H.A. Helfgott recently proved Goldbach's weak conjecture here: http://arxiv.org/pdf/1305.2897v2.pdf
In (1.1), he explains that he is trying to show that $$\sum_{n_1 + n_2 + n_3 = N}\Lambda(n_1)\Lambda(n_2)\Lambda(n_3) > 0$$ if $N\ge7$, (N, 2) = 1. If the paper is correct, he successfully proved this. However, doesn't this shows that all integers are the sum of 3 prime powers, since the Von Mangoldt function at n is nonzero if n is a prime power? The most common generating functions that I have seen for the circle method have been of the form: $$F_N(\alpha) = \sum_{p\le N}\ln p e^{2\pi i\alpha p} $$ rather than $$F_n(\alpha) = \sum_{k\le N}\Lambda(k)e^{2\pi i \alpha k}$$