论文标题
在Cuspidal Newforms的中心界限消失
Bounding Vanishing at the Central Point of Cuspidal Newforms
论文作者
论文摘要
Katz-sarnak密度猜想指出，$ L $ - 功能的家族的零是随机矩阵集团的特征值良好的。对于适当限制的测试功能，该对应关系可以使家庭在中心角度消失的秩序产生上限。我们将先前的结果概括为$ n $ \ textsuperscript {th}零件的居中时刻，以允许任意测试功能。在计算方面，我们使用改进的公式在消失的Cuspidal Newforms的范围内获得了明显的更好的界限，从而为边界质量设定了世界记录。我们还发现了更好的测试功能，可以进一步优化我们的界限。我们最初看到的改善就可以提高到$ 5 $ \ textsuperscript {th}顺序，并且随着排名的增长，我们的界限迅速提高（排名第10级的数量级好，排名第50位的数量级好了，而排名第50位的数量级以上超过四个数量级）。
The Katz-Sarnak Density Conjecture states that zeros of families of $L$-functions are well-modeled by eigenvalues of random matrix ensembles. For suitably restricted test functions, this correspondence yields upper bounds for the families' order of vanishing at the central point. We generalize previous results on the $n$\textsuperscript{th} centered moment of the distribution of zeros to allow arbitrary test functions. On the computational side, we use our improved formulas to obtain significantly better bounds on the order of vanishing for cuspidal newforms, setting world records for the quality of the bounds. We also discover better test functions that further optimize our bounds. We see improvement as early as the $5$\textsuperscript{th} order, and our bounds improve rapidly as the rank grows (more than one order of magnitude better for rank 10 and more than four orders of magnitude for rank 50).
