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.