Assuming the weak Hardy-Littlewood-Goldbach conjecture as stated in this paper,
does the density $d(\delta,\varepsilon)$ of integers $m$ below $n$ such that $$ \left\vert\frac{G(m)}{{\frak{S}}(m)m}-\delta\,\right\vert<\varepsilon $$ equal $d(2-\delta,\varepsilon)$?
Here $G(m)=\underset{m_1+m_2=m \\ \phantom{2.} 2∤m_1m_2 } \sum \Lambda(m_1)\Lambda(m_2)$, ${\frak{S}}(m)=2 \underset{p>2} \prod (1-\frac{1}{(p-1)^2}) \underset{p|m \\ p>2} \prod (1+\frac{1}{p-2}) $ and $\Lambda$ is the von Mangoldt function.
Moreover can one prove that the function $f_{\varepsilon}:\delta\mapsto d(\delta,\varepsilon)$ is strictly increasing on $(0,1)$ as $\varepsilon\to 0$?
Edit december 27th 2021: should $d(\delta,\varepsilon)$ be positive for some $\delta<1$ and all $\varepsilon>0$, would it imply the existence of an exponent $m_{\delta}>0$ (like, say, $m_{\delta}=1-\delta$) such that the exceptional set $E(x)$ in Goldbach's conjecture of even integers not exceeding $x$ which are not expressible as the sum of two primes fulfills $E(x)=\Omega(x^{m_{\delta}})$? By the way, would it be equivalent to the existence of a positive proportion of non trivial zeros of the Riemann zeta function of real part $\frac{\delta}{2}$?
Edit February 25th 2022: the paper Average Goldbach and Quasi-Riemann Hypothesis by Gautami Bhowmik suggests to take $m_{\delta}:=1-\frac{\delta}{2}$, as Hardy and Littlewood proved that under RH, $E(x)\ll_{\varepsilon}x^{\frac{1}{2}+\varepsilon}$. This would be consistent with $S(x)=\frac{x^{2}}{2}+O(x^{\frac{3}{2}})$ under RH where $S(x):=\sum_{n\leq x}G(n)$.
There may be a possibility to draw a parallel to RH that way: a given positive value of $d(\delta,\varepsilon)$ for all $\varepsilon>0$ boils down to the existence of the limit $l(\delta)$ of some sequence. Now consider the operation $\iota$ which permutes the subsequences of terms of odd indices and of even indices of a sequence. This operation preserves the limit, so that we can write $d_{\iota}(\delta,\varepsilon)=d(\delta,\varepsilon)$. Assuming $d_{r_{1}}(\delta,\varepsilon):=d(2-\delta,\varepsilon)$ preserves $d(\delta,\varepsilon)$ for all $(\delta,\varepsilon)$, we would get that $d(\delta,\varepsilon)$ is invariant under the group generated by $\iota$ and $r_{1}$, which is isomorphic to the Klein group. As RH boils down to the symmetry group of $\zeta$ being of order $2$, this would boil down to showing that the induced action of $r_{1}$ on the set of $\delta$ such that $d(\delta,\varepsilon)>0$ for all $\varepsilon>0$ coincides with the action thereon of the identity.
Edit January 4th, 2022: assuming there exists $0<\delta<1$ such that $d(\delta,\varepsilon)=d(2-\delta,\varepsilon)>0$, this would mean that infinitely many even numbers have less Goldbach decompositions than expected and asymptotically as many would have more Goldbach's decompositions than expected. This would mean that the prime numbers are distributed more irregularly than the assumption of RH would allow, giving rise to an error term in the prime number theorem of the form $x^{1/2+\kappa_{\delta}+\epsilon}$ for some $\kappa_{\delta}>0$. Can an expression thereof in terms of $\delta$ be obtained? Can we come up with a rigorous proof that RH implies that $d(\delta,\varepsilon)=0$ for $\delta<1$ as close to $1$ as we want?
If we could attach to some $\delta$ a given L-function $F_{\delta}$ so that its dual is attached to $2-\delta$, the functional equation might provide the desired equality $d(\delta,\varepsilon)=d(2-\delta,\varepsilon)$, and the self duality of zeta may by itself imply that $d(\delta,\varepsilon)=0$ for $\delta<1$ as close to $1$ as we want. One can also proceed the other way around and associate to some L-function $F$ the sequence $(\delta_{k}(F):=1+\frac{1}{\pi}\arg a_{k}(F))_{k>0}$ where $a_{k}(F)$ is the $k$-th coefficient of the Dirichlet series of $F$.
