Restriction theory of the Selberg sieve, with applications

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 ,⋯,a k and b 1 ,⋯,b k be positive integers. Write h(θ):=∑ n∈X e(nθ), 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.

Locations

• Journal de Théorie des Nombres de Bordeaux - View - PDF
• Journal de Théorie des Nombres de Bordeaux - View - PDF