Geodesic
An additive divisor problem in $\mathbb{Z}[i]$
Algebra and discrete mathematics, no. 1 (2003), pp. 103-110

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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.
Keywords: additive divisor problem; asymptotic formula. 
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O. V. Savasrtu; P. D. Varbanets. An additive divisor problem in $\mathbb{Z}[i]$. Algebra and discrete mathematics, no. 1 (2003), pp. 103-110. http://geodesic.mathdoc.fr/item/ADM_2003_1_a9/