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Dedekind zeta function

The Dedekind zeta function of an algebraic number field K, denoted \zeta_K(s), is a complex-valued function defined for \operatorname{Re}(s) > 1 as the Dirichlet series \zeta_K(s) = \sum_{\mathfrak{a}} \frac{1}{N(\mathfrak{a})^s}, where the sum runs over all nonzero ideals \mathfrak{a} in the ring of integers \mathcal{O}_K of K, and N(\mathfrak{a}) denotes the absolute norm of \mathfrak{a}.^[1]^[2]^[3] This function admits an Euler product decomposition \zeta_K(s) = \prod_{\mathfrak{p}} \left(1 - N(\mathfrak{p})^{-s}\right)^{-1}, where the product is over all prime ideals \mathfrak{p} of \mathcal{O}_K, mirroring the structure of the Riemann zeta function \zeta(s) for K = \mathbb{Q}.^[1]^[2] It converges absolutely in the half-plane \operatorname{Re}(s) > 1 and possesses a meromorphic continuation to the entire complex plane, featuring a simple pole at s = 1 with residue given by the analytic class number formula \operatorname{Res}_{s=1} \zeta_K(s) = \frac{2^{r_1} (2\pi)^{r_2} h R}{w \sqrt{|\Delta_K|}}, where r_1 and r_2 are the numbers of real and complex embeddings of K, h is the class number of \mathcal{O}_K, R is the regulator of the unit group, w is the number of roots of unity in K, and \Delta_K is the discriminant of K.^[1]^[2] Furthermore, \zeta_K(s) satisfies a functional equation relating its values at s and $1-s, involving the discriminant and embedding data: \Lambda_K(s) = |\Delta_K|^{s/2} \pi^{-r_1 s / 2} (2\pi)^{-r_2 s} \Gamma(s/2)^{r_1} \Gamma(s)^{r_2} \zeta_K(s) = \Lambda_K(1-s), where r_1 and r_2 are the numbers of real and complex embeddings of K.^[1] In the special case of quadratic fields K = \mathbb{Q}(\sqrt{D}), \zeta_K(s) factors as \zeta(s) L(s, \chi_D), where \chi_D is the Dirichlet character associated to the discriminant D, highlighting its connections to prime ideal distributions and L-functions.^[3]^[2] The Dedekind zeta function plays a central role in algebraic number theory, encoding arithmetic invariants such as the class number and providing tools for studying the distribution of prime ideals, with generalizations extending to Artin L-functions and broader contexts in arithmetic geometry.^[1]^[2]

Definition

Ideal-theoretic formulation

The Dedekind zeta function is fundamentally defined in the context of algebraic number theory, where a number field K is a finite extension of the rational numbers \mathbb{Q}, with degree n = [K : \mathbb{Q}] denoting the dimension of K as a vector space over \mathbb{Q}. The ring of integers \mathcal{O}_K of K is the integral closure of \mathbb{Z} in K, consisting of all elements of K that are roots of monic polynomials with integer coefficients. Integral ideals of \mathcal{O}_K are nonzero additive subgroups \mathfrak{a} \subseteq \mathcal{O}_K that are finitely generated as \mathcal{O}_K-modules and closed under multiplication by elements of \mathcal{O}_K. The norm N(\mathfrak{a}) of such an ideal \mathfrak{a} is defined as the cardinality of the finite quotient ring \mathcal{O}_K / \mathfrak{a}, which is a positive integer invariant under the Galois group of K/\mathbb{Q}.^[1] The Dedekind zeta function \zeta_K(s) attached to K is given by the Dirichlet series

\zeta_K(s) = \sum_{\mathfrak{a} \neq (0)} \frac{1}{N(\mathfrak{a})^s},

where the sum runs over all nonzero integral ideals \mathfrak{a} of \mathcal{O}_K, and s \in \mathbb{C} is a complex variable. This series converges absolutely in the half-plane \operatorname{Re}(s) > 1, owing to the fact that the number of integral ideals with norm at most x grows asymptotically like c_K x for some constant c_K > 0 depending on K, ensuring the partial sums behave sufficiently like an integral \int_1^\infty t^{-s} \, dt.^[1]^[4] This formulation generalizes the Riemann zeta function \zeta(s), as the case K = \mathbb{Q} yields \mathcal{O}_K = \mathbb{Z}, whose nonzero principal ideals are (n) for positive integers n with N((n)) = n, reducing \zeta_\mathbb{Q}(s) to the classical series \sum_{n=1}^\infty n^{-s}. The ideal-theoretic perspective underscores the arithmetic nature of \zeta_K(s), capturing the distribution of ideals in \mathcal{O}_K in a manner analogous to how \zeta(s) encodes the primes in \mathbb{Z}.^[1]

Euler product expansion

The Euler product expansion of the Dedekind zeta function \zeta_K(s) for a number field K with ring of integers \mathcal{O}_K is given by

\zeta_K(s) = \prod_{\mathfrak{p}} \left(1 - N(\mathfrak{p})^{-s}\right)^{-1},

where the product runs over all non-zero prime ideals \mathfrak{p} of \mathcal{O}_K, and N(\mathfrak{p}) denotes the norm of \mathfrak{p}.^[1]^[5] This representation holds for \operatorname{[Re](/page/Re)}(s) > 1, where the series and product converge absolutely.^[6] The derivation relies on the unique factorization theorem for ideals in the Dedekind domain \mathcal{O}_K, which states that every non-zero ideal \mathfrak{a} factors uniquely as \mathfrak{a} = \prod_{\mathfrak{p}} \mathfrak{p}^{k_{\mathfrak{p}}} with k_{\mathfrak{p}} \geq 0 and finitely many non-zero.^[5] The norm is multiplicative, so N(\mathfrak{a}) = \prod_{\mathfrak{p}} N(\mathfrak{p})^{k_{\mathfrak{p}}}, and thus

\zeta_K(s) = \sum_{\mathfrak{a} \neq 0} N(\mathfrak{a})^{-s} = \prod_{\mathfrak{p}} \sum_{k=0}^{\infty} N(\mathfrak{p})^{-k s}.

Each inner sum is a geometric series \sum_{k=0}^{\infty} x^k = (1 - x)^{-1} with x = N(\mathfrak{p})^{-s} and |x| < 1 for \operatorname{Re}(s) > 1, yielding the Euler product.^[1]^[6] The absolute convergence follows from bounding the partial products and the density of prime ideals, analogous to the Riemann zeta function case.^[7] This product structure connects to the decomposition of rational primes p in \mathcal{O}_K. Each p \mathcal{O}_K factors as a product of prime ideals \mathfrak{p}_i^{e_i} with inertial degrees f_i, where the local Euler factor over primes above p is \prod_i (1 - N(\mathfrak{p}_i)^{-s})^{-1}.^[5] In the inert case, p remains prime with N(\mathfrak{p}) = p^f and e=1; in the split case, it decomposes into distinct primes with N(\mathfrak{p}_i) = p; and in the ramified case, there is multiplicity e > 1 typically at primes dividing the discriminant.^[1] These cases are determined by the splitting behavior in the Galois group of the extension.^[7] For quadratic fields K = \mathbb{Q}(\sqrt{d}) with discriminant \Delta_K, the Euler product reflects the prime decomposition via the Kronecker symbol \left( \frac{\Delta_K}{p} \right): primes split if =1, remain inert if =-1, and ramify if =0.^[1] This yields \zeta_K(s) = \zeta(s) L(s, \chi), where \chi is the primitive Dirichlet character modulo |\Delta_K| associated to \Delta_K, and the product over rational primes becomes \prod_p (1 - p^{-s})^{-1} \prod_p (1 - \chi(p) p^{-s})^{-1}.^[7] For example, in K = \mathbb{Q}(i) with \Delta_K = -4, the character \chi(p) = \left( \frac{-4}{p} \right) distinguishes split primes (e.g., p \equiv 1 \pmod{4}) from inert ones (e.g., p \equiv 3 \pmod{4}), directly tying the product's local factors to the field's arithmetic.^[1]

Analytic properties

Meromorphic continuation

The Dedekind zeta function \zeta_K(s), defined for \operatorname{Re}(s) > 1 via its Dirichlet series \sum_{\mathfrak{a}} 1/\mathrm{N}(\mathfrak{a})^s over nonzero ideals \mathfrak{a} of the ring of integers of a number field K, extends to a meromorphic function on the entire complex plane \mathbb{C}. This analytic continuation is unique and features a single simple pole at s = 1, with the function being holomorphic at all other points in \mathbb{C}.^[1] The continuation is achieved through integral representations, notably the Mellin transform of theta series associated to the ideals of \mathcal{O}_K. Specifically, one employs the completed zeta function Z_K(s) = \Gamma_K(s) \zeta_K(s), where \Gamma_K(s) incorporates Gamma factors reflecting the real and complex embeddings of K, and applies Poisson summation to derive the meromorphic extension. For quadratic fields, simpler methods relate \zeta_K(s) to products of the Riemann zeta function and Dirichlet L-functions, facilitating continuation via known properties of those functions. In general, these techniques ensure the meromorphic structure without additional poles.^[1]^[8] The pole at s = 1 is simple, arising from the Euler product's behavior near the abscissa of convergence, analogous to the Riemann zeta function. No other poles exist, as confirmed by the integral representations and the absence of further singularities in the Gamma factors or theta series transforms.^[1] In the critical strip $0 < \operatorname{Re}(s) < 1, \zeta_K(s) exhibits controlled asymptotic growth, with bounds such as |\zeta_K(\sigma + it)| \ll t^{A(1-\sigma) + \epsilon} for \sigma \in (0,1) and large t, derived from convexity principles applied to the functional equation (though the equation itself is not used here for continuation). These estimates highlight the function's boundedness away from the pole and zeros.^[9] Historically, Dedekind first established the meromorphic continuation for quadratic fields in 1877, using relations to Dirichlet characters and theta functions in his supplement to Dirichlet's Vorlesungen über Zahlentheorie. The general case for arbitrary number fields was later proven using advanced theta series and Hecke's integral methods in the early 20th century.^[10]^[11]

Functional equation

The Dedekind zeta function \zeta_K(s) of a number field K of degree n = [K : \mathbb{Q}] satisfies a functional equation that relates its values at s and $1 - s. This equation is formulated using a completed version of \zeta_K(s) that incorporates gamma factors accounting for the infinite places of K. The meromorphic continuation of \zeta_K(s) to the complex plane, as discussed previously, is essential for the equation to hold globally. The completed Dedekind zeta function is given by

\Lambda_K(s) = |d_K|^{s/2} \left( \pi^{-s/2} \Gamma\left( \frac{s}{2} \right) \right)^{r_1} \left( (2\pi)^{-s} \Gamma(s) \right)^{r_2} \zeta_K(s),

where d_K denotes the discriminant of K, r_1 is the number of real infinite places (real embeddings), and r_2 is the number of complex infinite places (pairs of complex conjugate embeddings). This \Lambda_K(s) is entire except for simple poles at s = 0 and s = 1, and it satisfies the functional equation

\Lambda_K(s) = \Lambda_K(1 - s).

The equality holds without an additional root number factor, as the self-dual nature of the Dedekind zeta function yields a root number of +1. The functional equation was first established by Erich Hecke using theta series associated to ideals in the ring of integers of K. A sketch of the derivation proceeds via the construction of theta functions \theta_{\mathfrak{a}}(\tau) = \sum_{\mathbf{x} \in \mathfrak{a}} e^{\pi i \operatorname{Tr}_{K/\mathbb{Q}}(\mathbf{x}^2 / \tau)} for ideals \mathfrak{a} of \mathcal{O}_K, where the trace reflects the embeddings. Applying the Poisson summation formula to these functions on the adele ring yields the transformation law \theta_{\mathfrak{a}}(\tau) = (N\mathfrak{a} / |\tau|^{n/2}) \theta_{\mathfrak{a}^\vee}(-1/\tau), where \mathfrak{a}^\vee is the codifferent ideal. Summing over ideals and relating to the Mellin transform produces the gamma factors and the reflection principle, leading to the equation for \Lambda_K(s). A modern adelic proof, due to John Tate, interprets \zeta_K(s) as a local Euler product and uses integration over the ideles to derive the equation uniformly. In contrast to the Riemann zeta function \zeta(s), where n=1, r_1=1, r_2=0, and the completed form is \pi^{-s/2} \Gamma(s/2) \zeta(s), the Dedekind case generalizes with multiple gamma factors determined by the signature (r_1, r_2). The degree n = r_1 + 2r_2 influences the archimedean contribution, affecting the growth of \zeta_K(s) and the location of its pole at s=1. The functional equation induces a symmetry in the non-trivial zeros of \zeta_K(s): if \rho is a zero, then so is $1 - \overline{\rho} and $1 - \rho. Consequently, all non-trivial zeros lie in the critical strip $0 < \operatorname{Re}(s) < 1 and are symmetric with respect to the critical line \operatorname{Re}(s) = 1/2. The generalized Riemann hypothesis asserts that all non-trivial zeros lie on this line, with implications for the distribution of primes in ideals of \mathcal{O}_K.

Special values

Residue at the pole s=1

The Dedekind zeta function \zeta_K(s) of a number field K possesses a simple pole at s=1, and the residue at this pole is a fundamental arithmetic invariant expressed by the analytic class number formula:

\operatorname{Res}_{s=1} \zeta_K(s) = \frac{2^{r_1} (2\pi)^{r_2} h_K R_K}{w_K \sqrt{|d_K|}},

where r_1 denotes the number of real embeddings of K, $2r_2 the number of complex embeddings, d_K the discriminant of K, and the other terms are defined below.^[1]^[12] This formula arises from the meromorphic continuation of \zeta_K(s) and encodes deep connections between analytic and arithmetic properties of K.^[2] Here, h_K is the class number of K, which equals the order of the ideal class group of the ring of integers \mathcal{O}_K and measures the extent to which unique factorization fails in \mathcal{O}_K.^[12]^[2] The regulator R_K is the determinant of the matrix formed by the logarithms of the absolute values of the embeddings of a fundamental system of units in the unit group \mathcal{O}_K^\times, providing a volume measure for the unit lattice in the logarithmic embedding space.^[1]^[12] Finally, w_K is the number of roots of unity in \mathcal{O}_K^\times, accounting for the torsion subgroup of the unit group.^[1]^[2] The residue equals the limit \lim_{s \to 1^+} (s-1) \zeta_K(s), which converges due to the simple nature of the pole and reflects the growth of the partial sums of the Dirichlet series for \zeta_K(s) near s=1.^[12]^[2] For the quadratic field K = \mathbb{Q}(\sqrt{d}) with square-free integer d < 0 (an imaginary quadratic field), the formula simplifies because r_1 = 0, r_2 = 1, and the regulator R_K = 1 by convention, yielding

\operatorname{Res}_{s=1} \zeta_K(s) = \frac{2\pi h_K}{w_K \sqrt{|d_K|}},

where d_K = 4d if d \equiv 2,3 \pmod{4} or d_K = d if d \equiv 1 \pmod{4}.^[1]^[2] A concrete computation occurs for K = \mathbb{Q}(i), where d = -1, so d_K = -4, h_K = 1, and w_K = 4; substituting gives \operatorname{Res}_{s=1} \zeta_K(s) = \pi / 4.^[1]

Values at non-positive integers

The Dedekind zeta function \zeta_K(s) of a number field K attains rational values at all non-positive integers s = 0, -1, -2, \dots. These values vanish at negative even integers s = -2, -4, -6, \dots and, unless K is totally real, also at negative odd integers s = -1, -3, -5, \dots; the only non-vanishing cases occur for totally real K at negative odd integers, where the values are non-zero rationals.^[13]^[14] At s = 0, \zeta_K(s) has a zero of order r = r_1 + r_2 - 1, where r_1 and r_2 are the numbers of real and pairs of complex embeddings of K, respectively. The leading coefficient in the Laurent expansion is given by

\lim_{s \to 0} \frac{\zeta_K(s)}{s^r} = -\frac{h_K R_K}{w_K},

with h_K the class number of K, R_K the regulator of the unit group, and w_K the number of roots of unity in K.^[15] For imaginary quadratic fields K = \mathbb{Q}(\sqrt{-d}), where r_1 = 0 and r_2 = 1, this simplifies to r = 0 and \zeta_K(0) = -h_K / w_K, since R_K = 1 by convention for fields of unit rank zero; this directly expresses the class number in terms of the zeta value and relates to class numbers of associated real quadratic fields via the factorization \zeta_K(s) = \zeta(s) L(s, \chi_{-d}).^[16] For totally real fields, the non-zero values at negative odd integers s = -(2k-1) for k = 1, 2, \dots are rational numbers expressible in terms of generalized Bernoulli numbers of K, with explicit formulas involving products of gamma functions that generalize the classical relation \zeta(-(2k-1)) = (-1)^k B_{2k} / (2k) for the Riemann zeta function.^[17] p-adic interpretations of these values include Kummer congruences, which provide modular relations modulo primes p between \zeta_K(1 - 2m) and \zeta_K(1 - 2n) for distinct positive integers m, n, extending classical congruences for Bernoulli numbers and applicable to families of totally real fields.^[18] In cyclotomic fields, the values at s = 1 - k for positive integers k generate the Stickelberger ideals, which annihilate the p-primary parts of the class groups via the Stickelberger theorem, linking arithmetic invariants to these special values.^[19]

Connections to L-functions

Relation to Dirichlet L-functions

For an abelian extension K/\mathbb{Q}, the Dedekind zeta function \zeta_K(s) factors as a product of Dirichlet L-functions over the Dirichlet characters associated to the Galois group \mathrm{Gal}(K/\mathbb{Q}). Specifically, if K is the fixed field of a subgroup H of the group of Dirichlet characters modulo m (for the conductor m of the extension), then \zeta_K(s) = \prod_{\chi \in H} L(s, \chi), where the product runs over the characters in H and L(s, \chi) = \sum_{n=1}^\infty \chi(n) n^{-s} is the Dirichlet L-function for each primitive character \chi.^[4] This factorization arises from the orthogonality of characters and the structure of the ray class group, ensuring that the degree [K:\mathbb{Q}] equals the number of characters in H.^[4] A particularly explicit case occurs for quadratic extensions K = \mathbb{Q}(\sqrt{d}), where d is a square-free integer, the fundamental discriminant. Here, \zeta_K(s) = \zeta(s) L(s, \chi_d), with \zeta(s) the Riemann zeta function and \chi_d the quadratic Dirichlet character given by the Kronecker symbol \left( \frac{d}{\cdot} \right) modulo |d|.^[20] This decomposition reflects the splitting behavior of rational primes in the ring of integers of K: primes inert or ramified contribute factors aligned with the trivial character (via \zeta(s)), while split primes are captured by the non-trivial character \chi_d.^[4] The Euler product form of \zeta_K(s) = \prod_{\mathfrak{p}} (1 - N(\mathfrak{p})^{-s})^{-1} (over prime ideals \mathfrak{p} of \mathcal{O}_K) thus decomposes into \prod_p (1 - p^{-s})^{-1} \prod_p (1 - \chi_d(p) p^{-s})^{-1} for the quadratic case, where the local factors at each rational prime p encode the Frobenius conjugacy class via the character values, determining whether p splits, remains inert, or ramifies.^[20] This mirrors the prime factorization in the Euler products of the individual L-functions.^[4] Historically, this relation originated with Dirichlet's 1837 introduction of L(s, \chi_d) to study the distribution of primes in arithmetic progressions and class numbers of quadratic fields, later extended by Dedekind in 1877 to general number fields through his definition of the zeta function via ideals.^[10] Dedekind's generalization built on Dirichlet's ideas, replacing sums over integers with sums over ideals to handle non-principal ideals in the ring of integers.^[10] This factorization has key implications for the distribution of primes in quadratic fields: the pole of \zeta_K(s) at s=1 arises solely from \zeta(s), while the non-vanishing of L(1, \chi_d) \neq 0 (due to Dirichlet) ensures an asymptotic prime-counting formula \pi_K(x) \sim \mathrm{Li}(x) for the number of prime ideals of norm up to x, generalizing the prime number theorem to K.

Links to Artin and Hecke L-functions

The Dedekind zeta function of a Galois extension K/F of number fields admits a factorization into Artin L-functions associated to the irreducible representations of the Galois group \mathrm{Gal}(K/F). Specifically, if G = \mathrm{Gal}(K/F), then

\zeta_K(s) = \zeta_F(s) \prod_{\rho \in \widehat{G}, \rho \neq 1} L(s, \rho)^{ \dim \rho },

where \widehat{G} denotes the set of irreducible complex representations of G, \zeta_F(s) is the Dedekind zeta function of the base field F, and L(s, \rho) is the Artin L-function attached to \rho.^[21] This decomposition generalizes the abelian case and was established by Artin in 1923 using density arguments for prime ideals.^[21] Artin's holomorphy conjecture asserts that each non-trivial irreducible Artin L-function L(s, \rho) is entire (holomorphic everywhere in the complex plane).^[21] The conjecture holds in the abelian case by class field theory, where Artin L-functions reduce to Hecke L-functions (or Dirichlet L-functions over \mathbb{Q}) that are known to be entire except for possible poles at s=1 only for the trivial character.^[5] It is also proven for solvable Galois groups and specific non-solvable cases, including tetrahedral representations (isomorphic to A_4) via Langlands' work using automorphic forms and octahedral representations via Tunnell's extension of those methods.^[21] The general case remains open, with no known counterexamples, but refinements through the Langlands program predict that each L(s, \rho) corresponds to an automorphic L-function on \mathrm{GL}_n(\mathbb{A}_F), where n = \dim \rho, ensuring holomorphy and providing a deeper reciprocity law.^[21] Hecke L-functions provide another class of L-functions linked to the Dedekind zeta function, defined via grossencharacters (or Hecke characters) \psi, which are continuous homomorphisms from the idele class group of K to \mathbb{C}^\times satisfying a congruence condition modulo a fixed modulus.^[5] The associated L-function is given by the Euler product

L(s, \psi) = \prod_{\mathfrak{p}} \left(1 - \psi(\mathfrak{p}) N(\mathfrak{p})^{-s}\right)^{-1},

where the product runs over prime ideals \mathfrak{p} of the ring of integers of K unramified for \psi, and N(\mathfrak{p}) is the norm of \mathfrak{p}.^[5] The Dedekind zeta function \zeta_K(s) corresponds precisely to the Hecke L-function for the trivial grossencharacter \psi = 1.^[5] For a finite abelian extension L/K, class field theory yields the factorization

\zeta_L(s) = \prod_{\chi} L(s, \chi),

where the product is over all grossencharacters \chi of L (equivalently, characters of \mathrm{Gal}(L/K)).^[5] More generally, Hecke L-functions connect to automorphic representations on \mathrm{GL}_1(\mathbb{A}_K) via the Langlands correspondence, bridging to higher-rank groups in the non-abelian setting.^[21] An illustrative example is the cyclotomic field K = \mathbb{Q}(\zeta_m) for a positive integer m, where the Galois group \mathrm{Gal}(K/\mathbb{Q}) is isomorphic to (\mathbb{Z}/m\mathbb{Z})^\times. Here, \zeta_K(s) decomposes as

\zeta_K(s) = \prod_{\chi \bmod m} L(s, \chi),

with the product over all Dirichlet characters \chi modulo m, which coincide with the Hecke characters of K in this rational base case.^[1] This abelian factorization underscores the role of Hecke L-functions in explicitly resolving the Dedekind zeta function for cyclotomic extensions.^[1]

Arithmetic applications

Role in the class number formula

The analytic class number formula provides an explicit expression for the class number h_K of the ring of integers of a number field K in terms of the residue of its Dedekind zeta function \zeta_K(s) at s = 1:

h_K = \frac{w_K \sqrt{|d_K|}}{2^{r_1} (2\pi)^{r_2} R_K} \cdot \Res_{s=1} \zeta_K(s),

where r_1 and r_2 are the numbers of real and pairs of complex embeddings of K, w_K is the number of roots of unity in K, d_K is the discriminant of K, and R_K is the regulator of the unit group of K.^[4] This formula links the arithmetic invariants of K to the analytic behavior of \zeta_K(s) near its simple pole at s=1. The formula originated with Dedekind's work in 1877, where he computed the residue as a limit from the right for number fields using density estimates of ideals, without full analytic continuation.^[22] Landau extended this in 1903 by establishing the meromorphic continuation of \zeta_K(s) near s=1, confirming the residue interpretation for general number fields.^[22] Hecke completed the general proof in 1917, incorporating the functional equation and full meromorphic continuation to derive the formula in its modern form.^[22] A sketch of the proof begins with the Euler product representation of \zeta_K(s), leading to the behavior of its logarithmic derivative near s=1:

\frac{\zeta_K'(s)}{\zeta_K(s)} \sim \sum_{\mathfrak{p}} \frac{\log N(\mathfrak{p})}{N(\mathfrak{p})^s - 1},

where the sum is over prime ideals \mathfrak{p} of the ring of integers of K. This approximates the pole contribution, and integrating or using partial summation ties it to the asymptotic count of ideals of bounded norm, T(x) \sim \kappa x as x \to \infty, with \kappa the residue. Dirichlet's theorem on the density of prime ideals in ideal classes then connects this to the class number via the structure of the ideal class group.^[4] Effective versions of the formula yield bounds on h_K from zero-free regions of \zeta_K(s). For instance, zero-free regions away from \Re(s) = 1 imply upper bounds on the residue, hence on h_K. Siegel's theorem (1935) provides a key ineffective lower bound for imaginary quadratic fields: h_K \gg |d_K|^{1/2 - \epsilon} for any \epsilon > 0, using subconvexity estimates near s=1 to control exceptions in the class number formula.^[23] Applications include the Heegner–Stark–Baker method, which resolves the class number one problem for imaginary quadratic fields by combining the formula with modular forms and values of L-functions (factoring \zeta_K(s)) to show there are exactly nine such fields.^[24]

Arithmetically equivalent fields

Two number fields K and L are arithmetically equivalent if their Dedekind zeta functions are identical, that is, \zeta_K(s) = \zeta_L(s) for all complex s.^[25] This equality necessitates that K and L share the same degree over \mathbb{Q}, the same absolute discriminant, the same signature (comprising the number of real and pairs of complex embeddings), and the same number of roots of unity.^[26] Additionally, the unit rank is identical, as it is determined by the signature.^[27] The implications extend to arithmetic invariants derived from the zeta function: arithmetically equivalent fields have the same ideal class number h, regulator R, and count of roots of unity \mu, ensuring h R / \mu matches via the analytic class number formula, though individual h and R may differ if compensated accordingly.^[25] They also exhibit identical prime ideal norms, splitting laws over \mathbb{Q}, and overall prime distribution, reflecting the Euler product's uniqueness in encoding ramification and inertia.^[27] Non-isomorphic examples arise from Gassmann triples, group-theoretic configurations yielding equivalent zeta functions; the earliest such constructions, due to Gassmann in the 1930s, appear in degree 7, with no non-isomorphic pairs in degrees less than 7.^[28] For instance, in degree 8, pairs like

\mathbb{Q}(\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=8&&&citation_type=wikipedia}}{a})

and

\mathbb{Q}(\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=8&&&citation_type=wikipedia}}{16a})

for suitable a (e.g., a=3) provide non-isomorphic fields with matching zeta functions, class number 1, and discriminant -2^{24} \cdot 3^7.^[27] Similar examples exist in higher degrees, often involving radical extensions or fields with specific Galois closures.^[25] Classification is complete for abelian extensions, where equivalence implies isomorphism, but remains partial for non-abelian cases, relying on enumerating Gassmann triples up to conjugation.^[26] For degrees up to 15, possible class number ratios between equivalent pairs are restricted to prime powers dividing explicit bounds (e.g., $2^{14} \cdot 3^6 \cdot 5^3).^[27] Post-2000 computational efforts, including searches via Magma and the L-functions and Modular Forms Database (LMFDB), confirm the rarity of such pairs beyond degree 2, with finitely many identified up to degree 32 and open questions on infinite families in solvable but non-abelian settings.^[25]

References

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[10]
A history of L-functions (reviewed) - LMFDB
May 10, 2016 · Dedekind zeta functions In 1877 Dedekind began generalizing some of Dirichlet's work to number fields. His first paper on the topic is Über die ...
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[PDF] dedekind.pdf - andrew.cmu.ed
Apr 13, 2005 · Reprinted in (Dedekind, 1968), Volume 1, Chapter XV, pages 202–237. Richard Dedekind. Über die Theorie der ganzen algebraischen Zahlen. Sup ...
[12]
[PDF] Introduction to L-functions: Dedekind zeta functions
Artin L-functions: Properties. (1) L(s,ρ) converges absolutely and uniformly for Re(s) > 1. (2) If (ρ,V) is the trivial representation, then. L(s,ρ) = ζK (s).
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[PDF] A note on values of the Dedekind zeta-function at odd positive integers
Feb 25, 2021 · The Dedekind zeta function satisfies a functional equation in the same spirit as the Riemann zeta-function, namely,. ξK(s) = ξK(1 − s) ...
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Numerical Evaluation at Negative Integers of the Dedekind Zeta ...
It is known that the values ζ K (-n) of the Dedekind zeta function ζ K (s) of K are rational numbers for all non-negative integers n≥ 1. We develop a ...
[15]
Introduction - American Mathematical Society
ζK(s) sr1+r2−1. = −. hKRK. wK. , where r1 is the number of real embeddings of K, r2 is the number of conjugate pairs of complex embeddings of K, wK is ...
[16]
[PDF] arXiv:2105.04141v1 [math.NT] 10 May 2021
May 10, 2021 · In this article, we study special values of the Dedekind zeta function over an imaginary quadratic field. The values of the Dedekind zeta ...
[17]
values of Dedekind zeta functions of real quadratic number fields at ...
Mar 22, 2013 · values of Dedekind zeta functions of real quadratic number fields at negative integers. Let K K be a real quadratic number field of discriminant ...
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[PDF] On Some Congruences of Zeta and L–values at Negative Odd Integers
Abstract. In this thesis, we establish congruences for values of Dedekind Zeta func- tions attached to a specific family of totally real fields.
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A Kummer Congruence for the Hurwitz-Herglotz Function .
Facts about certain zeta functions. In the proof of Theorem 1.4 it will be important to calculate q- expansions. The following facts will beneeded. For proofs ...<|separator|>
[20]
[PDF] Kimball Martin - 3 Zeta and L-functions
... Dirichlet L-functions.) ⇤Riemann's zeta function was studied before Riemann ... C} denote the space of complex valued functions from a finite abelian group G into ...
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[PDF] On Artin L-functions - OSU Math
The Weber L-functions also figured into the factorization of the Dedekind zeta function of a relative abelian extension K/k through characters of the ...
[22]
History of the analytic class number formula - MathOverflow
Sep 9, 2014 · The (general) analytic class number formula gives a value for the residue of the Dedekind zeta function of a number field at the point s=1.Dedekind Zeta function: behaviour at 1 - MathOverflowDerivative of a function related to Dedekind zeta functionMore results from mathoverflow.net<|control11|><|separator|>
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[PDF] class numbers of quadratic fields
CLASS NUMBERS OF QUADRATIC FIELDS. 5. Dedekind zeta function of Q(. √ d), that h(p) > 1 for p > 1013. In [6], Biro provided an unconditional proof that h(4m2 ...
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[PDF] Class numbers of imaginary quadratic fields Mark Watkins ...
The first complete results were for N = 1 by Heegner, Baker, and Stark. ... The Dedekind zeta function ζ−d(s) of Q(√−d) is the product of ζ(s) and L(s ...
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[PDF] Stronger arithmetic equivalence - arXiv
In addition to having the same Dedekind zeta function (the usual notion of arithmetic equivalence), number fields that are equivalent in any of these stronger ...
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[PDF] ARITHMETIC EQUIVALENCE AND ISOSPECTRALITY Let K be a ...
The Dedekind zeta function encodes many features of the number field K: it has a simple pole at s = 1 whose residue is intimately related to several invariants ...
[27]
[PDF] On arithmetically equivalent number fields of small degree
{ 1 4 , 1 3 , 1 2, 1, 2, 3, 4}. The first known instances of pairs of arithmetically equivalent number fields with different class numbers were generated using ...
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[PDF] A New Condition for Arithmetic Equivalence.
Today we say that Gassmann gave the first example of non-isomorphic arithmetically equivalent fields. Page 12. 2. Perlis [PI] and Komatsu [Kol] have studied ...