Effective bounds for Huber's constant and Faltings's delta function

Abstract

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} .