Question
This is a "spin-off" question of: Reformulation of Goldbach's Conjecture as optimization problem correct?
I was wondering if a function existed such that:
$$ G(x)^2 = (\sum_{i=1}^\infty x^{b_i})^2 =\sum_{m=3}^\infty A_m x^{2m}$$
Where $A_m$ is an integer $\neq 0$ and $b_i$ is an arbitrary natural number.
And grows less quicker than:
$$ \lim_{x \to 1^-} (\sum_{i=1}^\infty x^{b_i})^2 < \frac{1}{((1-x)\ln(1-x))^2} $$
Background
We know $\sum x^{p_i} \sim \frac{1}{(x-1)\ln(1-x)} $
as $ x \nearrow 1 $ and $p_i$ is the $i$'th prime
as explained in the last page of http://mathstat.dal.ca/~antoniov/notes/boundaryasymp.pdf
And if Goldbach's conjecture is true:
$$ (\sum_{i=1}^\infty x^{p_i})^2 =\sum_{m=3}^\infty A_m x^{2m}$$
Where $A_m$ is an integer $\neq 0$
So I was wondering if there were another set of numbers $b_i$ that could satisfy Goldbach's conjecture but grow slower than $\sum x^{p_i}$