AI 中文总结
研究三次和四次大筛法的非正交性，利用久保田狄利克雷级数二阶矩平均林德勒夫上界及亚纯θ函数兰金 - 塞尔伯格卷积傅里叶系数平均下界，采用扎吉尔正则化方法，还对特定赫克特征族提出猜想。
AI 中文摘要
我们无条件地证明了三次和四次大筛法并非完全正交。完全正交性的主要障碍来自高斯和所表现出的偏差。我们的证明需要两个主要输入：久保田狄利克雷级数二阶矩的平均林德勒夫上界，以及某个亚纯θ函数的兰金 - 塞尔伯格卷积的傅里叶系数的紧密平均下界。在四次情形中，后一个输入尤为重要，因为关于亚纯θ函数的傅里叶系数所知甚少。为确立这两个输入，我们采用了扎吉尔（1981年）提出的兰金 - 塞尔伯格正则化方法。除了本文所考虑的三次和四次情形，我们预计在数域\(K \supset \mathbb{Q}(\zeta_n)\)上每个固定阶\(n \geq 3\)的赫克特征族也并非完全正交。我们为每个\(n\)的这些集合的算子范数提供了一个精确的猜想。
英文摘要
We show unconditionally that the cubic and quartic large sieves are not perfectly orthogonal. The main obstruction to perfect orthogonality comes from the bias exhibited by Gauss sums. Our proof requires two main inputs: a Lindelöf-on-average upper bound for the second moment of Kubota's Dirichlet series, and a tight average lower bound for the Fourier coefficients of a certain Rankin-Selberg convolution of metaplectic theta functions. The latter input is particularly important in the quartic case, where much less is known about Fourier coefficients of metaplectic theta functions. To establish both of these inputs, we adapt a Rankin-Selberg regularization method due to Zagier (1981). In addition to the cubic and quartic cases considered in this paper, we expect that the family of Hecke characters of each fixed order $n \geq 3$ over a number field $K \supset \mathbb{Q}(ζ_n)$ is not perfectly orthogonal. We provide a precise conjecture for the operator norm of these ensembles for each $n$.
Comments41 pages
