An -Function-Free Proof of Vinogradov’s Three Primes Theorem

Abstract

We give a new proof of Vinogradov's three primes theorem, which asserts that all sufficiently large odd positive integers can be written as the sum of three primes. Existing proofs rely on the theory of L-functions, either explicitly or implicitly. Our proof is sieve theoretical and uses a transference principle, the idea of which was first developed by Green [Ann. of Math. (2008), . To make our argument work, we also develop an additive combinatorial result concerning popular sums, which may be of independent interest.

A density version of the Vinogradov three primes theorem

“…One nice consequence of breaking the density 1/2 barrier is that it opens the door for giving a new proof of Vinogradov's theorem without using the theory of L-functions. This will be carried out in a forthcoming paper [10].…”

Waring-Goldbach problem for fourth powers in short intervals

“…k ≪ q 1− 1 k 2 (q 2 , t k ) 1 k ;we can also deduce from[Shao, Lemma A3] that d 1 ,d 2 |P t|[d 1 ,d 2 ] ρ d 1 ρ d 2 [d 1 , d 2 ] ≪ J −1 t −1+ε .…”

“…We'll construct a standard Selberg's sieve majorant as follows. Following the notations in [Shao,p.15], let P be the product of all primes p < z and (p, W ) = 1. Let ρ d be weights which are supported on d < z and satisfy |ρ d | ≤ 1 and ρ 1 = 1.…”

A robust version of Freiman's 3k–4 Theorem and applications

“…The second corollary deals with popular sums; see [10,Lemma 3.4] and [12,Theorem 2.4]. and n > 2η −1/2 .…”