Let's recall: Von Mangoldt function $\Lambda$ is a following function: $$\Lambda: \mathbb{N}\rightarrow \mathbb{R}$$ $$ \Lambda(n) = \left\{ \begin{array}{ll} \log p & \textrm{if $n = p^k$ for some prime $p$}\\ 0 & \textrm{otherwise}\\ \end{array} \right. $$ Ternary Goldbach Conjecture states that: $$W(N):=\sum _{k_1+k_2+k_3 = N}\chi(k_1)\chi(k_2)\chi(k_3)>0$$ For odd numbers greater than $5$. Here $\chi$ is characteristic function for prime numbers.
But here is the problem. In proof of this concjecture we see a quite different expression. $$G(N) = \sum_{k_1+k_2+k_3 = N}\Lambda(k_1)\Lambda(k_2)\Lambda(k_3)$$ This sum is proved to be sufficiently big.
My question is:
How are $W(N)$ and $G(N)$ connected? How to estimate from below the first expression ($W(N)$) using just $G(N)$?
Thank you in advance.