On Chen's theorem, Goldbach's conjecture and almost prime twins II

TL;DR

This paper advances understanding of Goldbach's conjecture and twin primes by providing lower bounds on the number of primes related to almost prime representations of even integers, using advanced sieve techniques.

Contribution

It establishes near-optimal bounds for primes related to Chen's theorem, improving previous results with new sieve methods and distribution level estimates.

Findings

Lower bound for primes with almost prime complements near the conjectured constant

Results on twin prime problem and additive representations

Enhanced sieve techniques applied to prime number problems

Abstract

Let $N$ denote a sufficiently large even integer and $x$ denote a sufficiently large integer, we define $D_{1,2}(N)$ as the number of primes $p$ that such that $N - p$ has at most 2 prime factors. In this paper, we show that $D_{1,2}(N) \geqslant 1.9728 \frac{C(N) N}{(\log N)^2}$, which is rather near to the asymptotic constant $2$ in Hardy--Littlewood conjecture for Goldbach's conjecture. We also get similar results on twin prime problem and additive representations of integers. The proof combines various techniques in sieve methods, such as weighted sieve, Chen's switching principle, new distribution levels proved by Lichtman and Pascadi, Chen's double sieve and Harman's sieve.