Prime geodesic theorem for higher-dimensional hyperbolic manifold

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 $\infty$.