Gaussian integers with small prime factors

Source abstract

Let ψ G ( x t , x ) denote the number of Gaussian integers with norm not exceeding x 2 t whose Gaussian prime factors have norm not exceeding x 2 . Previous estimates have required restrictions on the parameter t with respect to x . The purpose of this note is to present asymptotic estimates for ψ G ( x t , x ) for all ranges of the parameter t with respect to x .