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It is well known that there are infinitely many primes of the form $a^2+b^2$ (namely all primes congruent to $1$ modulo $4$). On the other hand, Euler raised the problem as to whether there are infinitely many primes of the form $a^2+1$, which is still open (a positive answer is a special case of Bunyakovsky's conjecture). I am wondering whether there are known results in between these cases.

To be precise, I am interested in the following problem: Are there infinitely many primes $p$ of the form $a^2+b^2$ such that $b$ is bounded by a (slowly growing) function of $p$, for example $b = o(\sqrt{p})$?

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Hecke showed that there are infinitely many primes of the form $p=a^2+b^2$ with $a = o(\sqrt{p})$. Ankeny improved this to $o(\log p)$, conditional on the Extended Riemann Hypothesis. Harman and Lewis obtain $o(p^{0.119})$ unconditionally. (Disclaimer: I copied the Hecke reference from Ankeny's bibliography without checking it.) The latter two papers cite other related work.

I'd like to point out that I am not an expert here -- this was the result of googling

"Gaussian prime" angle

and then chasing references through MathSciNet and ResearchGate.

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