Prime geodesic theorem via the explicit formula of \(\Psi\) for hyperbolic 3-manifolds
Let \(\Gamma< \text{PSL} (2,\mathbb{C})\) be a cofinite group acting on the upper half-space model \(\mathbb{H}\) of hyperbolic three-space, and let \(\pi_\Gamma(x)\) denote the number of associated prime geodesics of length at most \(\log x\), i.e., the number of primitive hyperbolic or loxodromic conjugacy classes \(\{P_0\}\) in \(\Gamma\) satisfying \(N(P_0)\leq x\) (where \(N(P_0)\) denotes the norm of \(P_0\)). Denoting by \(s_0=2\), \(s_1,\dots, s_M\) the \(s\)-values belonging to the small eigenvalues \(0= \lambda_0, \lambda_1,\dots, \lambda_M\leq 1\) of the Laplacian on \(\Gamma \setminus \mathbb{H}\) the function \(\pi_\Gamma\) satisfies an asymptotic law of the form \[ \pi_\Gamma(x)= \text{li}(x^2)+ \sum_{k=1}^M \text{li}(x^{s_k})+ R(x) \] where li denotes the logarithmic integral and where \(R(x)\) denotes an error term. When \(\Gamma\) is cocompact or when \(\Gamma\) is cofinite and satisfies an extra condition the author shows that \[ R(x)= \Omega_\pm \Biggl( \frac {x(\log\log x)^{1/3}} {\log x}\Biggr) \quad\text{as}\quad x\to \infty. \] The proof is based on the Selberg zeta-function and the usual techniques applied in the proof of the prime number theorem. In this way the author also proves: When \(\Gamma\) is cocompact or \(\Gamma= \text{PSL} (2,{\mathfrak o}_K)\) with \(K\) an imaginary quadratic number field, then \[ R(x)= O(x^{(5/3)+ \varepsilon}) \quad\text{as}\quad x\to \infty. \]
scientific article; zbMATH DE number 1800027