Prime geodesic theorem for higher-dimensional hyperbolic manifold

Abstract

For a ( d + 1 ) (d+1) -dimensional hyperbolic manifold M \mathcal {M} , we consider an estimate of the error term of the prime geodesic theorem. Put the fundamental group Γ \Gamma of M \mathcal {M} to be a discrete subgroup of S O e ( d + 1 , 1 ) 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 Ω ± ( x d / 2 ( log ⁡ log ⁡ x ) 1 / ( d + 1 ) log ⁡ x ) \Omega _{\pm }(\tfrac {x^{d/2}(\log \log x)^{1/(d+1)}}{\log x}) as x x goes to ∞ \infty .