We use B. Randol’s method to improve the error term in the prime geodesic theorem for a noncompact Riemann surface having at least one cusp. The case considered is a general one, corresponding to a Fuchsian group of the first kind and a multiplier system with a weight on it.

We prove that the meromorphic continuations of the Ruelle and Selberg zeta functions considered by Bunke and Olbrich are of finite order not larger than the dimension of the underlaying compact, odd-dimensional, locally symmetric space.

Abstract We derive approximate formulas for the logarithmic derivative of the Selberg and the Ruelle zeta functions over compact, even-dimensional, locally symmetric spaces of real rank one. The obtained formulas are given in terms of zeta singularities.

For compact, even-dimensional, locally symmetric spaces, we obtain precise estimates on the number of singularities of Selberg's and Ruelle's zeta functions considered by U. Bunke and M. Olbrich.

In this paper we pay our particular attention to the error term in the prime geodesic theorem for compact symmetric spaces represented as double coset spaces of the special linear group of order four over real numbers. It is known that in the case of compact locally symmetric Riemannian manifolds of strictly negative sectional curvature, the corresponding error term depends on classification of Riemannian symmetric spaces of real rank one. In particular, the error term is a function depending on the dimension of the underlying locally symmetric space. In this research we prove that the error term in the case at hand is a function depending on the degree of the polynomial that appears in the functional equation of the corresponding Selberg zeta function.