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RESEARCH ARTICLE

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

O. V. Savasrtu^a, P. D. Varbanets<sup>b

a</sup> ul. Dvoryanskaya 2, Dept. of computer algebra and discrete mathematics, Odessa national university, Odessa 65026, Ukraine
^b ul. Solnechnaya. 7/9 apt.. 18, Odessa. 65009 Ukraine

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.

Keywords: additive divisor problem; asymptotic formula.

MSC: 11N37, 11R42

Received: 22.02.2003

Language: English

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