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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 ?

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    $\begingroup$ Montgomery and Vaughan showed that the exceptional set in Goldbach's conjecture contains at most $O(x^{1-\delta})$ elements for some $\delta >0$. See matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27126.pdf . $\endgroup$
    – Lucia
    Jul 15, 2014 at 23:09
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    $\begingroup$ What does "cte" mean? $\endgroup$ Jul 16, 2014 at 0:19
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    $\begingroup$ @GerryMyerson: "cte" means "constant". I'm sorry, the standard abbreviation is rather "cst"... $\endgroup$ Jul 16, 2014 at 0:20
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    $\begingroup$ $\text{const}$ or $O(1)$ would have required less head-scratching for me. $\endgroup$
    – Ben Barber
    Jul 16, 2014 at 11:18
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    $\begingroup$ I just read your application. I believe it follows more directly, namely a variant of Vinogradov's proof for the ternary Goldbach problem should yield that: any odd number $p$ can be written as $p'+q'-q$ with three primes $p',q',q$. In general, solving a linear equation in three prime variables avoiding the obvious obstructions (e.g. modulo $2$) can be done by the circle method. There is no need to switch to a binary problem. $\endgroup$
    – GH from MO
    Jul 16, 2014 at 17:12

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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 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.

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  • $\begingroup$ Let $τ_2(x)$ be the number of even numbers $2<2n<x$ that can't be written as a sum of two distinct pairs of primes. Do you know if we have similar bounds for $τ$ and $τ_2$ ? Perhaps this question requires to open another post, what do you think? $\endgroup$ Jul 15, 2014 at 23:56
  • $\begingroup$ @Sébastien: The mentioned results not only prove that most even numbers can be written as a sum of two primes, but also that they can be written as a sum of two primes in many ways. Look up Vaughan's book and the Montgomery-Vaughan paper (referred in my response and Lucia's comment). $\endgroup$
    – GH from MO
    Jul 16, 2014 at 0:08
  • $\begingroup$ Ok, thank you! Do you know if these results allow to keep similar bounds for $\tau_2$ (see the application I've edited) ? I will look the paper more closely. $\endgroup$ Jul 16, 2014 at 0:17
  • $\begingroup$ Yes, see the added section in my response. $\endgroup$
    – GH from MO
    Jul 16, 2014 at 19:28
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    $\begingroup$ @AdamRubinson Yes, it is widely open. The best known result is that $\tau(x)<x^{0.72}$ for $x$ sufficiently large. See arxiv.org/abs/1804.09084 $\endgroup$
    – GH from MO
    Jun 19, 2023 at 22:09

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