Divisor problem in special sets of Gaussian integers

Abstract

Let $A_{1}$ and $A_{2}$ be fixed sets of gaussian integers. We denote by $\tau_{A_{1}, A_{2}}(\omega)$ the number of representations of $\omega$ in form $\omega=\alpha\beta$, where $\alpha \in A_{1}, \beta \in A_{2}$. We construct the asymptotical formula for summatory function $\tau_{A_{1}, A_{2}}(\omega)$ in case, when $\omega$ lie in the arithmetic progression, $A_{1}$ is a fixed sector of complex plane,  $A_{2}=\mathbb{Z}[i]$.