Primes of the form $p^2+n{q}^2$

Abstract

Suppose that $n$ is $0$ or $4$ modulo $6$. We show that there are infinitely many primes of the form $p^2+n{q}^2$ with both $p$ and $q$ prime, and obtain an asymptotic for their number. In particular, when $n=4$ we verify the `Gaussian primes conjecture' of Friedlander and Iwaniec. We study the problem using the method of Type I/II sums in the number field $\mathbf{Q}(\sqrt{-n})$. The main innovation is in the treatment of the Type II sums, where we make heavy use of two recent developments in the theory of Gowers norms in additive combinatorics: quantitative versions of so-called concatenation theorems, due to Kuca and to Kuca--Kravitz-Leng, and the quasipolynomial inverse theorem of Leng, Sah and the second author.

arXiv comment59 pages, accepted for publication in Acta Math. v3 corrects an error with a partial summation argument in Section 8.3 and a minor inaccuracy in Section 8.7