Abstract.
Effective estimates for the lattice point discrepancy of certain planar and three-dimensional domains. This paper provides estimates, with explicit constants, for the lattice point discrepancy of o-symmetric ellipse discs and ellipsoids in ℝ3, as well as of three-dimensional convex bodies which are invariant under rotations around one coordinate axis and have a smooth boundary of finite nonzero Gaussian curvature throughout.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V Bentkus F Götze (1997) ArticleTitleOn the lattice point problem for ellipsoids. Acta Arith 80 101–125
S Bochner (1932) ArticleTitleDie Poisson’sche Summenformel in mehreren Veränderlichen. Math Ann 106 56–63 Occurrence Handle10.1007/BF01455876
F Chamizo (1998) ArticleTitleLattice points in bodies of revolution. Acta Arith 85 265–277
F Chamizo H Iwaniec (1995) ArticleTitleOn the sphere problem. Rev Mat Iberoamericana 11 417–429
Fricker F (1982) Einführung in die Gitterpunktlehre. Birkhäuser: Basel
Götze F (2004) Lattice point problems and values of quadratic forms. Preprint, Univ. Bielefeld
Gradshteyn IS, Ryzhik IM (1994) In: Jeffrey A (ed) Table of Integrals, Series, and Products, 5th ed. San Diego: Academic Press
Heath-Brown DR (1999) Lattice points in the sphere. In: Györy K et al. (eds) Number Theory in Progress, vol 2, 883–892. Berlin: de Gruyter
Hlawka E (1950) Über Integrale auf konvexen Körpern I. Monatsh Math 54: 1–36; II, ibid. 54: 81–99
M Huxley (2003) ArticleTitleExponential sums and lattice points III. Proc London Math Soc III. Ser 87 591–609 Occurrence Handle10.1112/S0024611503014485
Krätzel E (1988) Lattice points. Berlin: VEB Deutscher Verlag der Wissenschaften
Krätzel E (2000) Analytische Funktionen in der Zahlentheorie. Wiesbaden: Teubner
E Krätzel (2004) ArticleTitleLattice points in convex planar domains. Monatsh Math 143 145–162 Occurrence Handle10.1007/s00605-003-0146-y
W Müller (1999) ArticleTitleLattice points in large convex bodies. Monatsh Math 128 315–330 Occurrence Handle10.1007/s006050050066
Soft Warehouse, Inc. (1995) Derive, Version 3.11, Honolulu (Hawaii)
JG Van der Corput (1923) ArticleTitleZahlentheoretische Abschätzungen mit Anwendungen auf Gitterpunktprobleme. Math Z 17 250–259 Occurrence Handle10.1007/BF01504346
IM Vinogradov (1963) ArticleTitleOn the number of integer points in a sphere (Russian). Izv Akad Nauk SSSR Ser Mat 27 957–968
Walfisz A (1963) Weylsche Exponentialsummen in der neueren Zahlentheorie. Berlin: VEB
Author information
Authors and Affiliations
Additional information
Der zweitgenannte Autor denkt für die Unterstützung durch den Österreichischen Forschungs förderungs fonds (FWF), Projekt Nr. P18079-N12.
Rights and permissions
About this article
Cite this article
Krätzel, E., Nowak, W. Effektive Abschätzungen für den Gitterrest gewisser ebener und dreidimensionaler Bereiche. Mh Math 146, 21–35 (2005). https://doi.org/10.1007/s00605-004-0291-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-004-0291-y