Vinogradov's theorem (1937): Every sufficiently large odd integer $N$ is a sum of three primes. The number of representations is: $r_3(N) = \sum_{\substack{p_1 + p_2 + p_3 = N}} (\log p_1)(\log p_2)(\log p_3) = \frac{1}{2}\mathfrak{S}(N) N^2 + O(N^2/(\log N)^A)$ for any $A > 0$, where $\mathfrak{S}(N) = \prod_{p|N}\frac{(p-1)^2 - 1}{(p-1)^3}\prod_{p\nmid N}\left(1 + \frac{1}{(p-1)^3}\right) > 0$ for odd $N > 5$.

The binary Goldbach conjecture: every even $N > 2$ is a sum of two primes. The circle method gives: $r_2(N) = \sum_{p_1+p_2=N} \log p_1 \log p_2 = \mathfrak{S}_2(N) N + E(N)$ where the singular series $\mathfrak{S}_2(N) = 2\prod_{p>2}\frac{p(p-2)}{(p-1)^2}\prod_{\substack{p|N \\ p>2}}\frac{p-1}{p-2} > 0$ for even $N$. The error term $E(N)$ is controlled by exceptional zeros of Dirichlet L-functions.