nt.number theory - Primes of the form a^2+1

The fact that the Riemann zeta function $zeta(s)$ and its brethren have a pole at $s=1$ is responsible for the infinitude of large classes of primes (all primes, primes in arithmetic progression; primes represented by a quadratic form). We cannot hope proving the infinitude of primes $p = a^2+1$ in this way because the series $sum 1/p$, summed over these primes, converges. This implies that the corresponding Euler product

$$zeta_G(s)= prod_{p = a^2+1} frac1{1 - p^{-s}}$$
converges for $s = 1$. But if we could show that $zeta_G(s)$ has a pole at, say,
$s = frac12$, then the desired result would follow. Now I know that there are heuristics on the number of primes of the form $p = a^2+1$ below $x$ (by Hardy and Littlewood?)

Can these heuristics be explained by hypothetical properties of $zeta_G(s)$ (or a related Dirichlet series), or can the domain of convergence of $zeta_G(s)$ be derived from such asymptotics?

BTW, here's a little known conjecture by Goldbach on these primes: let $A$ be the set of all numbers $a$ for which $a^2+1$ is prime ($A =${1, 2, 4, 6, 10, $ldots$}). Then every
$a in A$ ($a > 1$) can be written in the form $a = b+c$ for $b, c in A$. I haven't seen this discussed anywhere.

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