Let \(\gamma\) be a prime closed geodesic on a compact hyperbolic surface of genus \(g\) uniformized by \(\Gamma1\). Then, for any \(\eta\in T(f)\) and \(\delta>0\), we have \[ \#\{\gamma: l(\gamma)\leq T,\;[\gamma]= \alpha,\;|r_m(f,\gamma)-\eta|\leq \delta\}\sim C\frac{e^T} {T^{{\mathfrak g}+2}}, \] where \(C>0\) is a constant independent of \(\alpha\) and \(\eta\). The most important special cases of this result are when \(T(f)\) is a lattice or when \(T(f)= \mathbb{C}\). In these cases we can make the slightly more precise statements below. In each case \(\beta_f: \mathbb{R}^{2{\mathfrak g}+2}\to \mathbb{R}\) is a certain ``thermodynamic'' function (depending only on \(f\)). Special cases. (1) If \(T(f)\) is a lattice in \(\mathbb{C}\) then, for any \(\eta\in T(f)\), we have \[ \#\{\gamma: l(\gamma)\leq T,\;[\gamma]= \alpha,\;r_m(f,\gamma)= \eta\}\sim \frac{|\mathbb{C}/T(f)|} {(2\pi)^{{\mathfrak g}+1} \sqrt{\det \nabla^2\beta_f(0)}} \frac{e^T} {T^{{\mathfrak g}+2}}, \] where \(|\mathbb{C}/ T(f)|\) denotes the area of a fundamental domain for \(T(f)\). (2) If \(T(f)= \mathbb{C}\) then, for any \(\eta\in \mathbb{C}\) and \(\delta>0\), we have \[ \#\{\gamma: l(\gamma)\leq T,\;[\gamma]=\alpha,\;|r_m(f,\gamma)-\eta|\leq \delta\}\sim \frac{\pi\delta^2} {(2\pi)^{{\mathfrak g}+1} \sqrt{\det\nabla^2 \beta_f(0)}} \frac{e^T} {T^{{\mathfrak g}+2}}. \] Corollary. For any \(\delta>0\), we have \[ \lim_{T\to\infty} \frac{1}{\pi(T,\alpha)} \#\{\gamma: l(\gamma)\leq T,\;[\gamma]= \alpha,\;|r_m(f,\gamma)- \varepsilon(f)|\leq \delta\}=0, \] i.e., the closed geodesics with period close to \(\varepsilon(f)\) have zero density in \(\{\gamma: [\gamma]= \alpha\}\). Remark. The restriction to a fixed homology class \(\alpha\) in (*) is crucial, even though the result is independent of \(\alpha\). Without this restriction, the averages vanish''.