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What are the current research results on the estimation of the upper bound of $D(N)$ in Chen Jingrun's theorem?

Including but not limited to Chen Jingrun's improvement 7.8342 and Wu Jie's improvement 7.8209. enter image description here

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    $\begingroup$ It would be better if you clarify the definition of $D(N)$ $\endgroup$ Commented Sep 12, 2022 at 14:00
  • $\begingroup$ I do not know about $D(N)$, but the state-of-the-art for the related twin prime problem is Lichtman's recent preprint on twin primes, building on Maynard's work, see arxiv.org/abs/2109.02851 . The table in p. 3 lists previous works, including Chen's. $\endgroup$ Commented Sep 12, 2022 at 14:23
  • $\begingroup$ Thank you for your reply, but what I want to get is the upper bound estimation of $D(N)$. All the information you mentioned in this article is about the upper bounds on $π_2 (x)/Π(x)$. $\endgroup$ Commented Sep 13, 2022 at 12:19
  • $\begingroup$ In addition, how to modify the weight in the weighted sieve method? $\endgroup$ Commented Sep 17, 2022 at 13:54

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