Connected components of the ranges of twisted divisor functions on number fields

Technical Summary: Connected Components of the Ranges of Twisted Divisor Functions on Number Fields

Problem Statement
This paper investigates the topological structure of the image of twisted divisor functions defined on the monoid of integral ideals of a number field. Let $K$ be a finite extension of $\mathbb{Q}$, $\mathcal{O}_K$ its ring of integers, and $\mathcal{I}_K$ the monoid of integral ideals. For a complex parameter $r$ and a Dirichlet character $\chi$, the twisted ideal divisor function is defined as:
$$\sigma_{r,K,\chi}(I) = \sum_{J|I} N(J)^{-r}\chi(N(J))$$
where $N(J)$ denotes the ideal norm. The primary object of study is the closure of the image, $\overline{\sigma_{r,K,\chi}(\mathcal{I}_K)}$, and specifically the number of its connected components, denoted $C_{r,K,\chi}$.

While the case $K=\mathbb{Q}$ and $\chi$ being the trivial character (the standard divisor function $\sigma_r$) has been studied extensively (e.g., by Laatsch, Defant, Sanna, and Zubrilina), this work generalizes these inquiries to arbitrary number fields and twisted characters. The central questions addressed are:

1. Is the number of connected components $C_{r,K,\chi}$ finite for real $r > 1$ and real-valued $\chi$?
2. What values can $C_{r,K,\chi}$ assume as the character $\chi$ varies (for fixed $K$ and $r$)?
3. How does $C_{r,K}$ behave as the number field $K$ varies, particularly regarding lower bounds and unboundedness?

Methodology
The paper employs a combination of analytic number theory, algebraic number theory, and topological arguments. Key methodological components include:

• Partitioning and Gap Analysis: The core technique involves partitioning the domain (ideals or integers) based on the divisibility of small prime factors. By analyzing the "gaps" in the range of the divisor function created by these partitions, the authors establish lower bounds on the number of connected components. This relies on showing that the images of disjoint partition sets have disjoint closures.
• Chebotarev's Density Theorem: To handle general number fields $K$, the authors utilize Chebotarev's density theorem to estimate the distribution of prime ideals with specific splitting behaviors (e.g., primes that split completely vs. those that remain inert). This allows for the construction of fields with desired norm distributions.
• Dedekind Zeta Function Coefficients: The growth of the number of ideals of a given norm, $a_K(n)$, is bounded using coefficients of the Dedekind zeta function $\zeta_K(s)$. These bounds are crucial for establishing the existence of "gaps" in the range for large $r$.
• Constructive Field Theory: To prove unboundedness results for fixed degree extensions, the paper constructs specific number fields using the Dedekind-Kummer theorem. By selecting monic irreducible polynomials with prescribed factorization types modulo specific primes, the authors ensure that chosen rational primes split completely or remain inert in the constructed field $K$.
• Recursive Prime Selection: The construction of "r-mighty norms" (norms that generate gaps) involves recursively selecting sequences of rational primes that are sufficiently spaced apart to satisfy specific inequalities involving the parameter $r$.

Key Contributions and Results

1. Finiteness of Components (Theorem 1.2):
   The paper proves that for any real $r > 1$, finite extension $K/\mathbb{Q}$, and real-valued Dirichlet character $\chi$, the number of connected components $C_{r,K,\chi}$ is finite. This extends Sanna's previous result for $K=\mathbb{Q}$ to the general number field setting. The proof utilizes a downward induction argument on the prime ideals of $\mathcal{O}_K$, establishing that the image is a finite union of disjoint closed intervals.

2. Realization of Component Counts for $K=\mathbb{Q}$ (Theorems 1.3 & 1.4):

   • For a fixed $r > 1$, as the character $\chi$ varies, the value $C_{r,\mathbb{Q},\chi}$ assumes all sufficiently large positive integers.
   • There exists a threshold $r_0$ such that for all $r > r_0$, $C_{r,\mathbb{Q},\chi}$ assumes every positive integer value as $\chi$ varies. This is demonstrated by constructing principal characters with moduli composed of specific sets of primes to control the number of "jumps" in the function's range.

3. Improved Lower Bounds (Theorem 1.5):
   For $r \ge 3$ and a character $\chi$ of modulus $m$, the paper establishes a new lower bound:
   $$C_{r,\mathbb{Q},\chi} \ge 2^{\pi_m(2r-2) - \delta_m}$$
   where $\pi_m(x)$ counts primes $\le x$ coprime to $m$, and $\delta_m$ depends on the parity of $m$. This improves upon previous bounds by Achenjang and Berger.

4. Exponential Lower Bounds for General Fields (Theorem 1.6):
   For finite Galois extensions $K/\mathbb{Q}$, the paper derives a new exponential lower bound for $C_{r,K}$ as $r \to \infty$. Specifically, for any $\epsilon > 0$, there exist constants such that:
   $$\log C_{r,K} > \mu_{K,\epsilon} \left( \frac{r^{1/(1+\epsilon)}}{\log r} \right)$$
   This implies that for a fixed field $K$, the number of components is unbounded as $r$ increases.

5. Unboundedness over Fixed Degree Extensions (Theorem 1.7):
   Fixing the parameter $r > 1$ and the degree $s \ge 2$, the paper proves that the set of values $\{C_{r,K} : [K:\mathbb{Q}]=s\}$ is unbounded. By constructing degree-$s$ extensions where specific sequences of primes split completely (creating small norms) while others remain inert (creating large norms), the authors generate an arbitrary number of gaps in the range, thereby realizing arbitrarily large numbers of connected components.

Significance and Claims
The paper claims to provide a comprehensive generalization of the topological study of divisor function ranges from the rational integers to arbitrary number fields. Its significance lies in:

• Generalization: Extending finiteness results and component counting from $\mathbb{Q}$ to general number fields $\mathcal{I}_K$.
• Quantitative Precision: Providing explicit, improved lower bounds for the number of components, including exponential growth rates in terms of $r$ and the degree of the field.
• Completeness of Range: Demonstrating that the number of connected components is not only finite but can be controlled to take on any sufficiently large integer value (and all integers for large $r$) by varying the character.
• Structural Insight: Revealing that the topology of the range is deeply tied to the splitting behavior of primes in the number field and the arithmetic properties of the character's modulus.

The authors do not claim to solve the exact value of $C_{r,K,\chi}$ for all parameters but rather establish the finiteness, unboundedness, and realizability of these values, offering a framework for understanding the topological complexity of divisor functions in algebraic number theory.