论文标题
算术密度的分区理论公式，II
Partition-theoretic formulas for arithmetic densities, II
论文作者
论文摘要
In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers $\mathbb N$ as limiting values of $q$-series as $q\to ζ$ a root of unity (instead of using the usual Dirichlet series to compute densities), replacing multiplicative structures of $\mathbb整数分区中的类似结构$ \ Mathcal P $。在最近的工作中，王获得了对Alladi原始定理的广泛概括，其中将质数的亚集的算术密度计算为源自Dirichlet卷积引起的Dirichlet序列值。在这里，作者证明了王的扩展名也有一个分区理论模拟，可为任何$ \ mathbb n $的任何子集产生新的$ q $ series密度公式。为此，我们概述了从第一原理计算$ q $ series密度计算的理论，基于统计量，我们称为给定子集的“ $ q $密度”。该理论反过来产生了无限的算术密度的无限族。
In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers $\mathbb N$ as limiting values of $q$-series as $q\to ζ$ a root of unity (instead of using the usual Dirichlet series to compute densities), replacing multiplicative structures of $\mathbb N$ by analogous structures in the integer partitions $\mathcal P$. In recent work, Wang obtains a wide generalization of Alladi's original theorem, in which arithmetic densities of subsets of prime numbers are computed as values of Dirichlet series arising from Dirichlet convolutions. Here the authors prove that Wang's extension has a partition-theoretic analogue as well, yielding new $q$-series density formulas for any subset of $\mathbb N$. To do so, we outline a theory of $q$-series density calculations from first principles, based on a statistic we call the "$q$-density" of a given subset. This theory in turn yields infinite families of further formulas for arithmetic densities.
