射线类群中k-自由理想的分布
The distribution of $k$-free ideals in ray class groups
AI总结：
该研究将算术级数中k-自由整数分布问题推广到任意数域，借助射线类群建立相关渐近公式，证明k-自由理想在射线类中等分布，改进了两类群的误差估计并恢复经典公式，还给出了计算实例。
AI中文摘要：
本文将研究算术级数中k-自由整数分布的经典问题推广到任意数域的情形。利用射线类群的语言，我们建立了有界范数的k-自由理想在给定射线类中数量的渐近公式及误差项。特别地，我们的结果表明k-自由理想在射线类中是等分布的。我们还通过使用Landau给出的更精确的理想计数渐近式，在理想类群和窄类群的情形下得到了改进的误差估计。我们的结果恢复了在有理数域$\boldsymbol{\text{Q}}$上的Gegenbauer和Cohen--Robinson经典公式，并将Benkowski、Nymann和Sittinger的前期工作推广到射线类群的情形。我们还给出了说明渐近公式以及k-自由理想在射线类中等分布的显式计算实例。
英文摘要：
In this paper, we extend the classical problem of studying the distribution of $k$-free integers in arithmetic progressions to the setting of arbitrary number fields. Using the language of ray class groups, we establish asymptotic formulas, together with error terms, for the number of $k$-free ideals of bounded norm lying in a given ray class. In particular, our results show that $k$-free ideals are equidistributed among ray classes. We also obtain improved error estimates in the cases of ideal class groups and narrow class groups by using sharper ideal counting asymptotics due to Landau. Our results recover the classical formulas of Gegenbauer and Cohen--Robinson over $\mathbb{Q}$ and extend previous work of Benkowski, Nymann, and Sittinger to the setting of ray class groups. We also present explicit computational examples that illustrate the asymptotic formulas and the equidistribution of $k$-free ideals among ray classes.
