发表机构
The University of Nottingham; Boston College（诺丁汉大学; 波士顿学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究由模符号形成的克洛斯特曼和的和，利用陶伯型方法给出相关拉马努金和的和的估计，定义类似塞尔伯格zeta函数的函数并确定其解析延拓，证明和的抵消结论，解释与特征值问题联系并给出数值证据。
AI 中文摘要
我们研究由模符号形成的克洛斯特曼和的和。利用陶伯型方法，首先给出由模符号形成的拉马努金和的（里斯）和的估计。进一步定义一个类似于塞尔伯格zeta函数的zeta函数，确定其到\(\Re(s)>1/2\)的解析延拓，给出其增长估计并用于证明这些扭曲克洛斯特曼和的和的抵消结论。解释此构造与特征值\(1/4\)问题的联系并提出林尼克猜想的类似物。最后给出数值证据表明存在抵消且带模符号的克洛斯特曼和与经典克洛斯特曼和不相关。
英文摘要
We study sums of Kloosterman sums formed with a modular symbol. Employing Tauberian methods, we first give an estimate for a (Riesz) sum of Ramanujan sums formed with a modular symbol. We further define a zeta function that is analogous to the Selberg zeta function, establish its continuation to $\Re(s)>1/2$, give estimates for its growth and use this to prove a cancellation statement for sums of these twisted Kloosterman sums. We explain the connection of this construction to the eigenvalue 1/4 problem and formulate an analogue of Linnik's conjecture. Finally, we present numerical evidence that there is cancellation and also that the Kloosterman sums with a modular symbol are not correlated with classical Kloosterman sums.
