Mathematics – Number Theory
Scientific paper
2004-05-30
Mathematics
Number Theory
36 pages. To appear in Journal de theorie des nombres de Bordeaux; French abstract added, and several minor amendments made
Scientific paper
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'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.
Green Ben
Tao Terence
No associations
LandOfFree
Restriction theory of the Selberg sieve, with applications does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Restriction theory of the Selberg sieve, with applications, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Restriction theory of the Selberg sieve, with applications will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-34982