Katz and Sarnak conjectured that the statistics of low-lying zeros of various family of $L$-functions matched with the scaling limit of eigenvalues from the random matrix theory. In this paper we confirm this statistic for a family of primitive Dirichlet $L$-functions matches up with corresponding statistic in the random unitary ensemble, in a range that includes the off-diagonal contribution. To estimate the $n$-level density of zeros of the $L$-functions, we use the asymptotic large sieve method developed by Conrey, Iwaniec and Soundararajan. For the random matrix side, a formula from Conrey and Snaith allows us to solve the matchup problem.

中文翻译：

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原始狄利克雷 L 函数的低位零点的 n 级密度

Katz 和 Sarnak 推测，各种 $L$ 函数族的低位零点的统计量与随机矩阵理论中特征值的标度限制相匹配。在本文中，我们确认了原始 Dirichlet $L$ 函数族的这个统计量与随机酉系综中的相应统计量相匹配，在一个包括非对角线贡献的范围内。为了估计 $L$ 函数的 $n$ 级零点密度，我们使用由 Conrey、Iwaniec 和 Soundararajan 开发的渐近大筛法。对于随机矩阵方面，Conrey 和 Snaith 的公式允许我们解决匹配问题。