  Let $\Gamma$ be a cofinite Fuchsian group acting on hyperbolic two-space
$\HH.$ Let $M=\Gamma \setminus \HH $ be the corresponding quotient space. For
$\gamma,$ a closed geodesic of $M$, let $l(\gamma)$ denote its length. The
prime geodesic counting function $\pi_{M}(u)$ is defined as the number of
$\Gamma$-inconjugate, primitive, closed geodesics $\gamma $ such that
$e^{l(\gamma)} \leq u.$ The \emph{prime geodesic theorem} implies:
  $$\pi_{M}(u)=\sum_{0 \leq \lambda_{M,j} \leq 1/4} \text{li}(u^{s_{M,j}}) +
O_{M}(\frac{u^{3/4}}{\log{u}}), $$ where $0=\lambda_{M,0} < \lambda_{M,1} <...$
are the eigenvalues of the hyperbolic Laplacian acting on the space of smooth
functions on $M$ and $s_{M,j} = \frac{1}{2}+\sqrt{\frac{1}{4} - \lambda_{M,j}}.
$ Let $C_{M}$ be smallest implied constant so that $$|\pi_{M}(u)-\sum_{0 \leq
\lambda_{M,j} \leq 1/4} \text{li}(u^{s_{M,j}})|\leq
C_{M}\frac{u^{3/4}}{\log{u}} \quad \text{\text{for all} $u > 1.$}$$
  We call the (absolute) constant $C_{M}$ the Huber constant. The objective of
this paper is to give an effectively computable upper bound of $C_{M}$ for an
arbitrary cofinite Fuchsian group. As a corollary we estimate the Huber
constant for $\PSL(2,\ZZ),$ we obtain $C_{M} \leq 16,607,349,020,658 \approx
\exp(30.44086643)$.
