Geodesic
Asymptotic formula for the number of points of a lattice in the circle on the Lobachevsky plane
Diskretnaya Matematika, Tome 18 (2006) no. 4, pp. 9-17 
G. I. Arkhipov; V. N. Chubarikov. Asymptotic formula for the number of points of a lattice in the circle on the Lobachevsky plane. Diskretnaya Matematika, Tome 18 (2006) no. 4, pp. 9-17. http://geodesic.mathdoc.fr/item/DM_2006_18_4_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

We define the distance $d=d(z,z')$ between points $z=x+iy$ and $z'=x'+iy'$ in the upper half-plane, setting $$ d=\ln\biggl(\frac{u+2+\sqrt{u^2+4u}}2\biggr), $$ where $$ u=\frac{|z-z'|^2}{yy'}\,. $$ The circle $K(z_0,T)$ with centre in a point $z_0$ consists of the points $z$ satisfying the inequality $d(z,z_0)\leq T$. Let $N(z_0,T)$ be the number of elements $\gamma$ of the modular group $\mathit{PSL}_2(\mathbf Z)$ such that the point $\gamma z_0$ lies in the circle $K(z_0,T)$. In this paper, we refine the remainder term in the asymptotic formula for $N(z_0,T)$.

[1] Arkhipov G. I., Chubarikov V. N., “O chisle tochek reshetki v trekhmernom prostranstve Lobachevskogo”, Vestnik Moskovskogo un-ta. Ser. 1, Matematika. Mekhanika, 2 (1994), 24–27 | MR | Zbl

[2] Arkhipov G. I., Chubarikov V. N., “Otsenka snizu pervogo sobstvennogo znacheniya operatora Beltrami–Laplasa v trekhmernom prostranstve”, Trudy seminara im. I. G. Petrovskogo, 22 (2002), 22–36

[3] Arkhipov G. I., Sadovnichii V. A., Chubarikov V. N., Lektsii po matematicheskomu analizu, Drofa, Moskva, 2004

[4] Vinogradov I. M., Elementy vysshei matematiki, Vysshaya shkola, Moskva, 1999

[5] Vinogradov I. M., Osobye varianty metoda trigonometricheskikh summ, Nauka, Moskva, 1976 | MR

[6] Grunevald F., Mennike I., Elstrodt Yu., Gruppy, deistvuyuschie na giperbolicheskom prostranstve, MTsNMO, Moskva, 2003

[7] Delsarte J., “Sur le gitter fuchsien”, C. R. Acad. Sci. Paris, 214 (1942), 147–149 | MR | Zbl

[8] Delsarte J., “Le gitter fuchsien”, Oeuvres de Jean Delsarte II, Paris, 1971, 829–845

[9] Graham S. W., Kolesnik G., Van der Corput's method of exponential sums, Cambridge Univ. Press, Cambridge, 1991 | MR

[10] Huber H., “Über eine neue Klasse automorphe Funktionen und eine Gitterpunktproblem in der hyperbolischen Ebene”, Comment. Math. Helv., 30 (1956), 20–62 | DOI | MR

[11] Huber H., “Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen. I”, Math. Ann., 138 (1959), 1–26 | DOI | MR | Zbl

[12] Huber H., “Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen. II”, Math. Ann., 142 (1961), 385–398 | DOI | MR | Zbl

[13] Huber H., “Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen. II”, Naehtrag zu Math. Annalen 142, 385–398 (1961), Math. Ann., 143 (1961), 463–464 | DOI | MR | Zbl

[14] Huxley N. M., Area, lattice points, and exponential sums, Oxford Univ. Press, Oxford, 1996 | MR | Zbl

[15] Lax P. D., Phillips R. S., “The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces”, J. Funct. Anal., 46:3 (1982), 280–350 | DOI | MR | Zbl

[16] Levitan B. M., “Asimptoticheskie formuly dlya chisla tochek reshetki v prostranstvakh Evklida i Lobachevskogo”, Uspekhi matem. nauk, 42:3(255) (1987), 13–38 | MR | Zbl

[17] Sugi M., “Tselye tochki v oblastyakh na ploskosti Lobachevskogo”, Tr. MIAN, 207, 1994, 299–325 | Zbl

[18] Khooli K., Primeneniya metodov resheta v teorii chisel, Nauka, Moskva, 1987 | MR