Piatetski-Shapiro meets Chebotarev
Volume 167 / 2015
Acta Arithmetica 167 (2015), 301-325
MSC: Primary 11L07; Secondary 11R45, 11B83.
DOI: 10.4064/aa167-4-1
Abstract
Let $K$ be a finite Galois extension of the field ${\mathbb Q}$ of rational numbers. We prove an asymptotic formula for the number of Piatetski-Shapiro primes not exceeding a given quantity for which the associated Frobenius class of automorphisms coincides with any given conjugacy class in the Galois group of $K/{\mathbb Q}$. In particular, this shows that there are infinitely many Piatetski-Shapiro primes of the form $a^2 + n b^2$ for any given natural number $n$.
