Dedekind Zeta Coefficients

• Dedekind zeta coefficients are defined via hₛ([K]) = |ζ*_K(s)|, capturing the arithmetic complexity of number fields through special zeta values.
• For values with Re(s) < 0, the Northcott property holds, ensuring finitely many fields satisfy a bounded height, while for Re(s) ≥ 1/2 the accumulation of zeta values precludes diophantine finiteness.
• Techniques like the resonance method and random Euler product models construct fields with extreme Dedekind heights, highlighting their intricate interplay of analytic and algebraic properties.

The Dedekind zeta coefficients at special values—the “Dedekind height” hₛ([K]) = |ζ_K(s)| where ζ_K(s) is the (possibly normalized or modified) Dedekind zeta function at a prescribed complex point s—capture the arithmetic complexity of a number field K. Their distribution is deeply intertwined with diophantine finiteness properties (such as the Northcott and Bogomolov property) and the analytic structure of zeta functions throughout the complex plane (Caro et al., 28 Feb 2025).

1. Diophantine Properties: Northcott and Bogomolov Behavior

The Northcott property for a function f on number fields is the assertion that the set $\{ [K] : f([K]) \leq B \}$ is finite for any real B. The Bogomolov property is stronger and implies the existence of a gap above the minimal value (i.e., the infimum is an isolated point of the set of values). For the Dedekind height, the results are:

• If $\operatorname{Re}(s) < 0$ (with s not a negative integer), the height function $[K] \mapsto (|ζ^*_K(s)|, [K : ℚ])$ has the Northcott property, i.e., for each fixed degree, the special values do not accumulate too densely. This generalizes earlier work to arbitrary complex $s$ with negative real part.
• For $σ = \operatorname{Re}(s) \geq 1/2$, neither the Northcott nor the Bogomolov property holds: for every $B$ greater than the infimum (which is $0$ for $1/2 \leq σ < 1$, and $1$ for $σ > 1$), there are infinitely many fields K with $h_s([K])$ between $0$ and $B$ (or arbitrarily close to $1$ for $σ > 1$). Thus, the set of heights accumulates at $0$ (or $1$), indicating that there is no “gap” above the minimal value.

This dichotomy underscores a critical distinction between the left half-plane (nice diophantine behavior) and the critical/right half-planes (rich accumulation phenomena).

2. Behavior of Dedekind Heights Across the Complex Plane

The analytic behavior of Dedekind zeta values at special arguments exhibits sharp contrasts depending on $σ = \operatorname{Re}(s)$:

• For $σ \geq 1/2$, especially for $1/2 \leq σ \leq 1$, there exist infinite families of quadratic fields (using Artin's factorization $$\zeta_{\mathbb{Q}(\sqrt{d})}(s) = \zeta(s) L(s, \chi_d)$$) where $|L(s, \chi_d)|$ becomes arbitrarily small, forcing $|\zeta_K(s)|$ to be arbitrarily close to zero. This is established by resonance methods and results on random Euler products.
• At $σ > 1$, the Euler product representation

$$\zeta_K(σ) = 1 + \sum_{n=2}^{\infty} \frac{a_K(n)}{n^{σ}}$$

with $a_K(n)$ nonnegative (counting the number of ideals of norm $n$) yields $\zeta_K(σ) \geq 1$ for any field K. However, by constructing fields with prescribed ramification and inertial behavior (e.g., via explicit polynomials controlling splitting), one can achieve $\zeta_K(σ)$ arbitrarily close to $1$ as the degree increases, or arbitrarily large via the complementary construction.

3. Techniques for Constructing Fields with Extreme Dedekind Heights

For the critical and right half-plane, two key methods produce families of number fields with prescribed zeta values:

• Resonance Method (Soundararajan): For quadratic fields, one constructs a “resonator” $R(8d) = \sum_{\ell \leq N} r(\ell) \chi_{8d}(\ell)$ with carefully chosen coefficients $r(\ell)$ to force small values of $L(s, \chi_d)$. This leads to moment calculations using sums over discriminants, yielding explicit asymptotic formulas for first and second moments.
• Random Euler Product Model: For $1/2 < σ \leq 1$, the statistical behavior of $L(σ, \chi_d)$ is shown to match that of random Euler products, with the distribution of small values having positive density according to results of Granville–Soundararajan and Lamzouri.
• Polynomial Construction for σ > 1: By fixing the splitting or inertial behavior of small primes in number fields of large degree (achieved by imposing congruence conditions on a generating polynomial), one can force the Euler factors to behave so that $\zeta_K(σ)$ approaches the Riemann zeta function at a large multiple of σ; for instance, for certain families, $\zeta_K(σ) \to 1$ as degree increases.

4. Generalization of the Northcott Property to the Left Half-Plane

For values $s$ with $\operatorname{Re}(s) < 0$, the results leverage the functional equation:

$$\zeta_K(s) = \zeta_K(1-s) \cdot \text{(Γ-factors)} \cdot |\Delta_K|^{1/2 - s}$$

to show that, upon bounding the degree $[K:ℚ]$, the values $|\zeta^*_K(s)|$ cannot cluster except possibly over subfamilies with bounded root discriminant. This extends the Northcott property to the pair $(|\zeta^*_K(s)|, [K:ℚ])$ for all $s$ with negative real part, generalizing previous results that were available only for integral s.

5. Implications for Dedekind Zeta Coefficients

The distribution and accumulation phenomena for special values of Dedekind zeta functions have direct implications for their coefficients in the Dirichlet series expansion and in their use as diophantine invariants:

• For $σ \geq 1/2$, both arbitrarily small and arbitrarily large values for $\zeta_K(s)$ can be attained, even among quadratic fields, so the spread of special values is unbounded—thereby precluding Northcott or Bogomolov properties.
• For $σ > 1$, all coefficients $a_K(n)$ are non-negative and generate monotonically increasing Euler products, yet the distribution of small heights can be made dense near $1$ by explicit constructions, again negating the Bogomolov property.
• The moment computations and resonance-based inequalities give explicit bounds on how small $|\zeta_K(s)|$ can be along infinite families, further illustrating the “wild” arithmetic behavior of the coefficients.
• On the left half-plane, the Northcott property constrains the arithmetic, so that the special values are diophantine invariants with finiteness properties when paired with bounded degree.

6. Broader Arithmetic and Diophantine Significance

The results of (Caro et al., 28 Feb 2025) demonstrate profound differences in the arithmetic and diophantine utility of Dedekind zeta coefficients across the (complex) s-plane:

• In the left half-plane, the special values $\zeta_K(s)$ serve as effective invariants for height finiteness results, supporting diophantine finiteness theorems analogous to Northcott’s theorem in diophantine geometry.
• In and to the right of the critical strip, the “height” associated to the zeta value is insufficient to control finiteness or even to guarantee the existence of a minimal gap—reflecting a high degree of arithmetic entropy.
• The techniques employed—resonance methods, random Euler products, and explicit field constructions—reveal the deep analytic and algebraic interconnections underpinning the behavior of Dedekind zeta coefficients.
• This global view elucidates the inapplicability of zeta special values as diophantine invariants in much of the classical critical region, while isolating exactly where such invariants retain their arithmetic power.

Table: Diophantine Properties of Dedekind Zeta Heights

Region for $s$	Northcott Property	Bogomolov Property
$\operatorname{Re}(s) < 0$	Holds (with bounded degree)	Not isolated in unbounded degree
$1/2 \leq \operatorname{Re}(s) \leq 1$	Fails	Fails
$\operatorname{Re}(s) > 1$	Fails	Fails, accumulation at $1$

This panorama reinforces that Dedekind zeta coefficients provide a compelling case paper for the arithmetic geometry of L-functions, coupling explicit field-theoretic and analytic constructions to diophantine and statistical properties on the spectrum of number fields.