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

2024/05/09 by Runbo Li, Li, Runbo · 1 citation
Mathematics · #Analytic Number Theory Research #Limits and Structures in Graph Theory #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2405.05727

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 (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.

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