Gaussian integers with small prime factors (Q1256505)

Let \(\psi_G(x^t,x)\) be the number of Gaussian integers whose norm do not exceed \(x^{2t}\) but all of whose Gaussian prime factors have norms not exceeding \(x^2\). The author studies the function \(\psi_G(x^t,x)\)). The results carry over with modifications the famous results of N. G. De Bruijn to the Gaussian number field, i.e. the set of numbers \(a+ ib\) where \(a\) and \(b\) are rational with the usual structure of the four fundamental operations.