---
title: Varying Cubic Dedekind Zeta Functions
url: https://www.emergentmind.com/topics/varying-cubic-dedekind-zeta-functions
type: topic
---

# Varying Cubic Dedekind Zeta Functions

A varying cubic Dedekind zeta function refers to families of Dedekind zeta functions $\zeta_{K_c}(s)$ associated to cubic number fields $K_c$ that are parametrized by an indexing set (often algebraic integers with special properties), with attention to the behavior of special values, zeros, residues, and value-distributions as the underlying cubic field varies. Contemporary research rigorously analyzes the statistical properties and limiting distributions induced by this variation, particularly in relation to Artin $L$-functions, value-distribution phenomena, residue asymptotics, and explicit bounds in families. Key results apply both to Galois and non-Galois cubic fields and often integrate techniques from analytic number theory, representation theory, and arithmetic geometry.

## 1. Cubic Dedekind Zeta Functions and Parametrized Families

For a cubic number field $K$, the Dedekind zeta function is defined by
\[
\zeta_K(s) = \sum_{\mathfrak{a} \subset \mathcal{O}_K} \frac{1}{N(\mathfrak{a})^s}
\]
where the sum ranges over nonzero integral ideals $\mathfrak{a}$ in the ring of integers $\mathcal{O}_K$, and $N(\mathfrak{a})$ denotes the norm. In the context of varying families, $K = k(c^{1/3})$ is constructed over $k = \mathbb{Q}(\sqrt{-3})$ for $c \in \mathfrak{O}_k$ square-free and congruent to $1$ modulo $\langle 9 \rangle$ [1805.00724]. This parametrization yields a family of non-Galois or Galois cubic extensions with controlled arithmetic invariants.

The variation is studied through the ensemble $\{K_c\}_{c}$, with each field linked to a Dedekind zeta function $\zeta_{K_c}(s)$. The quotient $L_c(s) = \zeta_{K_c}(s)/\zeta_k(s)$ corresponds to a product of Artin $L$-functions attached to cubic Hecke characters $\chi_c$.

## 2. Value-Distribution Phenomena and Characteristic Functions

For fixed $\sigma > \frac{1}{2}$, key random variables are:
- $\,\log L_c(\sigma)$
- $\,L'_c/L_c(\sigma)$

The value-distribution of these quantities, as $c$ varies, is captured by an asymptotic distribution function $F_\sigma(z)$ defined by
\[
F_\sigma(z) = \lim_{Y\to\infty} \frac{1}{N(Y)} \#\{c \in \mathcal{C} : N(c)\leq Y,\, \mathcal{L}_c(\sigma)\leq z\}
\]
where $\mathcal{L}_c(s)$ is either $\log L_c(s)$ or $L'_c/L_c(s)$ [1805.00724]. The characteristic function $\varphi_\sigma(y)$ of $F_\sigma$ is computed explicitly as a convergent Euler product over prime ideals $\mathfrak{p} \subset \mathfrak{O}_k$, with separate cases for the logarithm and logarithmic derivative. The characteristic function satisfies super-Gaussian decay,
\[
|\varphi_\sigma(y)| \leq \exp\big(-C |y|^{1/\sigma-\varepsilon}\big)
\]
ensuring smooth probability densities $M_\sigma(t)$ via Fourier inversion.

## 3. Arithmetic Selection Constraints and Analytic Techniques

The congruence condition $c \equiv 1 \pmod{\langle 9\rangle}$ and square-freeness ensure that $\chi_c$ is a primitive Hecke character with minimal conductor, facilitating orthogonality in averaging and primitive $L$-function behavior [1805.00724].

Analysis utilizes:
- Exponential sum averaging $S_Y(y)$ with Mellin inversion and contour shift to access value-distribution statistics.
- Zero-density theorems and zero-free regions for $L(s,\chi_c)$; in rectangles $\Re s > 1/2 + \delta$, at most $O(Y^\delta)$ fields lack zero-freeness.
- Large sieve inequalities for cubic characters control off-diagonal contributions.
- Orthogonality relations over ray-class groups detect character values efficiently.

## 4. Asymptotic Formulas and Limiting Distributions

Limit theorems yield probability laws for log-values and logarithmic derivatives as $K_c$ varies, encapsulating dense arithmetic information on Dedekind zeta functions in the cubic family [1805.00724]. These results provide:
- Explicit limiting distributions for the error term in the Brauer–Siegel theorem, where
\[
E(c) := \log(h_cR_c) - \frac{1}{2} \log|D_c|
\]
has limiting law $F_1$ shifted by a constant, with $h_c$ and $R_c$ the class number and regulator, and $D_c$ the discriminant.
- Limiting distributions for the Euler–Kronecker constants $\gamma_{K_c}$ via
\[
\gamma_{K_c} = \lim_{s\to 1} \left(\frac{\zeta'_{K_c}(s)}{\zeta_{K_c}(s)} + \frac{1}{s-1}\right)
\]
as $c$ varies.

## 5. Explicit Asymptotic Residue Bounds and Zero-Density Results

In general cubic fields, explicit upper and lower bounds for residues are established under GRH [2101.11853]. For $K$ of degree $n_K=3$ and discriminant $\Delta$,
\[
\frac{1}{e^{17.54} \log\log|\Delta|} \leq \kappa_K \leq e^{18.87} (\log\log|\Delta|)^2
\]
with $\kappa_K = \operatorname{Res}_{s=1} \zeta_K(s)$ and all constants explicit; the proof invokes Duke's short-sum theorem and optimized bounds for associated Artin $L$-functions [2101.11853].

Zero-density results for cubic Dedekind zeta functions quantify the number of zeros in regions of the critical strip. For $N_K(T)$ the count of zeros $\rho$ with $|\Im \rho| \leq T$,
\[
N_K(T) = \frac{T}{\pi} (3 \log T + L - 3 \log(2\pi e)) + O(L + \log T)
\]
with $L = \log d_K$ [1201.3932]. Explicit bounds also apply to zeros with $\Re \rho \geq \sigma$.

## 6. Mean Residue Asymptotics via Adjoint Zeta Constructions

Analysis of the mean density of residues $\operatorname{Res}_{s=1} \zeta_E(s)$ in totally real cubic fields $E$ is formulated via adjoint zeta functions for $GL(3)$ acting on $gl_3$. Using regularized trace formulas, one obtains
\[
\sum_{E : m_2(E)\leq X} \operatorname{Res}\zeta_E(1) = O(X^{5/2})
\]
where $m_2(E)$ is the second successive minimum of the trace form quadratic $Q_E$ on $O_E$ [1308.5394]. The corresponding Dirichlet series admits a rightmost simple pole at $t=\frac{5}{2}$, and Tauberian methods confirm sharp upper and lower bounds, with asymptotic matching up to arbitrarily small loss.

## 7. Central Values and Sign-Distribution in Non-Galois Cubic Fields

In families of non-Galois cubic fields (S$_3$-fields), results show that the Dedekind zeta function can attain negative central values. For any fixed set of local specifications, the logarithmic density $d_X(X)$ of fields $K$ with $\zeta_K(\frac{1}{2}) < 0$ satisfies
\[
\liminf_{X\to\infty} d_X(X) \geq 95/128 \approx 0.67368
\]
with explicit construction methods using the Delone–Faddeev parametrization, Shintani zeta functions with congruence weights, and sieve techniques [2107.10900]. These phenomena are consistent with density conjectures for Artin $L$-functions and models for low-lying zeros in families.

## Table: Summary of Key Results for Varying Cubic Dedekind Zeta Functions

| Focus                                 | Main Quantitative Result                         | Reference         |
|---------------------------------------|--------------------------------------------------|-------------------|
| Value-distribution ($\log L_c$, $L'/L$)      | Limiting probability laws, explicit Euler products| [1805.00724]      |
| Residue bounds $\kappa_K$             | Explicit GRH-based bounds for cubic fields        | [2101.11853]      |
| Zero-density ($N_K(T)$, $N_K(\sigma,T)$)  | Explicit counts, Deuring–Heilbronn phenomenon     | [1201.3932]       |
| Mean residue asymptotics              | $O(X^{5/2})$ bounds for $\sum \operatorname{Res}\zeta_E(1)$ | [1308.5394]      |
| Negative central value density        | $d_X(X) \geq 95/128$ for $\zeta_K(1/2)<0$        | [2107.10900]      |

These results characterize the statistical and analytic variations in Dedekind zeta functions and their invariants across cubic field families, leveraging advanced techniques including trace formulas, sieve methods, and deep zero-density estimates. This framework connects value-distribution theory, effective class number and regulator bounds, and density problems in the analytic theory of $L$-functions.

Source: https://www.emergentmind.com/topics/varying-cubic-dedekind-zeta-functions