Geodesic
Multiplicative characteristics of function for the number of classes of primitive hyperbolic elements in the group $\Gamma_0(N)$ by level~$N$
Dalʹnevostočnyj matematičeskij žurnal, Tome 9 (2009) no. 1, pp. 48-73.

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An arithmetical forms of Selberg's trace formula and Selberg's zeta-function for the congruence subgroup $\Gamma_0(N)$, explicit expression for the number of classes of primitive hyperbolic elements in the congruence subgroup level $N$ in terms of the number of classes of primitive elements in the congruence subgroup level $N_1=N/P^i$, $(N,N_1)=1$ and sharp upper bound of the number classes by level $N$ are obtained. 
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V. V. Golovchanskii; M. N. Smotrov. Multiplicative characteristics of function for  the number of classes of primitive hyperbolic elements in the group $\Gamma_0(N)$ by level~$N$. Dalʹnevostočnyj matematičeskij žurnal, Tome 9 (2009) no. 1, pp. 48-73. http://geodesic.mathdoc.fr/item/DVMG_2009_9_1_a5/

[1] V. V. Golovchanskii, M. N. Smotrov, “Yavnaya formula chisla klassov primitivnykh giperbolicheskikh elementov gruppy $\Gamma_0(N)$”, Matem. sb., 199:7 (2008), 63–84 | DOI | MR

[2] N. V. Kuznetsov, “Raspredelenie norm primitivnykh giperbolicheskikh klassov modulyarnoi gruppy i asimptoticheskie formuly dlya sobstvennykh znachenii operatora Laplasa – Beltrami na fundamentalnoi oblasti modulyarnoi gruppy”, DAN, 242:1 (1978), 40–43 | MR | Zbl

[3] M. Peter, “The correlation between multiplicities of closed geodesics on the modular surface”, Comm. Math. Phys., 225 (2002), 171–189 | DOI | MR | Zbl

[4] V. Lukianov, A central limit theorem for congruence subgroups of the modular group, Ph. D. Thesis, Tel Aviv Univ., 2005

[5] T. Arakawa, S. Koyama, M. Nakasuji, “Arithmetic foms of Selberg zeta functioms with applications to prime geodesic theorem”, Proc. Japan Acad., 78:A (2002), 120–125 | DOI | MR | Zbl

[6] Y. Hashimoto, “Arithmetic expressions of Selberg's zeta functions for congruence subgroups”, J. Number Theory, 122 (2007), 324–335 | DOI | MR | Zbl

[7] W. Luo, Z. Rudnick, P. Sarnak, “On Selberg's eigenvalue conjecture”, Geom. Funct. Anal., 5:2 (1995), 387–401 | DOI | MR | Zbl