Distribution of lattice points in the shifted balls

Ilgar  Jabbarov; Jeyhun  Abdullayev

doi:10.59400/jam2152

Distribution of lattice points in the shifted balls

Ilgar Jabbarov Department of Informatics and Algebra, Ganja State University, Ganja 429, Azerbaijan
Jeyhun Abdullayev Department of Mathematical Analysis, Ganja State University, Ganja 429, Azerbaijan

Article ID: 2152

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DOI: https://doi.org/10.59400/jam2152

Keywords: lattice points; shifted ball; Fourier series; trigonometric integral; eigenvalue

Abstract

In this work we study the mean value of the difference between the number of integer points and the volume of a ball as a function of the center of a ball in the unit cube [0, 1]³, applying new method. This mean value is estimated by its possible exact value. Using methods of Fourier analysis, we lead the question to the estimates of double trigonometric integrals. This method allows consider the question on lattice points in domains of arbitrary nature without any symmetry.

References

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[2]Виноградов ИМ. Special variants of the trigonometric sum method (Russian). Москва: Наука; 1976.

[3]Chen JR. Improvement on the asymptotic formulas for the lattice points in a region of the three dimensions (II). Sci. Sinica; 1963. 751-764.9.

[4]Chamizo F, Iwaniec H. On the Sphere Problem. Revista Matemática Iberoamericana. 1995; 11(2): 417-429. doi: 10.4171/rmi/178

[5]Heath-Brown DR. Lattice points in the sphere. Number Theory in Progress; 1999.

[6]Arkhipova LG. Number of lattice points in a ball. Vestnik Moskov. Univ. Ser. 1. Mat. Mekh; 2008.

[7]Tsang KM. Counting Lattice Points in The Sphere. Bulletin of the London Mathematical Society. 2000; 32(6): 679-688. doi: 10.1112/s0024609300007505

[8]VinogradovА I, SkriganovM M. The number of lattice points inside the ball with variable center, analytic number theory and the theory of functions, 2, Zap. Nan cˇ n. Sem. Leningrad Otdal. Mat. Inst. Steklov (LOMI); 1979.

[9]Bleher P, Bourgain J. Distribution of the error term for the number of lattice points inside a shifted ball. Analytic Number Theory; 1995.

[10]JabbarovI Sh, AslanovaN Sh, JeferliE. On the Number of Lattice Points in the Shifted Circles. AZJM; 2020.

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[13]Джаббаров ИШ. On evaluations of trigonometric integrals (Russian). Чебышевский сборник; 2010

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Published

2024-12-30

How to Cite

Jabbarov, I., & Abdullayev, J. (2024). Distribution of lattice points in the shifted balls. Journal of AppliedMath, 2(6), 2152. https://doi.org/10.59400/jam2152

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