Zur Gitterpunktlehre der euklidischen Ebene

AbstractOn lattice points in the Euclidean planeFor a large parameter T, let D(T) denote a domain in the (x,y)-plane bounded by a smooth (closed) Jordan curve C(T) which is defined by φ(x/T,y/T) = 0 where φ(u, v) is an analytic function with grad φc ≠ (0, 0) on C(1). Denote further by A(T) the number of lattice points (of the unit lattice Z2) in D(T) and by V(T) its area and put P(T) =A(T)- V(T). Then it had been proved by J.G. Van der Corput [5] that P(T)= O(Tθ) with θ<23, provided that the curvature of C(T) vanishes nowhere, and recently by Y. Colin de Verdière [3] that P(T) = O(T1-1/n) if the curvature of C(T) has only zeroes of order ≦n-2 (n≧3). In this paper, under the hypothesis that C(T) has rational slope in each point with curvature 0, the contribution of these points to P(T) is evaluated explicitely up to an error term O(Tθ) with θ