Restriction theory of the Selberg sieve, with applications
| dc.creator | Green, Ben | |
| dc.creator | Tao, Terence | |
| dc.date | 2004-05-30 | |
| dc.date | 2005-05-10 | |
| dc.date.accessioned | 2026-07-07T05:08:44Z | |
| dc.date.available | 2026-07-07T05:08:44Z | |
| dc.description | 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. | |
| dc.description | 36 pages. To appear in Journal de theorie des nombres de Bordeaux; French abstract added, and several minor amendments made | |
| dc.identifier | https://arxiv.org/abs/math/0405581 | |
| dc.identifier | http://arxiv.org/abs/math/0405581 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/71383 | |
| dc.subject | Number Theory | |
| dc.subject | Classical Analysis and ODEs | |
| dc.title | Restriction theory of the Selberg sieve, with applications | |
| dc.type | text |