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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     author = {V. V. Golovchanskii and M. N. Smotrov},
     title = {Multiplicative characteristics of function for  the number of classes of primitive hyperbolic elements in the group $\Gamma_0(N)$ by level~$N$},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {48--73},
     publisher = {mathdoc},
     volume = {9},
     number = {1},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DVMG_2009_9_1_a5/}
}
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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/