Prime geodesic theorem via the explicit formula of $\Psi $ for hyperbolic 3-manifolds

Abstract

We obtain a lower bound for the error term of the prime geodesic theorem for hyperbolic 3-manifolds. Our result is $\Omega_{\pm}(x(\log\log x)^{1/3} / \log x)$. We also generalize Sarnak's upper bound $O(x^{(5/3) + \varepsilon})$ to compact manifolds.