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 $$\frac{3}{4}$$ 3 4 to $$\frac{7}{10}$$ 7 10 outside a set of finite logarithmic measure. Recent advances in results of the latter type and the methods applied are briefly discussed.

中文翻译：

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黎曼曲面的加拉格尔型质数测地线定理

我们考虑具有有限多个不等价抛物线元素和任意权重的酉乘法系统的第一类余有限 Fuchsian 群。通过对相应非紧双曲曲面上的质数测地线定理的 Gallagher-Koyama 方法，我们将误差项中的指数从 $$\frac{3}{4}$$ 3 4 减少到 $$\frac{7}{7}{ 10}$$ 7 10 一组有限对数测度之外。简要讨论了后一种类型的结果和所应用的方法的最新进展。