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Prime geodesic theorem via the explicit formula of $\Psi $ for hyperbolic 3-manifolds
We obtain a lower bound for the error term of the prime geodesic theorem for hyperbolic 3-manifolds.Our result is Ω ± (x(log log x) 1/3 / log x).We also generalize Sarnak's upper bound O(x (5/3)+ε ) to compact manifolds.