论文标题
立方$ l $的不变 - 功能
Non-vanishing for cubic $L$--functions
论文作者
论文摘要
我们证明，与$ \ mathbb {f} _q [t] $相关的$ l $ functions的$ l $ functions比例不存在，而在关键点$ s = 1/2 $上不会消失。这是通过使用作者在立方$ l $ functions的第一刻进行的技术来计算第一个动荡时刻的方法，并在莱斯特·拉兹维尔（Lester-Radziwill）的工作基础上获得了第二个动荡时刻的尖锐上限，从而从莱斯特·拉兹维尔（Lester-Radziwill）的作品中构建了这一点，从而从soundararajan，harper and harper and radziwill and radziwill and radziwill and radziwill soundararajan and of Sounder Idea又有了进一步的想法。当$ q \ equiv 2 \ pmod {3} $时，我们在非kummer设置中工作，但是当$ q \ equiv 1 \ equiv 1 \ equiv 1 \ pmod {3} $以及数字字段案例时，我们的结果可以转换为Kummer设置（假设广义Riemann假设）。我们的非逐渐播种的正比例是显式的，但非常小，这是因为在第二瞬间的上限上的隐含常数非常大。
We prove that there is a positive proportion of $L$-functions associated to cubic characters over $\mathbb{F}_q[T]$ that do not vanish at the critical point $s=1/2$. This is achieved by computing the first mollified moment using techniques previously developed by the authors in their work on the first moment of cubic $L$-functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Lester-Radziwill, which in turn develops further ideas from the work of Soundararajan, Harper, and Radziwill-Soundararajan. We work in the non-Kummer setting when $q \equiv 2\pmod{3}$, but our results could be translated into the Kummer setting when $q\equiv 1\pmod{3}$ as well as into the number field case (assuming the Generalized Riemann Hypothesis). Our positive proportion of non-vanishing is explicit, but extremely small, due to the fact that the implied constant in the upper bound for the mollified second moment is very large.
