Lattice points in a circle: An improved mean-square asymptotics
We investigate the number of lattice points in special three-dimensional convex bodies. They are called convex bodies of pseudo revolution, because we have in one special case a body of revolution and in another case even a super sphere. These bodies have lines at the boundary, where all points have Gaussian curvature zero. We consider the influence of these points to the lattice rest in the asymptotic representation of the number of lattice points.
In this article we consider the number of lattice points in -dimensional super spheres with even power . We give an asymptotic expansion of the -fold anti-derivative of for sufficiently large . From this we deduce a new estimation for the error term in the asymptotic representation of for .
Some interesting lattices can be constructed using association schemes. We illustrate this by a unimodular lattice without roots of dimension 28 which admits as its automorphism group.