AI 中文总结
研究大亏格随机双曲曲面上同调类中闭测地线分布，通过将曲面视为模空间随机点，研究闭测地线加权计数函数波动，得出大亏格极限下不同模的方差情况，与胡利猜想对比并给出差异解释。
AI 中文摘要
我们研究大亏格随机双曲曲面上同调类中闭测地线的分布。将曲面视为配备韦伊 - 彼得森概率测度的模空间中的随机点，我们研究模\(q\)下同调类中闭测地线加权计数函数的波动。我们表明，在大亏格极限下，对于每个\(q>2\)的模，方差渐近于\(X\log X\)，当\(q = 2\)时有一个特殊的因子\(2\)。这与算术级数中素数的胡利猜想形成对比，后者方差预期为\(X\log q\)。通过将我们的结果与有限域上函数域的相应理论进行比较，我们给出了这种差异的一种解释。
英文摘要
We study the distribution of closed geodesics in homology classes on random hyperbolic surfaces of large genus. Viewing the surface as a random point in moduli space equipped with the Weil--Petersson probability measure, we investigate the fluctuations of the weighted counting function of closed geodesics in homology classes modulo $q$. We show that, in the large genus limit, the variance is asymptotic to $X\log X$ for every modulus $q>2$, with an exceptional factor of two when $q=2$. We relate our findings to a conjecture of Hooley, and to work of Friedlander and Goldston, on primes in arithmetic progressions. We also introduce a statistical model based on the random allocation of weighted balls which exhibits analogous behavior.
CommentsRevision, introduced a statistical model based on the random allocation of weighted balls
