Average number of lattice points in a disk

Sujay  Jayakar; Robert S. Strichartz

doi:10.3934/cpaa.2016.15.1

Article Contents

2016, Volume 15, Issue 1: 1-8. Doi: 10.3934/cpaa.2016.15.1

This issue Previous Article Next Article Average error for spectral asymptotics on surfaces

Average number of lattice points in a disk

Sujay Jayakar^1, and
Robert S. Strichartz^2,

1.
3633 19th St., San Francisco, CA 94110
2.
Mathematics Department, Malott Hall, Cornell Univeristy, Ithaca, NY 14853

Received: November 2014

Revised: October 2015

Published: January 2016

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Abstract

The difference between the number of lattice points in a disk of radius $\sqrt{t}/2\pi$ and the area of the disk $t/4\pi$ is equal to the error in the Weyl asymptotic estimate for the eigenvalue counting function of the Laplacian on the standard flat torus. We give a sharp asymptotic expression for the average value of the difference over the interval $0 \leq t \leq R$. We obtain similar results for families of ellipses. We also obtain relations to the eigenvalue counting function for the Klein bottle and projective plane.

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Mathematics Subject Classification: Primary: 35J05; Secondary: 42B99.

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References

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