Using the Ratios Conjecture, we write down precise formulas with lower order terms for the one and the two level densities of zeros of quadratic Dirichlet $L$--functions over function fields. We denote the various terms arising as Type-$0$, Type-I and Type-II contributions. When the support of the Fourier transform of the test function is sufficiently restricted, we rigorously compute the Type-$0$ and Type-I terms and confirm that they match the conjectured answer. When the restrictions on the support are relaxed, our results suggest that Type-II contributions become important in the two level density.

中文翻译：

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I 型对函数域上二次狄利克雷 L 函数的一级和二级密度的贡献

使用比率猜想，我们为函数域上的二次狄利克雷 $L$ 函数的一阶和二阶零点密度写下了具有低阶项的精确公式。我们将各种术语表示为 Type-$0$、Type-I 和 Type-II 贡献。当测试函数的傅立叶变换的支持被充分限制时，我们严格计算 Type-$0$ 和 Type-I 项并确认它们与推测的答案匹配。当对支持的限制放宽时，我们的结果表明 II 型贡献在两级密度中变得重要。