I am currently studying analytic number theory and my teacher suggested to ask here if the following sum
$S = \sum_{p} x^p = x^2 + x^3 + x^5 + x^7 + ...$
Where $p$ is a prime number is known and if it has an asymptotic behaviour. My motivation behind it relies on the following. If you square the sum
$S^2 = (\sum_{p} x^p)^2$
You get
$S^2 = \sum_{p,q} a_{p+q}x^{p+q}$
Where $p$ and $q$ are primes and $a_{p+q}$ is an integer.
It is not hard to see that Goldbach's conjecture is equivalent to claim that the $2k^{th}$ derivative of $S^2$ at $x=0$ is different than $0$, for all positive integer $k$.
My questions are the following:
Does $S$ have an asymptotic behavior?
Have there been attempts of proving Goldbach's conjecture trying to apply the circle method to $S^2$?
Thank you
For Question 1, Since every primes $\ge 5$ are of the form $6k \pm 1$, by summing up the geometric sequences $x^{6k-1} + x^{6k+1}$ for $k = 1,2,\ldots, \infty$ and adding $x^{2} + x^{3}$, and taking advantage of the fact the the density of primes among the first few numbers of these form is high we get
$$ \sum_{p \ge 2} x^p = \frac{1 + x + x^3 + x^5 -x^6 - x^7}{x^4(x - x^7)} + O(x^{25}) $$