16
$\begingroup$

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.

  • 0
    @Charles Judging from (1.4) I don't see think $l$ is meant to always be prime. However, (1.3) seems so general that one could just as easily restrict $l$ to be prime by setting $\lambda_l$ to be the prime indicator function. Also, the amount of duplicate representation should be minimal since the sum of squares decomposition is unique for primes, so only something like $p^2+2^2$ gets counted twice.2014-01-08

1 Answers 1

3

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.