Lattice Points in Planar Convex Domains

We investigate the number of lattice points in planar convex domains. We give estimates of the remainder in the asymptotic representation with numerical constants, which are astonishingly small. We consider convex planar domains whose boundary has nonvanishing curvature throughout. Here the curvature of the curve of boundary plays an important role. Further, we consider the number of lattice points in domains which are bounded by superellipses. These curves have isolated points with curvature zero.