By a closer inspection of the Friedman-Jorgenson-Kramer algorithm related to the prime geodesic theorem on cofinite Fuchsian groups of the first kind, we refine the constants therein. The newly obtained effective upper bound for Huber’s constant is in the modular surface case approximately 74000 74000 -times smaller than the previously claimed one. The degree of reduction in the case of an upper bound for Faltings’s delta function ranges from 10 8 10^{8} to 10 16 10^{16} .

We correct the exponent in the error term of the prime geodesic theorem for hyperbolic 3‐manifolds and in Park's theorem for higher dimensions [ , ].

Taking the Iwaniec explicit formula as a starting point, we give a short proof of a more precise $\frac{2}{3}$ bound for the exponent in the error term of the Gallagher-type prime geodesic theorem for the modular surface.

Through the Selberg zeta approach, we reduce the exponent in the error term of the prime geodesic theorem for cocompact Kleinian groups or Bianchi groups from Sarnak's 53 to 32 . At the cost of excluding a set of finite logarithmic measure, the bound is further improved to 139 .

Taking the Iwaniec explicit formula as a starting point, we give a short proof of a more precise 2 3 bound for the exponent in the error term of the Gallagher-type prime geodesic theorem for the modular surface.

Under the generalized Lindel\"{o}f hypothesis, the exponent in the error term of the prime geodesic theorem for the modular surface is reduced to $\frac{5}{8}+\varepsilon $ outside a set of finite logarithmic measure.