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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{ADM_2020_29_2_a10,
     author = {S. Varbanets and Ya. Vorobyov},
     title = {Norm of {Gaussian} integers in arithmetical progressions and narrow sectors},
     journal = {Algebra and discrete mathematics},
     pages = {259--270},
     publisher = {mathdoc},
     volume = {29},
     number = {2},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2020_29_2_a10/}
}
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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/