An additive divisor problem in $\mathbb{Z}[i]$

Abstract

Let $\tau(\alpha)$ be the number of divisors of the Gaussian integer $\alpha$. An asymptotic formula for the summatory function $\sum\limits_{N(\alpha)\leq x}\tau(\alpha)\tau(\alpha+\beta)$ is obtained under the condition $N(\beta)\leq x^{3/8}$. This is a generalization of the well-known additive divisor problem for the natural numbers.