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 L2-Lprestriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime k-tuples. Let a1,., akand b1,., bkbe positive integers. Write h(θ):= Σn∈Xe(nθ), where X is the set of all n ≤ N such that the numbers a1n+b1,., akn+bkare all prime. We obtain upper bounds for ||h||Lp(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 p1< p2< p3of primes, such that pi+ 2 is either a prime or a product of two primes for each i = 1, 2, 3.