Gaussian almost primes in almost all narrow sectors

Abstract

We show that almost all sectors of the disc $\{z\in \mathbb{C}:|z{|}^2\le X\}$ of area $(\log X{)}^{15.1}$ contain products of exactly two Gaussian primes, and that almost all sectors of area $(\log X{)}^{1+\epsilon }$ 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.

53 pages; referee comments incorporated; to appear in Rev. Mat. Iberoam