Varying Cubic Dedekind Zeta Functions

• Varying cubic Dedekind zeta functions are families linked to cubic number fields, defined by series over ideals and exhibiting distinctive special values, zeros, and residue asymptotics.
• The study employs analytic number theory techniques including explicit Euler products, Mellin inversion, and zero-density theorems to characterize limiting distributions and error terms.
• Applications encompass refined asymptotic formulas for class numbers, regulators, and Euler–Kronecker constants, providing new insights into arithmetic invariants in cubic field families.

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 $\langle 9 \rangle$ (Akbary et al., 2018). 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 $L'_c/L_c(s)$ (Akbary et al., 2018). 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 (Akbary et al., 2018).

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 (Akbary et al., 2018). 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 (Garcia et al., 2021). 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 (Garcia et al., 2021).

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 $L = \log d_K$ (Kadiri et al., 2012). 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 $O_E$ (Matz, 2013). 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 (Shankar et al., 2021). 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	(Akbary et al., 2018)
Residue bounds $\kappa_K$	Explicit GRH-based bounds for cubic fields	(Garcia et al., 2021)
Zero-density ($N_K(T)$, $N_K(\sigma,T)$)	Explicit counts, Deuring–Heilbronn phenomenon	(Kadiri et al., 2012)
Mean residue asymptotics	$O(X^{5/2})$ bounds for $\sum \operatorname{Res}\zeta_E(1)$	(Matz, 2013)
Negative central value density	$d_X(X) \geq 95/128$ for $\zeta_K(1/2)<0$	(Shankar et al., 2021)

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.