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

Published online by Cambridge University Press:  14 November 2011

Werner Georg Nowak
Werner Georg Nowak
Affiliation:
Institut für Mathematik der Universität für Bodenkultur, Gregor Mendel-Straße 33, A-1180 Wien, Austria
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Synopsis

Let 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 estimate

is 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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 100 , Issue 3-4 , 1985 , pp. 295 - 299
Copyright
Copyright © Royal Society of Edinburgh 1985

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References

1Bochner, S.. Die Poissonsche Summenformel in mehreren Veränderlichen. Math. Ann. 106 (1932), 56–63.CrossRefGoogle Scholar
2Bonnesen, T. and Fenchel, W.. Theorie der konvexen Körper (Berlin: Springer, 1934).Google Scholar
3Corput, J. G. van der. Neue zahlentheoretische Abschätzungen. Math. Ann. 89 (1923), 215–254.CrossRefGoogle Scholar
4Fricker, F.. Einführung in die Gitterpunktlehre (Basel: Birkhäuser, 1982).CrossRefGoogle Scholar
5Gangadharan, K. S.. Two classical lattice point problems. Proc. Cambridge Philos. Soc. 57 (1961), 699–721.CrossRefGoogle Scholar
6Hafner, J. L.. New omega theorems for two classical lattice point problems. Invent. Math. 63 (1981), 181–186.CrossRefGoogle Scholar
7Hardy, G. H.. On the Dirichlet's divisor problem. Proc. London Math. Soc. 15 (1916), 1–25.Google Scholar
8Hlawka, E.. Integrale auf konvexen Korpern I. Mönatsh. Math. 54 (1950), 1–36.CrossRefGoogle Scholar
9Hlawka, E.. Integrale auf konvexen Korpern II. Mönatsh. Math. 54 (1950), 81–99.CrossRefGoogle Scholar
10Ingham, A. E.. On two classical lattice point problems. Proc. Cambridge Philos. Soc. 36 (1940), 131–138.CrossRefGoogle Scholar
11Kátai, I. and Corradi, K.. Egy megiegyzés K. S. Gangadharan “Two classical lattice point problems” cimu dolgozata hoz (In Hungarian). MTA III Ostaly Közlemenyei 17 (1967), 89–97.Google Scholar
12Nowak, W. G.. Zur Gitterpunktlehre der euklidischen Ebene. Indag. Math. 46 (1984), 209–223.CrossRefGoogle Scholar
13Oberhettinger, F.. Tabellen zur Fourier Transformation (Berlin: Springer, 1957).CrossRefGoogle Scholar