RESEARCH ARTICLE

Norm of Gaussian integers in arithmetical progressions and narrow sectors

S. Varbanets^a, Ya. Vorobyov<sup>b

a</sup> Odessa I.I. Mechnikov National University, Dvoryanskaya str. 2, 65026 Odessa, Ukraine
^b Izmail State Humanities University, Izmail, Repina str. 12, 68610 Izmail, Ukraine

Abstract: 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.

MSC: 11L07, 11T23

Received: 20.01.2020

Language: English

DOI: 10.12958/adm1529

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