Average Number of Lattice Points in a Disk

2012/06/07 by Jayakar, Sujay, Strichartz, Robert S.
#35J05 #42B99 (Primary) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1206.1613

Abstract

The difference between the number of lattice points in a disk of radius √(t)/2π and the area of the disk t/4π 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 ≤ t ≤ 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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