Geodesic
Primitive Points in Rational Polygons
Canadian mathematical bulletin, Tome 63 (2020) no. 4, pp. 850-870

Voir la notice de l'article provenant de la source Cambridge University Press

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 $\unicode[STIX]{x1D719}(n)$.
DOI : 10.4153/S0008439520000090
Mots-clés : Primitive points in polygons, visible points, Euler’s Totient function, Error term, rational polygons 
Bárány, Imre; Martin, Greg; Naslund, Eric; Robins, Sinai. Primitive Points in Rational Polygons. Canadian mathematical bulletin, Tome 63 (2020) no. 4, pp. 850-870. doi: 10.4153/S0008439520000090
@article{10_4153_S0008439520000090,
     author = {B\'ar\'any, Imre and Martin, Greg and Naslund, Eric and Robins, Sinai},
     title = {Primitive {Points} in {Rational} {Polygons}},
     journal = {Canadian mathematical bulletin},
     pages = {850--870},
     year = {2020},
     volume = {63},
     number = {4},
     doi = {10.4153/S0008439520000090},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439520000090/}
}
TY  - JOUR
AU  - Bárány, Imre
AU  - Martin, Greg
AU  - Naslund, Eric
AU  - Robins, Sinai
TI  - Primitive Points in Rational Polygons
JO  - Canadian mathematical bulletin
PY  - 2020
SP  - 850
EP  - 870
VL  - 63
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/S0008439520000090/
DO  - 10.4153/S0008439520000090
ID  - 10_4153_S0008439520000090
ER  - 
%0 Journal Article
%A Bárány, Imre
%A Martin, Greg
%A Naslund, Eric
%A Robins, Sinai
%T Primitive Points in Rational Polygons
%J Canadian mathematical bulletin
%D 2020
%P 850-870
%V 63
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/S0008439520000090/
%R 10.4153/S0008439520000090
%F 10_4153_S0008439520000090

[1] Apostol, T. M., Introduction to analytic number theory. Undergraduate Texts in Mathematics, Springer-Verlag, New York–Heidelberg, 1976. Google Scholar

[2] Baker, R. C., Primitive lattice points in planar domains. Acta Arith. 142(2010), 3, 267–302. https://doi.org/10.4064/aa142-3-4 Google Scholar | DOI

[3] Davenport, H., Multiplicative number theory, Third Ed., Graduate Texts in Mathematics, 74, Springer-Verlag, New York, 2000. Google Scholar

[4] 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-1951-043-3 Google Scholar | DOI

[5] 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

[6] Jameson, G. J. O., The prime number theorem. London Mathematical Society Student Texts, 53, Cambridge University Press, Cambridge, 2003. https://doi.org/10.1017/CBO9781139164986 Google Scholar | DOI

[7] 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)00013-1 Google Scholar | DOI

[8] Le Quang, N. and Robins, S., Macdonald’s solid-angle sum for real dilations of rational polygons. Preprint. . Google Scholar

[9] Mertens, F., Über einige asymptotische Gesetze der Zahlentheorie. J. Reine Angew. Math. 77(1874), 289–338. https://doi.org/10.1515/crll.1874.77.289 Google Scholar

[10] 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/BF02837826 Google Scholar | DOI

[11] 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

[12] 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

[13] 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-1-4-305-311 Google Scholar | DOI

[14] 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/s1-5.2.95 Google Scholar | DOI

[15] 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.1851.42.348 Google Scholar

[16] Rogers, C. A., Existence theorems in the geometry of numbers. Ann. of Math. 48(1947), 994–1002. https://doi.org/10.2307/1969390 Google Scholar | DOI

[17] Walfisz, A., Weylsche Exponentialsummen in der neueren Zahlentheorie. Mathematische Forschungsberichte, XV, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963. Google Scholar

Cité par Sources :