Average Number of Lattice Points in a Disk

The difference between the number of lattice points in a disk of radius $\sqrt{t}\mathrm{/}2\pi$ and the area of the disk $t\mathrm{/}4\pi$ 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 $0\le t\le R$. We obtain similar results for families of ellipses. We also obtain relations to the eigenvalue counting function for the Klein bottle and projective plane.