Let $\mathcal{P}$ denote the set of primes. Define the function $r_2(N)$ to be the number of ways to write $N$ as a sum of two not necessarily distinct primes (where order matters). Then the famous Goldbach Conjecture can be phrased as following:
$r_2(2N) > 0$ for all $N \geq 1$.
A related function, $r_3(N)$ which is the number of ways $N$ can be written as the sum of three not necessarily distinct primes (order matters), is used in the following classic result by I.M. Vinogradov:
Vinogradov's Theorem: Define $\mathfrak{S}(N) = \displaystyle \sum_{q=1}^\infty \frac{\mu(q)c_q(N)}{\varphi(q)^3}$ where $\mu$ is the Mobius function, $c_q$ is the Ramanujun sum, and $\varphi$ the Euler totient function. Then for odd $N$, we have:
$r_3(N) = \displaystyle \mathfrak{S}(N) \frac{N^2}{2(\log(N))^3}\left(1 + O\left(\frac{\log \log(N)}{\log(N)}\right)\right)$.
This clearly shows that $r_3(N) > 0$ for all sufficiently large odd $N$, which is to say that all sufficiently large positive odd integers $N$ can be written as the sum of three primes. Goldbach's Conjecture would be proved if we can find some arithmetic function $f(N)$ which tends to infinity such that $r_2(N) >> f(N)$.
It has been confirmed that the Goldbach Conjecture holds for the first $10^{18}$ even integers (according to Wikipedia) by exhaustive computer search. But my question is, is there anything known about $r_2(N)$ for these cases? In particular I am asking whether or not there are any examples where $r_2(N)$ is unexpectedly small. If this happens infinitely often, for instance, then attempting to prove Goldbach's Conjecture by finding a lower bound that tends to infinity would be infeasible.
Another formulation would be, is there any heuristic evidence to suggest that there exist a positive integer $M$ such that for infinitely positive integers $N$ we have $r_2(N) \leq M$?
Best Answer
I think the heuristic evidence suggests quite the opposite, that $r_2(N)$ increases without bound. See http://www.ieeta.pt/~tos/goldbach.html for both theoretical background and computational results.