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Prime geodesic theorem for higher-dimensional hyperbolic manifold

Author: Maki Nakasuji
Journal: Trans. Amer. Math. Soc. 358 (2006), 3285-3303
MSC (2000): Primary 11M36, 11F72
Published electronically: March 24, 2006
MathSciNet review: 2218976
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Abstract: For a $ (d+1)$-dimensional hyperbolic manifold $ \mathcal{M}$, we consider an estimate of the error term of the prime geodesic theorem. Put the fundamental group $ \Gamma$ of $ \mathcal{M}$ to be a discrete subgroup of $ SO_e(d+1, 1)$ with cofinite volume. When the contribution of the discrete spectrum of the Laplace-Beltrami operator is larger than that of the continuous spectrum in Weyl's law, we obtained a lower estimate $ \Omega_{\pm}(\tfrac{x^{d/2}(\log\log x)^{1/(d+1)}}{\log x})$ as $ x$ goes to $ \infty$.

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Additional Information

Maki Nakasuji
Affiliation: Department of Mathematics, Keio University, 3-14-1 Hiyoshi, 223-8522, Japan

Keywords: Prime geodesic theorem, Selberg zeta function, Weyl's law
Received by editor(s): April 22, 2003
Published electronically: March 24, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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