AUTHORS: Dzenan Gusic
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In this paper we investigate the Selberg zeta functions and the Ruelle zeta functions associated with locally homogeneous bundles over compact locally symmetric spaces of rank one. Our basic object will be a compact locally symmetric Riemannian manifold with negative sectional curvature. In particular, our research will be restricted to compact, odd-dimensional, real hyperbolic spaces. For this class of spaces, the Titchmarsh-Landau style approximate formulas for the logarithmic derivative of the aforementioned zeta functions are derived. As expected in this setting, the obtained formulas are given in terms of zeros of the attached Selberg zeta functions. Our results follow from the fact that these zeta functions can be represented as quotients of two entire functions of order not larger than the dimension of the underlying compact, odd-dimensional, locally symmetric space, and the application of suitably chosen Weyl asymptotic law. The obtained formulas can be further applied in the proof of the corresponding prime geodesic theorem.
KEYWORDS: Zeta functions of Selberg and Ruelle, logarithmic derivative, locally symmetric spaces, approximate formulas
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