Let $\tau(x)$ be the number of even numbers $2<2n<x$ which can't be written as a sum of two primes. 


*Goldbach's conjecture*: $\tau(x) = 0$  

*Asymptotic Goldbach's conjecture*: $\tau(x) = O(1) $


*Weak asymptotic Goldbach's conjecture*: $\tau(x) = \omicron (\frac{x}{ln(x)}) $ 


**Question**: Is this weak asymptotic Goldbach's conjecture open ?  

What's the better estimate known? 




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**Application**: Given an odd prime number $p$, there are odd prime numbers $q$, $p'$, $q'$ such that $\{p,q\} \neq \{ p',q'\}$ and $p+q = p'+q'$. 


As explained [here](https://math.stackexchange.com/questions/868224/is-this-goldbach-type-problem-easy-to-solve/868478#868478), we need something slightly stronger for this application: $\tau\_2(x) = \omicron (x/ln(x)) $, with $τ\_2(x)$ the number of even numbers $2<2n<x$ that can't be written as a sum of two distinct pairs of primes. Is it known ?



Is this weak asymptotic Goldbach's conjecture open?
The weak asymptotic Goldbach conjecture was proved by Chudakov in 1937 (based on the groundbreaking work of Vinogradov). Better bounds are known, see Lucia's comment. The preprint [arXiv:1804.09084](https://arxiv.org/abs/1804.09084) by Pintz shows that $\tau(x)<x^{0.72}$ for $x$ sufficiently large.


You can find the original reference and a modern treatment in Vaughan's book "The Hardy-Littlewood method".


**Added.** My response and Lucia's comment hold verbatim for $τ\_2$. In fact Montgomery-Vaughan proved that for all but $O(x^{1−δ})$ even integers $x/2<2n<x$, the number of representations $2n=p+q$ is at least $x^{1−3δ}$. Here $δ$ is any sufficiently small positive number, and the implied constant depends only on $δ$. See Section $8$ in the Montgomery-Vaughan paper.


