The author builds on the work of L. Guillopé and M. Zworski [Ann. Math. (2) 145, 597–660 (1997; Zbl 0898.58054) and Geom. Funct. Anal. 9, 1156–1168 (1999; Zbl 0947.58022)] concerning the wave trace formula and the bounds on the number of resonances. He proves a prime orbit theorem for an arbitrary Riemann surface of finite geometry and infinite volume. That is, let \(m = \Gamma\setminus H^2\) be a geometrically finite Riemann surface such that \(\delta>{1\over 2}\) and \(f\geq 1\), then as \(T\to+\infty\), \[ N(T)= \text{li}(e^{\delta T})+ \sum^p_{k-1} \text{li}(e^{\alpha_kT})+ O(e^{(\delta/2+ 1/4)T}), \] where the exponents \(1/2< \alpha \leq\alpha_{p-1}\leq\cdots < \alpha_0=\delta\) are related to the point spectrum of \(\Delta_M\) included in \((0,1/4)\) by the formula \(\alpha_k(1- \alpha_k)=\lambda_k\).