An Ω-estimate for the lattice rest of a convex planar domain

Abstract

SynopsisLet D be a compact convex planar domain containing the origin, the boundary of which is of class C∞ and has finite non-vanishing curvature throughout. For the number A(i) of lattice points in the “blown up” domain √tD, the estimateis established. This is a generalization of Hardy's classical result for the circle problem. The proof is based on asymptotic formulae for certain exponential integrals due to E. Hlawka.