Advertisement

manuscripta mathematica

, Volume 39, Issue 2–3, pp 277–285 | Cite as

Gitterpunkte in Lemniskatenscheiben

  • Werner Georg Nowak
Article

Abstract

Let G (R) denote the number of lattice points (ξ,η)ε **Z2 in the domain B(R) bounded by the lemniscate (x2+y2)2=2R2 (x2−y2) and put P(R)=G(R)-R2(R2 being the area of B(R)). The purpose of this paper is to determine the order of magnitude of P(R) for large R by proving the asymptotic relation (3). An analogous result is given for the “eight curve” x4=R2 (x2−y2).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literatur

  1. [1]
    FRICKER, F.: Einführung in die Gitterpunktlehre. Basel-Boston-Stuttgart: Birkhäuser 1982CrossRefzbMATHGoogle Scholar
  2. [2]
    KRÄTZEL, E.: Eine Verallgemeinerung des Kreisproblems. Arch.Math.18, 181–187(1967)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    KRÄTZEL, E.: Bemerkungen zu einem Gitterpunktsproblem. Math.Ann.179, 90–96(1969)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    LAWRENCE, J.D.: A catalog of special plane curves. New York: Dover Publ. 1972zbMATHGoogle Scholar
  5. [5]
    NOWAK, W.G.: A non-convex generalization of the circle problem. J. reine angew.Math.314, 136–145(1980)zbMATHGoogle Scholar
  6. [6]
    POPOV, V.N.: The number of integer points under a parabola. Math.Notes18, 1007–1010 (1975)CrossRefzbMATHGoogle Scholar
  7. [7]
    VAN DER CORPUT, J.G.: Neue zahlentheoretische Abschätzungen. Math.Ann.89, 215–254 (1923)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    WILTON, J.R.: An extended form of Dirichlet's divisor problem. Proc. London Math.Soc., II. Ser.36, 391–426 (1933)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Werner Georg Nowak
    • 1
  1. 1.Institut für Mathematik und angewandte Statistik der Universität für BodenkulturWienÖsterreich

Personalised recommendations