Dedekind Zeta Function in Number Theory

• The Dedekind zeta function is a central analytic invariant that generalizes the Riemann zeta function through a Dirichlet series and meromorphic continuation for number fields.
• Its analytic structure, including a functional equation and explicit zero-free regions, provides deep insights into class numbers, regulators, and the distribution of prime ideals.
• Special values and derivatives of the function reveal rich arithmetic complexities, underpinning advances in computational number theory, spectral analysis, and motivic studies.

The Dedekind zeta function is a central analytic invariant in algebraic number theory, capturing profound arithmetic information about a number field. For a field $K$ of degree $n = [K:\mathbb{Q}]$ with ring of integers $\mathcal{O}_K$, it is defined via the Dirichlet series

$$\zeta_K(s) = \sum_{\substack{0\neq \mathfrak{a}\subset \mathcal{O}_K}} \frac{1}{N(\mathfrak{a})^s}$$

for $\mathrm{Re}(s) > 1$, where $N(\mathfrak{a})$ is the absolute norm of the ideal $\mathfrak{a}$. Through analytic continuation and a functional equation, $\zeta_K(s)$ serves as a natural generalization of the Riemann zeta function, encoding in its special values, zeros, and residues fundamental invariants including the class number, regulator, and discriminant of $K$.

1. Analytic Structure and Functional Equation

The Dedekind zeta function admits Euler product and Dirichlet series expansions mirroring the Riemann zeta case and extends meromorphically to the complex plane with a simple pole at $s=1$. Its completed version is

$$\xi_K(s) = s(s-1) d_K^{s/2} \gamma_K(s) \zeta_K(s),$$

where $d_K$ is the absolute discriminant, and $\gamma_K(s)$ is a product of gamma factors determined by the archimedean places:

$$\gamma_K(s) = \pi^{-ns/2} \prod_{j=1}^{r_1} \Gamma\left( \frac{s}{2} \right)\prod_{k=1}^{r_2} \Gamma(s),$$

with $r_1$ and $r_2$ the number of real and complex embeddings. The critical functional equation

$\xi_K(s) = \xi_K(1-s)$

exhibits a symmetry analogous to the Riemann case and enables the paper of zeros and special values across the critical strip (Nguyen, 25 Sep 2025). The degree and discriminant enter as scaling factors crucial to both analytic and arithmetic estimates.

2. Zeros, Zero Distribution, and the Extended Riemann Hypothesis

The nontrivial zeros of $\zeta_K(s)$ in $0 < \mathrm{Re}(s) < 1$ encode deep arithmetic information and are the focus of critical conjectures. The Extended Riemann Hypothesis (ERH) posits that all such zeros lie on the line $\mathrm{Re}(s) = \frac{1}{2}$, unifying disparate fields in the paper of $L$-functions (Nguyen, 25 Sep 2025).

Classical results and recent advances include:

• Explicit Zero Density and Window Bounds: Under GRH, explicit formulas provide upper bounds for the number of zeros in an interval $[T-a, T+a]$, revealing that the density is governed by $\log |d_K| + n \log T$ with explicit lower-order corrections (Grenié et al., 2014, Kadiri et al., 2012).
• Zero-free Regions: Results give explicit zero-free regions near $\mathrm{Re}(s)=1$, crucial for applications to prime ideal theorems and conditional bounds in analytic number theory. For instance,

$$\sigma \geq 1 - \frac{1}{C_1\log d_K + C_2 n \log |t| + C_3 n + C_4}$$

where $C_i$ are explicitly given constants (Lee, 2020).

• Multiplicity: In the nonabelian Galois context, $\zeta_K(s)$ can have zeros of higher multiplicity, in contrast to the expected simplicity for Dirichlet L-functions; the order of zeros is tied to the maximal degree of irreducible representations in $\operatorname{Gal}(L/K)$ (Hu et al., 2021).
• Modular and Spectral Relations: Recent work establishes modular-type relations, expressing arithmetic sums in terms of spectral data—the zeros—generalizing identities due to Ramanujan, Hardy, and Littlewood (Dixit et al., 2022).
• Criterion Equivalent to ERH: An equivalence is established between ERH and a closed-form involving a series over the nontrivial zeros:

$$\sum_{\rho_K \neq 1/2} \frac{1}{|\frac{1}{2} - \rho_K|^2}$$

with the second derivative of a suitably normalized completed xi-function at $s=1/2$ (Nguyen, 25 Sep 2025).

3. Special Values, Residues, and Their Arithmetic Significance

The value and residue of $\zeta_K(s)$ at special points contain arithmetic invariants:

• Residue at $s=1$: The analytic class number formula expresses the residue in terms of the class number $h_K$, regulator $R_K$, discriminant, and the structure of units and roots of unity. Approximations of the residue benefit from advanced use of the explicit formula and prime splitting, with improvements reducing error constants significantly in computational approaches (Belabas et al., 2013).
• Central Values and Derivatives: Values and derivatives at $s=1/2$ possess strong arithmetical transcendence properties; for fields of degree $\leq3$, vanishing of $\zeta_K(1/2)$ forces vanishing of its derivative, while for almost all quadratic fields, the quotient $\zeta_K'(1/2)/\zeta_K(1/2)$ is transcendental (Kandhil, 2022).
• Special Values at Positive Integers: Extended analogues of Wilton's formula relate products $\zeta_K(u)\zeta_K(v)$ to Meijer G-functions and yield explicit, albeit intricate, formulas for $\zeta_K(s)$ at integer arguments, especially for quadratic fields (Banerjee et al., 2016). Identities for imaginary quadratic fields reminiscent of Ramanujan and Zagier provide explicit transformations and connect Dedekind zeta values at even and odd integers to Lambert series and special functions such as Kelvin and hypergeometric functions (Banerjee et al., 2021).
• Multiple Dedekind Zeta Values: Recent constructions introduce higher-order analogues via iterated integrals "on membranes," generalizing both the Dedekind zeta function and multiple zeta values, with conjectured period interpretations and rich connections to modular and automorphic forms (Horozov, 2011, Horozov, 2013).

4. Extreme and Minimal Values: Diophantine and Probabilistic Properties

The magnitude of $\zeta_K(s)$ across families of number fields and at specific $s$ encodes deep Diophantine properties reminiscent of height functions in arithmetic geometry:

• Northcott and Bogomolov Properties: For $\mathrm{Re}(s)<0$, the set of degrees paired with bounded $|\zeta_K^*(s)|$ is finite (Northcott property), generalizing to arbitrary complex $s$ in the left half-plane; but for $\mathrm{Re}(s)\geq 1/2$ the Bogomolov property fails, as the set of $|\zeta_K^*(s)|$ can be made arbitrarily small over infinite families (Généreux et al., 2022, Caro et al., 28 Feb 2025).
• Resonance and Probabilistic Methods: For $1/2 \leq \sigma < 1$, the resonance method is used to construct quadratic fields with extremely small $|\zeta_K(\sigma)|$, with quantitative bounds showing that the values can approach zero as $K$ varies. For $1/2<\sigma\leq 1$, probabilistic models of random Euler products (due to Granville–Soundararajan and Lamzouri) are employed to compare distributions and infer the density of such small values (Caro et al., 28 Feb 2025).
• Behavior at $\sigma>1$: The absolute convergence of the Dirichlet series enables explicit construction of fields where $\zeta_K(\sigma)$ is as close as desired to $1$, or arbitrarily large, by controlling the splitting behavior of small primes.

5. Extreme Values of Derivatives

The paper of maximal values (and $\Omega$-results) for derivatives of $\zeta_K(s)$ on and near the critical line has advanced using harmonic analysis and the resonance method:

• Cyclotomic Fields: Sharp lower bounds for $\max_{t\in[0,T]} |\zeta_K^{(\ell)}(1/2+it)|$ in cyclotomic fields $\mathbb{Q}(\zeta_d)$ are established, with exponents depending on $\varphi(d)$, explicitly of shape

$$\exp\left( (1+o(1)) \varphi(d) \sqrt{\frac{\log T \log\log\log T}{\log\log T}} \right)$$

for fixed $\ell$ (Li et al., 14 Jun 2025).

• Methods: The approach uses advanced convolution formulas and GCD sum estimates, generalizing prior work for the Riemann zeta function to the setting of higher degree and more structured Dedekind zeta functions.

6. Applications and Ongoing Research Directions

Dedekind zeta functions are a nexus for analytic and algebraic investigations:

• Algorithmic Number Theory: Efficient calculations of class groups and regulators (e.g. via Buchmann's algorithm) leverage improved estimates for residues of $\zeta_K(s)$ at $s=1$, with reductions in error constants translating to computational savings (Belabas et al., 2013).
• Spectral and Quantum Analogy: The distribution and multiplicity of zeros bear on quantum chaos and statistical physics analogues, with modular-type identities linking arithmetic and spectral data (Dixit et al., 2022).
• Categorification and Functoriality: Recent categorical approaches re-express the factorization of $\zeta_K(s)$ in the language of incidence algebras and objective linear algebra, providing structural understanding and generalizations, especially for quadratic fields (Aycock et al., 2022).
• Period and Motive Connections: Multiple Dedekind zeta values, conjectured to be mixed Tate periods, suggest links to motivic Galois groups and mixed motive cohomology (Horozov, 2011).
• Transcendence and Irrationality: At special points such as $s=1/2$, the transcendence of derivatives, as well as relations to generalized Bernoulli numbers and theta functions, continue to reveal new facets of their arithmetic complexity (Kandhil, 2022).

The analytic, algebraic, Diophantine, and probabilistic structures encapsulated by the Dedekind zeta function continue to generate foundational research, including investigations of its low-lying zeros (Kala, 17 Jan 2024), new equivalences for ERH via explicit series over zeros (Nguyen, 25 Sep 2025), and the interplay of special values with finiteness properties in number field towers (Caro et al., 28 Feb 2025). Ongoing work addresses extensions to nonabelian fields, categorical frameworks, and arithmetic statistics in families of $L$-functions where Dedekind zeta functions serve as test objects for conjectures at the heart of modern number theory.