发表机构
Elmhurst University; Hanoi University of Science and Technology; Lake Forest College（埃尔姆赫斯特大学; 河内科学技术大学; 湖森林学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用有限Frobenius环上的超特征理论统一推广Ramanujan和的算术性质，并给出判定有限交换环为Frobenius的新准则。
AI 中文摘要
Ramanujan和的理论在数学的多个子领域中一直发挥着基础性作用。它们出现在zeta函数特殊值理论、谱图理论、表示论以及解析数论中。近期工作表明，经典的Ramanujan和也可以通过环$\mathbb{Z}/n$的超特征理论解释为超傅里叶变换。在本文中，基于我们先前关于任意有限Frobenius环上超特征的工作，我们进一步探索Ramanujan和的算术性质。我们的方法提供了一个统一框架，推广了文献中关于这些和的各种结果。作为副产品，我们还描述了一个判定有限交换环是否为Frobenius环的新准则，这可能对代数学界具有独立意义。
英文摘要
The theory of Ramanujan sums has been playing a fundamental role in several subfields of mathematics. They appear in the theory of special values of zeta functions, spectral graph theory, representation theory, and analytic number theory. Recent work has shown that classical Ramanujan sums can also be interpreted as a super-Fourier transform via the theory of supercharacters for the ring $\mathbb{Z}/n$. In this article, building upon our recent work on supercharacters over an arbitrary finite Frobenius ring, we explore additional arithmetical properties of Ramanujan sums. Our approach provides a unified framework that generalizes various results in the literature regarding these sums. As a by-product, we also describe a new criterion for determining when a finite commutative ring is Frobenius, which could be of independent interest to the algebra community.
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