Bull. Korean Math. Soc. 49 (2012), No. 2, pp. 367–372 http://dx.doi.org/10.4134/BKMS.2012.49.2.367 ON THE ERROR TERM IN THE PRIME GEODESIC THEOREM Muharem Avdispahi´ c and Dˇ zenan Guˇ si´ c 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 Γ be a discrete co-finite torsion free subgroup of G = SO 0 (d, 1) satisfying the condition Γ ∩ P =Γ ∩ N (P ) for P ∈ P Γ , where P Γ is the set of Γ- conjugacy classes of Γ-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 Γ =Γ\G/K is a d-dimensional real hyperbolic manifold with cusps. As usual, let π Γ (x) be the number of prime geodesics C γ of length l γ ≤ log x on X Γ . Recall that prime geodesic C γ corresponds to the conjugacy class γ of a primitive hyperbolic element with the norm N (γ )= e l γ . The purpose of this short note is to prove that Park’s refinement of the prime geodesic theorem, due to Gangolli [5] and DeGeorge [3] in the compact case and to Gangolli-Warner [6] in the finite volume case, can be further improved to obtain the theorem in the following form. Theorem 1. Let X Γ be as above. Then π Γ (x)= ∑ 3 2 d 0 <s n (k)≤2d 0 (-1) k li ( x sn(k) ) + O ( x 3 2 d0 (log x) -1 ) as x → +∞, where d 0 = d-1 2 , (s n (k) - k) (2d 0 - k - s n (k)) is a small eigen- value in [ 0, 3 4 d 2 0 ] of ∆ k on π σ k ,λ n (k) with s n (k)= d 0 + iλ n (k) or s n (k)= d 0 - iλ n (k) in ( 3 2 d 0 , 2d 0 ] , ∆ k is the Laplacian acting on the space of k-forms over X Γ and π σ k ,λ n (k) is the principal series representation. Proof. Let Γ h (resp. PΓ h ) denote the set of the Γ-conjugacy classes of hy- perbolic (resp. primitive hyperbolic) elements in Γ. Set Λ(γ )= l γ0 , where Received November 5, 2010; Revised April 14, 2011. 2010 Mathematics Subject Classification. 11M36, 11F72. Key words and phrases. Ruelle zeta function, prime geodesic theorem. This work has been partially supported by a grant from the Federal Ministry of Science and Education of Bosnia and Herzegovina. c ⃝2012 The Korean Mathematical Society 367