A Remark on Chen’s Theorem with Small Primes

A Remark on a Remark by Macaulay or Enhancing Lazard Structural Theorem

Chen’s Primes and Ternary Goldbach Problem

In Iwaniec’s unpublished notes [10], the exponent 1/10 can be improved to 3/11. In [6], Green and Tao say a prime p is Chen’s prime if p ∈ P 2 . On the other hand, in 1937 Vinogradov [18] solved the ternary Goldbach problem and showed that every sufficiently large odd integer can be represented as the sum of three primes. Two years later, using Vinogradov’s method, van der Corput [2] proved tha...

Restriction Theory of the Selberg Sieve, with Applications

The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L–L restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime k-tuples. Let a1, . . . , ak and b1, . . . , bk be positive integers. Write h(θ) := ∑ n∈X e(nθ), where X is the set of all n 6 N such that the nu...

de Théorie des Nombres de Bordeaux 18 ( 2006 ) , 147 – 182 Restriction theory of the Selberg sieve , with applications par

The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L–L restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime k-tuples. Let a1, . . . , ak and b1, . . . , bk be positive integers. Write h(θ) := ∑ n∈X e(nθ), where X is the set of all n 6 N such that the nu...

Small Gaps between Primes Ii (preliminary)

We examine an idea for approximating prime tuples. 1. Statement of results (Preliminary) In the present work we will prove the following result. Let pn denote the nth prime. Then (1.1) lim inf n→∞ (pn+1 − pn) log pn(log log pn)−1 log log log log pn < ∞. Further we show that supposing the validity of the Bombieri–Vinogradov theorem up to Q ≤ X with any level θ > 1/2 we have bounded differences b...