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Journal Article
Prime Geodesic Theorem for Higher-Dimensional Hyperbolic Manifold
Maki Nakasuji
Transactions of the American Mathematical Society
Vol. 358, No. 8 (Aug., 2006), pp. 3285-3303
Published
by: American Mathematical Society
https://www.jstor.org/stable/3845331
Page Count: 19
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Topics: Mathematical theorems, Lie groups, Laplacians, Mathematical functions, Eigenvalues, Prime numbers, Continuous spectra, Determinants
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Abstract
For a (d + 1)-dimensional hyperbolic manifold $\cal M$, we consider an estimate of the error term of the prime geodesic theorem. Put the fundamental group Γ of $\cal M$, 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 $\Omega_\pm (\frac{{x^{d/2}(log log x)^{1/(d + 1)}}{log x}})$ as x goes to ∞.
Transactions of the American Mathematical Society
© 2006 American Mathematical Society