The ring of arithmetical functions with unitary convolution: the [n]-truncation

The ring of arithmetical functions with unitary convolution: General Truncations

Let A denote the ring of arithmetical functions with unitary convolution, and let V be a finite subset of the positive integers having the property that for every v in V, all unitary divisors of v lie in V. We study the truncation A_V, an artinian monomial quotient of a polynomial ring in finitely many indeterminates, isomorphic to the ``Artinified'' Stanley-Reisner ring C[\bar{Δ(V)}] of a certain simplicial complex Δ(V).

The ring of arithmetical functions with unitary convolution: Divisorial and topological properties

We study the ring of arithmetical functions with unitary convolution, giving an isomorphism to a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell-Everett between the ring of arithmetical functions with Dirichlet convolution and the power series ring on countably many variables. We topologize it with respect to a natural norm, and shove that all ideals are quasi-finite. Some elementary results on factorization into atoms are obtained. We prove the existence of an abundance of non-associate regular non-units.

数论函数环的截断（再探）

Truncations of the ring of number-theoretic functions, revisited

该研究重新探讨数论函数环的截断问题，证明截断环的单项式理想极小生成元计数等于勒让德筛选函数，解决了过往猜想，得到生成元总数的渐近阶与误差项，并修正了旧论文的错误与疏漏。

赫维茨级数环中的形变卷积、累积量变换与半群生成元

Deformed Convolution, Cumulant Transforms, and Semigroup Generators in Hurwitz Series Rings

该研究在赫维茨级数环上引入单参数卷积，推导其性质与应用，得到相关半群生成元的显式公式，拓展了自由概率与代数结构的相关理论。