TITLE: Is this weak asymptotic Goldbach's conjecture open?
QUESTION [7 upvotes]: 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?  

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, 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 ?

REPLY [12 votes]: 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. Pintz announced that the exceptional set up to $x$ has cardinality $O(x^{2/3})$, but he hasn't published that result yet.
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.