Geodesic
Norm of Gaussian integers in arithmetical progressions and narrow sectors
Algebra and discrete mathematics, Tome 29 (2020) no. 2, pp. 259-270.

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We proved the equidistribution of the Gaussian integer numbers in narrow sectors of the circle of radius $x^{\frac{1}{2}}$, $x\to\infty$, with the norms belonging to arithmetic progression $N(\alpha)\equiv\ell\pmod{q}$ with the common difference of an arithmetic progression $q$, $q\ll{x}^{\frac{2}{3}-\varepsilon}$.
Keywords: Gaussian integers, norm groups, Hecke $Z$-function, functional equation. 
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S. Varbanets; Ya. Vorobyov. Norm of Gaussian integers in arithmetical progressions and narrow sectors. Algebra and discrete mathematics, Tome 29 (2020) no. 2, pp. 259-270. http://geodesic.mathdoc.fr/item/ADM_2020_29_2_a10/

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