Under a previously studied condition on the argument of the Selberg zeta function on the critical line, we reach the critical exponent \(\frac{1}{2}\) in the error term of the prime geodesic theorem for the modular group PSL(2,\( \mathbb {Z}\)) outside a set of finite logarithmic measure. We also prove a conditional prime geodesic theorem of Hejhal’s type in this setting without the latter exclusion.

中文翻译：

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关于PSL（2，$$ \ mathbb {Z} $$ Z）的von Koch定理

在先前研究的关于临界线上Selberg zeta函数的自变量的条件下，对于模块化组PSL（）的本初测地定理的误差项，我们达到了临界指数\（\ frac {1} {2} \）。 2，\（\ mathbb {Z} \））在一组有限对数度量之外。我们还证明了在这种情况下Hejhal类型的条件素测短线定理，而没有排除后者。