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 
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/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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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