Abstract :
In [P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982) 229–247], it was proved that the Selberg zeta function for image is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of image. As applications, we study the Brun–Titchmarsh type prime geodesic theorem and the asymptotic formula of the sum of the class number.
