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. For t on the unit circle write h(t) := \sum_{n \in X} e(nt)$, where X is the set of all n <= N such that the numbers a_1n + b_1,..., 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&#39;s theorem, Roth&#39;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.