Gaussian almost primes in almost all narrow sectors

Abstract

We show that almost all sectors of the disc $\{z \in \mathbb{C}: |z|^2\leq X\}$ of area $(\log X)^{15.1}$ contain products of exactly two Gaussian primes, and that almost all sectors of area $(\log X)^{1 + \varepsilon}$ contain products of exactly three Gaussian primes. The argument is based on mean value theorems, large value estimates and pointwise bounds for Hecke character sums.