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Primitive Points in Rational Polygons

Part of:Additive number theory; partitionsDiscrete geometryGeometry of numbers

Published online by Cambridge University Press:  30 January 2020

Imre Bárány,
Greg Martin,
Eric Naslund and
Sinai Robins
Imre Bárány
Affiliation:
Rényi Institute of Mathematics, Hungarian Academy of Sciences, H-1364 Budapest, Pf. 127, HungaryDepartment of Mathematics, University College London, Gower Street, London, WC1E 6BT, England Email: barany@renyi.hu
Greg Martin
Affiliation:
Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada Email: gerg@math.ubc.canaslund.eric@gmail.com
Eric Naslund
Affiliation:
Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada Email: gerg@math.ubc.canaslund.eric@gmail.com
Sinai Robins
Affiliation:
Instituto de Mathematica e Estatistica, Universidade de São Paulo, 05508-090São Paulo, Brazil Email: srobins@ime.usp.br
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Abstract

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Let ${\mathcal{A}}$ be a star-shaped polygon in the plane, with rational vertices, containing the origin. The number of primitive lattice points in the dilate $t{\mathcal{A}}$ is asymptotically $\frac{6}{\unicode[STIX]{x1D70B}^{2}}\text{Area}(t{\mathcal{A}})$ as $t\rightarrow \infty$. We show that the error term is both $\unicode[STIX]{x1D6FA}_{\pm }(t\sqrt{\log \log t})$ and $O(t(\log t)^{2/3}(\log \log t)^{4/3})$. Both bounds extend (to the above class of polygons) known results for the isosceles right triangle, which appear in the literature as bounds for the error term in the summatory function for Euler’s $\unicode[STIX]{x1D719}(n)$.

Keywords

Primitive points in polygonsvisible pointsEuler’s Totient functionError termrational polygons

MSC classification

Primary:11H06: Lattices and convex bodies11P21: Lattice points in specified regions
Secondary:52C05: Lattices and convex bodies in $2$ dimensions

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 4 , December 2020, pp. 850 - 870
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Copyright
© Canadian Mathematical Society 2020

Footnotes

The first author was partially supported by ERC Advanced Research Grant no. 267165 (DISCONV) and by Hungarian National Science Grant K 111827. The second author was supported in part by a Natural Sciences and Engineering Research Council of Canada Discovery Grant. The fourth author was partially supported by ICERM, the Institute for Computational and Experimental Research in Mathematics, Brown University, and would like to thank the warm hospitality of the first author and the Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences.

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