The distribution of prime ideals of imaginary quadratic fields

Abstract

Let Q ( x , y ) Q(x, y) be a primitive positive definite quadratic form with integer coefficients. Then, for all ( s , t ) ∈ R 2 (s, t)\in \mathbb R^2 there exist ( m , n ) ∈ Z 2 (m, n) \in \mathbb Z^2 such that Q ( m , n ) Q(m, n) is prime and \[ Q ( m − s , n − t ) ≪ Q ( s , t ) 0.53 + 1. Q(m - s, n - t) \ll Q(s, t)^{0.53} + 1. \] This is deduced from another result giving an estimate for the number of prime ideals in an ideal class of an imaginary quadratic number field that fall in a given sector and whose norm lies in a short interval.