Maass形式对称平方L函数在短区间中的矩与非消失性

中文摘要 AI 辅助

最近，Li获得了Maass形式对称平方$L$-函数中心值在长度为$H \ge T^{18/19+\epsilon}$的短区间$(T-H, T+H)$上的三次矩的平均Lindelöf估计。我们改进了这一结果，证明该估计在$H \gg T^{6/7+\epsilon}$时成立。我们证明中的关键要素是Maass对称平方$L$-函数的第二扭曲矩的一个新的渐近公式。基于此公式，我们还改进了短区间中非消失中心$L$-值比例的较低界。此前，即使在Dirichlet $L$-函数的Lindelöf假设下，长度为$H = T^{\beta}$的区间中非消失值的比例也仅已知至少为$\frac{3\beta-1}{4}$。我们建立了无条件下界$\frac{7\beta-2}{8}$。

英文摘要

Recently, Li obtained a mean Lindelöf estimate for the cubic moment of the central values of Maass form symmetric square $L$-functions over short intervals $(T-H, T+H)$ of length $H \ge T^{18/19+ε}$. We improve this result by showing that the estimate holds for $H \gg T^{6/7+ε}$. The key ingredient in our proof is a new asymptotic formula for the second twisted moment of Maass symmetric square $L$-functions. Based on this formula, we also improve the lower bound for the proportion of non-vanishing central $L$-values in short intervals. Previously, even under the assumption of the Lindelöf hypothesis for Dirichlet $L$-functions, the proportion of non-vanishing values in intervals of length $H = T^β$ was only known to be at least $\frac{3β-1}{4}$. We establish an unconditional lower bound of $\frac{7β-2}{8}$.

发表机构

• Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）
• HSE University（高等经济大学）

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