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A 1997 paper of Étienne Fouvry and Henryk Iwaniec, Gaussian primes, concerns the prevalence of primes that are of the form $n^2+p^2$ for prime $p$. The asymptotic result is

$$\sum_{n^2+p^2\le x}\Lambda(p)\Lambda(n^2+p^2)=kx+O(x(\log x)^{-A})$$

with $A>0$ arbitrary and $$k=2\prod_{p>2}\left(1-\frac{\chi(p)}{(p-1)(p-\chi(p))}\right)\approx2.1564103447695$$ where $\chi(n)=(-1)^{(p-1)/2}$ is the nontrivial character mod 4. The big-O constant is uniform, depending only on the choice of $A$.

I would like to use this to find an asymptotic formula for $f(x):=|\mathcal{P}\cap\{n^2+p^2\le x\}|$. It looks like

$$f(x)=2k\frac{x}{(\log x)^2}(1+o(1))$$

but I'm not quite sure of my derivation, nor even of how to interpret the original result (are duplicate representations double-counted or not?). Can someone confirm or deny my calculation?

Bonus question: were Fouvry & Iwaniec the first to show that there are infinitely many of these primes? They cite Rieger, Coleman, Duke, and Pomykala as related results but none had both prime restrictions.

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If $p$ and $n^2+p^2$ are prime in the first sum, why bother with using the vonmangoldt function? You seem to have written their result down incorrecty – Ethan Feb 25 '13 at 7:21
@Ethan: In the sum $p$ is prime but $n^2+p^2$ need not be. See (1.5) in the paper I linked to, noting that $\ell$ is constrained to be prime. – Charles Feb 26 '13 at 0:11
$\Lambda(l)\Lambda(n^2+l^2)=0$, unless $l$ and $n^2+l^2$ are both prime powers, I don't see where it says $l$ is a prime. – Ethan Feb 26 '13 at 4:21
@Ethan: Last paragraph of page 249. – Charles Feb 26 '13 at 16:33
They are proving there are an infinite number of primes where $l$ is a prime, that doesn't imply that l is a prime in the latter sum. – Ethan Feb 26 '13 at 22:16
up vote 2 down vote accepted

Concerning the bonus question, the review in Math Reviews says the authors prove this result, and the reviewer doesn't mention anyone else having done it. I would take this as evidence that Fouvry and Iwaniec were the first.

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