The distribution of $k$-free ideals in ray class groups

Technical Summary: The Distribution of k-Free Ideals in Ray Class Groups

Problem Statement
This paper addresses the extension of a classical problem in analytic number theory—the distribution of $k$-free integers in arithmetic progressions—to the setting of arbitrary algebraic number fields. While the distribution of $k$-free integers (integers not divisible by the $k$-th power of any prime) in arithmetic progressions is well-understood over $\mathbb{Q}$ (with foundational results by Gegenbauer, Cohen, and Robinson), the analogous problem for integral ideals in general number fields, specifically within the framework of ray class groups, had not been systematically established. The authors aim to derive asymptotic formulas for the number of $k$-free ideals of bounded norm lying in a prescribed ray class, including precise error terms.

Methodology
The authors employ a combination of analytic number theory techniques adapted to number fields:

1. Ray Class Groups: They utilize ray class groups $Cl^{\mathfrak{m}}_K$ as the natural generalization of arithmetic progressions modulo $q$. This allows for the study of ideals coprime to a modulus $\mathfrak{m} = \mathfrak{m}_0\mathfrak{m}_\infty$.
2. Inclusion-Exclusion and Möbius Inversion: To count relatively $k$-free $r$-tuples of ideals, the authors apply the Principle of Inclusion-Exclusion. This involves expressing the counting function in terms of the Möbius function $\mu_K$ for the number field $K$, summing over square-free ideals.
3. Ideal Counting Asymptotics: The core of the error analysis relies on asymptotic formulas for the number of integral ideals in a given ray class with norm at most $X$. The authors distinguish between general ray class groups and the specific cases of the ordinary ideal class group ($\mathfrak{m}=(1)$) and the narrow ideal class group ($\mathfrak{m}=\mathfrak{m}^{full}_\infty$). For the latter two, they utilize sharper error estimates due to Landau, which depend on the degree $d$ of the number field.
4. Partial Summation: To handle the sums involving the Möbius function and the ideal counting functions, the authors apply partial summation techniques to derive bounds for the error terms based on the exponent $k$ and the degree $d$.

Key Contributions and Results
The primary result is Theorem 1.1, which provides an asymptotic formula for $Q^r_{k,K}(X; \mathbf{A})$, the number of relatively $k$-free $r$-tuples of ideals $(\mathfrak{a}_1, \dots, \mathfrak{a}_r)$ where each $\mathfrak{a}_i$ lies in a fixed ray class $A_i$ and has norm $\le X$.

The main term is given by:
Q^r_{k,K}(X; \mathbf{A}) = \frac{(\rho^{\mathfrak{m}}_K)^r}{\zeta_K(rk)} \prod_{\mathfrak{p}|\mathfrak{m}_0} \left(1 - \frac{1}{N(\mathfrak{p})^{rk})^{-1} X^r + E_{K,\mathfrak{m},k,r}(X)
where $\rho^{\mathfrak{m}}_K$ is the residue of the ray class zeta function at $s=1$.

Equidistribution: A central finding (Remark 1.2) is that the main term is independent of the specific tuple of ray classes $\mathbf{A}$. This establishes that relatively $k$-free ideals are equidistributed among the ray classes modulo $\mathfrak{m}$. Specifically, each ray class contains an asymptotic proportion of $1/h^{\mathfrak{m}}_K$ of all $k$-free ideals.

Error Terms: The paper provides detailed error estimates $E_{K,\mathfrak{m},k,r}(X)$ that depend on the degree $d = [K:\mathbb{Q}]$, the parameter $k$, and the type of modulus:

• General Ray Classes: The error is generally $O(X^{r - \delta_d(\mathfrak{m})})$ or $O(X^{1/k})$, where $\delta_d(\mathfrak{m}) = 1/d$.
• Ordinary and Narrow Class Groups: By leveraging Landau's sharper ideal counting estimates, the authors improve the error term for these specific cases. Here, $\delta_d(\mathfrak{m}) = 2/(d+1)$.
  • For $d=2$ and $k=2$, the error is $O(X^{1/2})$ for general moduli and remains $O(X^{1/2})$ for the ordinary/narrow class groups.
  • For $d=3$ and $k=2$, the error is $O(X^{1/2} \log X)$ for general moduli, improving to $O(X^{1/2} \log X)$ for the ordinary/narrow class groups (though the log factor persists in the improved bound for this specific case).
  • For $d \ge 4$, the error improves from $O(X^{1-1/d})$ in the general case to $O(X^{1-2/(d+1)})$ in the improved case.

Special Cases and Recoveries:

• Theorem 1.3 specializes the result to $r=1$ (single ideals) for general ray classes.
• Theorem 1.5 provides the improved estimates for the ordinary and narrow class groups.
• Corollary 1.4 demonstrates that setting $K=\mathbb{Q}$ and choosing the appropriate modulus recovers the classical formulas of Gegenbauer (for $k$-free integers) and Cohen–Robinson (for $k$-free integers in arithmetic progressions).
• Corollary 1.6 sums the results over all ideal classes to provide the asymptotic for the total number of $k$-free ideals in $K$.

Computational Verification
The authors implemented the counting functions in SageMath to provide explicit computational examples (Examples 1.8 and 1.9). These examples illustrate the equidistribution of square-free ideals across ideal classes in a cubic field and across ray classes in a real quadratic field. The data confirms the theoretical main terms and highlights symmetries in the counting functions arising from the Galois structure of the field (Remark 1.10).

Significance
The paper claims to extend previous work by Benkoski, Nymann, and Sittinger (who studied $k$-free tuples in the context of ideal classes) to the broader setting of ray class groups. By establishing these asymptotic formulas with explicit error terms, the work unifies the classical theory of $k$-free integers in arithmetic progressions with the modern theory of ideals in number fields. The results confirm that the equidistribution phenomenon observed in $\mathbb{Q}$ holds generally for number fields within the ray class framework, while also providing refined error bounds for the most fundamental class groups (ordinary and narrow).