Vinogradov's Theorem/Corollary 1

Corollary
Let


 * $\displaystyle r(N) = \sum_{p_1 + p_2 + p_3 = N}1$

with $p_1,p_2,p_3$ prime. Then for sufficiently large odd integers $N$:


 * $\displaystyle r(N) = \mathfrak S(N) \frac {N^2}{2 (\log N)^3}\left( 1 + \mathcal O\left( \frac{\log\log N}{\log N} \right) \right)$