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Primitive Points in Rational Polygons

Part of: Discrete geometry Additive number theory; partitions Geometry 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, Hungary Department 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

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 bodies 11P21: Lattice points in specified regions
Secondary: 52C05: Lattices and convex bodies in $2$ dimensions
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 4 , December 2020 , pp. 850 - 870
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Copyright
© Canadian Mathematical Society 2020

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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.

References

Apostol, T. M., Introduction to analytic number theory. Undergraduate Texts in Mathematics, Springer-Verlag, New York–Heidelberg, 1976.Google Scholar
Baker, R. C., Primitive lattice points in planar domains. Acta Arith. 142(2010), 3, 267–302. https://doi.org/10.4064/aa142-3-4CrossRefGoogle Scholar
Davenport, H., Multiplicative number theory, Third Ed., Graduate Texts in Mathematics, 74, Springer-Verlag, New York, 2000.Google Scholar
Erdős, P. and Shapiro, H. N., On the changes of sign of a certain error function. Canad. J. Math. 3(1951), 375–385. https://doi.org/10.4153/cjm-1951-043-3CrossRefGoogle Scholar
Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, Fifth ed., The Clarendon Press, Oxford University Press, 1979.Google Scholar
Jameson, G. J. O., The prime number theorem. London Mathematical Society Student Texts, 53, Cambridge University Press, Cambridge, 2003. https://doi.org/10.1017/CBO9781139164986CrossRefGoogle Scholar
Kranakis, E. and Pocchiola, M., Counting problems relating to a theorem of Dirichlet. Comput. Geom. 4(1994), 309–325. https://doi.org/10.1016/0925-7721(94)00013-1CrossRefGoogle Scholar
Le Quang, N. and Robins, S., Macdonald’s solid-angle sum for real dilations of rational polygons. Preprint. arxiv:1602.02681v1.Google Scholar
Mertens, F., Über einige asymptotische Gesetze der Zahlentheorie. J. Reine Angew. Math. 77(1874), 289–338. https://doi.org/10.1515/crll.1874.77.289Google Scholar
Montgomery, H. L., Fluctuations in the mean of Euler’s phi function. Proc. Indian Acad. Sci. Math. Sci. 97(1987), 1–3, 239–245. https://doi.org/10.1007/BF02837826CrossRefGoogle Scholar
Montgomery, H. L. and Vaughan, R. C., Multiplicative number theory. I. Classical theory. Cambridge Studies in Advanced Mathematics, 97, Cambridge University Press, Cambridge, 2007.Google Scholar
Niven, I., Zuckerman, H. S., and Montgomery, H. L., An introduction to the theory of numbers, Fifth ed., John Wiley & Sons, Inc., New York, 1991.Google Scholar
Nosarzewska, M., Évaluation de la différence entre l’aire d’une rǵion plane convexe et le nombre des points aux coordonnées entières couverts par elle. Colloq. Math. 1(1948), 305–311. https://doi.org/10.4064/cm-1-4-305-311CrossRefGoogle Scholar
Pillai, S. S. and Chowla, S. D., On the error terms in some asymptotic formulae in the theory of numbers (1). J. Lond. Math. Soc. 5(1930), 2, 95–101. https://doi.org/10.1112/jlms/s1-5.2.95CrossRefGoogle Scholar
Raabe, J. L., Zurückführung einiger Summen und bestimmten Integrale auf die Jacob Bernoullische Function. J. Reine Angew. Math. 42(1851), 348–376. https://doi.org/10.1515/crll.1851.42.348Google Scholar
Rogers, C. A., Existence theorems in the geometry of numbers. Ann. of Math. 48(1947), 994–1002. https://doi.org/10.2307/1969390CrossRefGoogle Scholar
Walfisz, A., Weylsche Exponentialsummen in der neueren Zahlentheorie. Mathematische Forschungsberichte, XV, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963.Google Scholar