A Prime Geodesic Theorem of Gallagher Type for Riemann Surfaces

Abstract

We consider a cofinite Fuchsian group of the first kind with finitely many inequivalent parabolic elements and a unitary multiplier system of an arbitrary weight on it. Through the Gallagher–Koyama approach to the prime geodesic theorem on the corresponding noncompact hyperbolic surface, we reduce the exponent in the error term from $$\frac{3}{4}$$ to $$\frac{7}{10}$$ outside a set of finite logarithmic measure. Recent advances in results of the latter type and the methods applied are briefly discussed.