Average Number of Lattice Points in a Disk

TL;DR

This paper provides a precise asymptotic analysis of the average difference between lattice point counts and disk areas, linking it to eigenvalue estimates on flat surfaces and extending results to ellipses, Klein bottle, and projective plane.

Contribution

It introduces a sharp asymptotic formula for the average lattice point discrepancy in disks and ellipses, connecting geometric counting problems with spectral eigenvalue estimates.

Findings

Derived a sharp asymptotic expression for the average lattice point difference.

Extended results to families of ellipses and related spectral problems.

Established relations to eigenvalue counting functions for Klein bottle and projective plane.

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.