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Chen Jing Run proved that every large enough even integer is either the sum of two primes or the sum of a prime and a semi-prime (that is, the product of two primes). Golbach's conjecture states that every even integer greater than 3 is the sum of two primes. Harald Helfgott (happy birthday to him) proved that every odd integer greater than 5 is the sum of three primes.

What is the best known upper bound for the number $\overline{\mathcal{G}}(x) $ defined as the number of even integers below $x$ that are the sum of a prime and a semi-prime but not the sum of two primes?

Many thanks in advance.

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  • $\begingroup$ Up to a constant (which is at most the number of "counterexamples" to Chen's theorem), by Chen's theorem $\overline{\mathcal G}(x)$ is the same as the number of counterexamples to Goldbach's conjecture, isn't it? $\endgroup$ – Wojowu Nov 25 '16 at 18:44
  • $\begingroup$ Yes, but I would like to know whether, for example, a combined approach using Chen's, Helfgott and Maynard-Tao's proofs lead to some explicit upper bound. $\endgroup$ – Sylvain JULIEN Nov 25 '16 at 18:48
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If I understand correctly, $\overline{\mathcal{G}}(x)$ is the usual exceptional set $E(x)$ except for possibly finitely many exceptions in the range $(4\cdot 10^{18},e^{e^{36}})$, where the Goldbach conjecture has not been verified, and an effective version of Chen's theorem does not apply. With that in mind, a classic result of Montgomery-Vaughan gives

$$E(x)<x^{1-\delta}$$

The best $\delta$ I know of is $0.121$, due to Wen Chao Lu in 2010.

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  • $\begingroup$ I don't think it is known that this set is the usual exceptional set, namely it differs by the number of possible $n$ which are neither sum of two primes not of a prime and semiprime. Chen's theorem states there are only finitely many of these though, so they are "essentially" the same. $\endgroup$ – Wojowu Nov 25 '16 at 22:23
  • $\begingroup$ You're right. I've edited the answer accordingly. $\endgroup$ – Myshkin Nov 28 '16 at 7:39

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