Restriction theory of the Selberg sieve, with applications

2004/05/30 by Ben Green, Terence Tao, Green, Ben +1 · 5 citations
Mathematics · #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT) #math.CA #math.NT

paper · pdf · doi:10.48550/arxiv.math/0405581

36 pages. To appear in Journal de theorie des nombres de Bordeaux; French abstract added, and several minor amendments made

openalex publication_date 2004/05/30 · arxiv created 2005/05/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L²-L^p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime k-tuples. Let a₁,...,a_k and b₁,...,b_k be positive integers. For t on the unit circle write h(t) := ∑_{n ∈ X} e(nt), where X is the set of all n <= N such that the numbers a₁n + b₁,..., a_kn + b_k are all prime. We obtain upper bounds for the L^p norm of h, p > 2, which are (conditionally on the 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 p₁ < p₂ < p₃ of primes, such that p_i + 2 is either a prime or a product of two primes for each i=1,2,3.

Citations

• Higher correlations of divisor sums related to primes III: k-correlations
• Small Gaps Between Primes I
• On Λ(p)-subsets of squares
• On Certain Sets of Integers

Cited by

• Restriction estimates with sifted integers
• Bounds for monochromatic solutions to \x+y,xy\
• Roth's Theorem in Super Smooth Numbers
• The trigonometric polynomial on sums of two squares, an additive problem and generalisation
• Small gaps between Goldbach primes

Related

• Variants of the Selberg sieve, and almost prime k-tuples
• A note on θ₂
• Notes on restriction theory in the primes
• On the theory of prime producing sieves
• Selberg's sieve of irregular density