On Goldbach conjecture

Question

Let $N$ a large natural number, let $\forall n\leq N,\, R_{2}\left(n\right)=\underset{p_{1}+p_{2}=n}{\sum}\log\left(p_{1}\right)\log\left(p_{2}\right)$ and let $S\left(\alpha\right)=\underset{p\leq N}{\sum}\log\left(p\right)e\left(p\alpha\right)$ where $p,\, p_{i}$ are prime numbers and $e\left(x\right)=e^{2\pi ix}$. We know that $\forall n\leq N$ $$R_{2}\left(n\right)=\int_{0}^{1}S(\alpha)^{2}e(-n\alpha)\textrm{d}\alpha=\left(\int_{\mathfrak{M}}+\int_{\mathfrak{m}}\right)S(\alpha)^{2}e(-n\alpha)\textrm{d}\alpha=$$ $$=R_{\mathfrak{M}}(n)+R_{\mathfrak{m}}(n)$$ where $R_{\mathfrak{M}}(n)$ is the contribution of major arcs and $R_{\mathfrak{m}}(n)$ is the contribution of minor arcs. As major arcs, minor arcs are disjoint $$\mathfrak{m}=\underset{i=1}{\overset{s}{\bigcup}}\mathfrak{m}_{i},\,\mathfrak{m}_{i}\cap\mathfrak{m}_{j}=\textrm{Ø}\, if\, i\neq j$$ say. Suppose that $\overline{\mathfrak{m}}\in\mathfrak{m}$ and suppose that $c\in\overline{\mathfrak{m}}$. I would like to know if it is possible to obtain an estimation like $$\left|\int_{\overline{\mathfrak{m}}}S(\alpha)^{2}e(-n\alpha)\textrm{d}\alpha\right|=\left|\underset{p_{1}\leq N}{\sum}\log p_{1}\underset{p_{2}\leq N}{\sum}\log p_{2}\int_{\overline{\mathfrak{m}}'}e\left(\left(p_{1}+p_{2}-n\right)\left(c+\eta\right)\right)\textrm{d}\eta\right|=$$ $$=\left|\underset{p_{1}\leq N}{\sum}\log p_{1}e\left(p_{1}c\right)\underset{p_{2}\leq N}{\sum}\log p_{2}e\left(p_{2}c\right)\int_{\overline{\mathfrak{m}}'}e\left(\left(p_{1}+p_{2}-n\right)\eta\right)\textrm{d}\eta\right|\ll\left|S\left(c\right)^{2}\right|F(N)$$ where $F(N)=O\left(N^{\delta}\right),\,\delta<1.$ If it isn't possible, which is the best possible limitation for $F(N)$? Thank you.