147-182},
publisher = {Université Bordeaux 1},
title = {Restriction theory of the Selberg sieve, with applications},
url = {http://eudml.org/doc/249661},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Green, Ben
AU - Tao, Terence
TI - Restriction theory of the Selberg sieve, with applications
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 1
SP - 147
EP - 182
AB - 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 $\Vert h \Vert _{L^p(\mathbb{T})}$, $p &gt; 2$, 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 $p_1 &lt; p_2 &lt; p_3$ of primes, such that $p_i + 2$ is either a prime or a product of two primes for each $i=1,2,3$.
LA - eng
KW - primes; arithmetic progressions; Selberg sieve
UR - http://eudml.org/doc/249661
ER -

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