COUNTING LATTICE POINTS IN THE SPHERE

KAI-MAN TSANG

doi:10.1112/S0024609300007505

Abstract

We consider the error term P3(R) = #{n ∈ ℤ3 :|n| [les ] R} − 4|3πR3 which occurs in the counting of lattice points in a sphere of radius R. By considering second and third power moments, we prove that P3(R) = Ω±(R√log R). An upper bound for the gap between the sign changes of P3(R) is also proved.

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Lau, Y.-K. and Tsang, K.-M. 2002. Large values of error terms of a class of arithmetical functions. Journal für die reine und angewandte Mathematik (Crelles Journal), Vol. 2002, Issue. 544,

Jakobson, Dmitry and Polterovich, Iosif 2005. Lower bounds for the spectral function and for the remainder in local Weyl’s law on manifolds. Electronic Research Announcements of the American Mathematical Society, Vol. 11, Issue. 9, p. 71.

Chamizo, Fernando Cristóbal, Elena and Ubis, Adrián 2007. Visible lattice points in the sphere. Journal of Number Theory, Vol. 126, Issue. 2, p. 200.

Chamizo, Fernando and Cristóbal, Elena 2012. The sphere problem and the L-functions. Acta Mathematica Hungarica, Vol. 135, Issue. 1-2, p. 97.

Narkiewicz, Władysław 2012. Rational Number Theory in the 20th Century. p. 13.

Meyer, Annika 2013. On the number of lattice points in a small sphere and a recursive lattice decoding algorithm. Designs, Codes and Cryptography, Vol. 66, Issue. 1-3, p. 375.

Bera, Sahadev Bhowmick, Partha and Bhattacharya, Bhargab B. 2014. Applied Algorithms. Vol. 8321, Issue. , p. 49.

Dabo-Niang, Sophie Hamdad, Leila Ternynck, Camille and Yao, Anne-Françoise 2014. A kernel spatial density estimation allowing for the analysis of spatial clustering. Application to Monsoon Asia Drought Atlas data. Stochastic Environmental Research and Risk Assessment, Vol. 28, Issue. 8, p. 2075.

Rosakis, Phoebus 2014. Continuum surface energy from a lattice model. Networks & Heterogeneous Media, Vol. 9, Issue. 3, p. 453.

Nowak, Werner Georg 2014. Topics in Mathematical Analysis and Applications. Vol. 94, Issue. , p. 583.

Dabo-Niang, Sophie Ternynck, Camille and Yao, Anne-Françoise 2016. Nonparametric prediction of spatial multivariate data. Journal of Nonparametric Statistics, Vol. 28, Issue. 2, p. 428.

Biswas, Ranita and Bhowmick, Partha 2016. From prima quadraginta octant to lattice sphere through primitive integer operations. Theoretical Computer Science, Vol. 624, Issue. , p. 56.

Bera, Sahadev Bhowmick, Partha and Bhattacharya, Bhargab B. 2016. On the Characterization of Absentee-Voxels in a Spherical Surface and Volume of Revolution in $${\mathbb Z}^3$$ Z 3. Journal of Mathematical Imaging and Vision, Vol. 56, Issue. 3, p. 535.

Biswas, Ranita Bhowmick, Partha and Brimkov, Valentin E. 2017. On the polyhedra of graceful spheres and circular geodesics. Discrete Applied Mathematics, Vol. 216, Issue. , p. 362.

Torquato, Salvatore 2018. Hyperuniform states of matter. Physics Reports, Vol. 745, Issue. , p. 1.

Askri, Aymen Othman, Ghaya Rekaya-Ben and Ghauch, Hadi 2019. Counting Lattice Points in the Sphere using Deep Neural Networks. p. 2053.

Fu, Bin Gu, Pengfei and Zhao, Yuming 2021. Approximate set union via approximate randomization. Theoretical Computer Science, Vol. 890, Issue. , p. 210.

Dabo‐Niang, Sophie Ternynck, Camille Thiam, Baba and Yao, Anne‐Françoise 2022. Geostatistical Functional Data Analysis. p. 175.

Surajkanta, Yumnam and Pal, Shyamosree 2023. Recognition of spherical segments using number theoretic properties of isothetic covers. Multimedia Tools and Applications, Vol. 82, Issue. 13, p. 19393.

Berend, Daniel Kumar, Rishi and Pollington, Andrew 2023. Simultaneous visibility in the integer lattice. Journal of Number Theory, Vol. 250, Issue. , p. 84.

Article contents

COUNTING LATTICE POINTS IN THE SPHERE

Abstract

Access options

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Article contents

COUNTING LATTICE POINTS IN THE SPHERE

Abstract

Access options

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests