ON THE ERROR TERM IN THE PRIME GEODESIC THEOREM

Abstract. Taking the integrated Chebyshev-type counting function of the appropriate order, we improve the error term in Park's prime geodesic theorem for hyperbolic manifolds with cusps. The obtained estimate co-incides with the best known result in the Riemann surfaces case. Let be a discrete co-nite torsion free subgroup of G = SO0 (d; 1) satisfying the condition \ P = \ N(P) for P 2 P, where P is the set of -conjugacy classes of -cuspidal parabolic subgroups in G and N(P) is the unipotent part of P. Denote by K a maximal compact subgroup of G. The manifold X = nG=K is a d-dimensional real hyperbolic manifold with cusps. As usual, let (x) be the number of prime geodesics C of length l log x on X. Recall that prime geodesic C corresponds to the conjugacy class of a primitive hyperbolic element with the norm N ( ) = el . The purpose of this short note is to prove that Park's renement of the prime geodesic theorem, due to Gangolli [5] and DeGeorge [3] in the compact case and to Gangolli-Warner [6] in the nite volume case, can be further improved to obtain the theorem in the following form. Theorem 1. Let X be as above. Then (x) =