Dedekind zeta function

Dedekind zeta function
Name	Dedekind zeta function
Field	Number theory
Introduced	19th century
Related	Riemann zeta function, Hecke L-series, Artin L-functions

Dedekind zeta function is a complex analytic object attached to a number field that generalizes the Riemann zeta function and encodes arithmetic of algebraic integers, ideals, and class groups. It was developed in the context of algebraic number theory by Richard Dedekind and subsequently studied by figures such as Ernst Kummer, Leopold Kronecker, and Heinrich Weber, and it connects to major conjectures and results involving Bernhard Riemann, André Weil, David Hilbert, and Emil Artin. The function plays a central role in class field theory, the Hilbert class field, and modern approaches to Langlands program-related reciprocity.

Definition and basic properties

For a number field K of degree n over Q, the Dedekind zeta function ζ_K(s) is initially defined for Re(s)>1 by a Dirichlet series summing over nonzero integral ideals of the ring of integers O_K, or equivalently via an Euler product over nonzero prime ideals. Its abscissa of absolute convergence equals 1, mirroring properties discovered for the Riemann zeta function by Bernhard Riemann and refined by Karl Weierstrass. The residue at s=1 relates to arithmetic invariants articulated by Dirichlet and formalized by Heinrich Weber and Leopold Kronecker in early class number investigations. Dedekind’s work influenced later contributions from Emil Artin and Emil Abel, and scholars like Ernst Eduard Kummer studied the interplay with cyclotomic fields and the Kummer–Vandiver conjecture.

Analytic continuation and functional equation

ζ_K(s) admits analytic continuation to a meromorphic function on the complex plane with a simple pole at s=1, a fact obtained through methods developed by Hecke and refined in the work of Erich Hecke, Goro Shimura, Atle Selberg, and André Weil. The completed Dedekind zeta function satisfies a functional equation involving the discriminant of K, the number of real and complex embeddings studied by David Hilbert in his Hilbert class field program, and Γ-factors related to the Gamma function as used by Adrien-Marie Legendre and Leonhard Euler. Results by John Tate and expositions in the framework of the Langlands program connect the analytic continuation to automorphic representations considered by Robert Langlands and James Arthur.

Euler product and relation to prime ideals

For Re(s)>1, ζ_K(s) factors as an Euler product over prime ideals 𝔭 of O_K, ζ_K(s)=∏_{𝔭}(1−N(𝔭)^{−s})^{−1}, reflecting the multiplicative structure of ideals analogous to the Unique factorization investigations of Richard Dedekind and investigations into prime splitting by Ernst Kummer and Leopold Kronecker. The norms N(𝔭) correlate with rational primes p and Frobenius elements in Galois extensions described by Évariste Galois and generalized through Emil Artin’s reciprocity. Chebotarev density theorem, proven after work by Niels Henrik Abel’s contemporaries and formalized by Nikolai Chebotaryov and Helmut Hasse, describes the distribution of primes in residue classes via the behavior of ζ_K(s) and its L-factor decompositions into Artin L-functions and Hecke L-series.

Special values and class number formula

The residue and special values of ζ_K(s) at s=1 and at nonpositive integers are governed by the analytic class number formula first formulated in settings by Dirichlet and extended by Heinrich Weber and Emil Artin. The class number h_K, regulator R_K, number of roots of unity w_K, and discriminant Δ_K appear in the leading term at s=1; these invariants were central to investigations by Leopold Kronecker, Heinrich Weber, and David Hilbert in his Zahlbericht. Special values at negative integers relate to algebraic K-theory and results conjectured by Bernard Dwork and John Tate, with concrete computations influenced by Kurt Mahler and explicit formulas used by D. H. Lehmer and Alan Baker. Stark conjectures linking derivatives of L-functions at s=0 to units in abelian extensions were proposed by Harold Stark and pursued by N. David Elkies and Don Zagier among others.

Zeros, conjectures and L-functions relation

Zeros of ζ_K(s) satisfy a functional symmetry and the nontrivial zeros lie in the critical strip 0Bernhard Riemann in his 1859 memoir. The Generalized Riemann Hypothesis for Dedekind zeta functions, posited in formulations influenced by Atle Selberg and Alan Turing’s computational verifications, asserts all nontrivial zeros have real part 1/2 and is connected to consequences considered by Enrico Bombieri and Hugh Montgomery in pair correlation studies. Relations between ζ_K(s) and automorphic L-functions via Artin reciprocity and the Langlands correspondence tie Dedekind zeta zeros to eigenvalues appearing in work by Andrew Wiles and Richard Taylor on modularity and to trace formula techniques developed by James Arthur and Robert Langlands.

Examples and computational aspects

For K=Q, ζ_K(s) reduces to the Riemann zeta function, whose values at negative integers were evaluated by Leonhard Euler; for quadratic fields studied by Carl Friedrich Gauss and Adrien-Marie Legendre, explicit class number computations trace back to Gauss’s Disquisitiones Arithmeticae and modern algorithms by H. Rademacher and C. L. Siegel. Computational methods for ζ_K(s) and related L-series utilize techniques from John von Neumann-era numerical analysis, the fast Fourier transform usage popularized by James Cooley and John Tukey, and rigorous interval arithmetic as applied by Alan Turing and Konrad Zuse-era computing. Software packages developed in environments influenced by projects at IBM, Microsoft Research, and academic groups at Princeton University and University of Cambridge implement algorithms from Michael Pohst and H. Cohen to compute regulators, discriminants, and zeros numerically, enabling explicit verification of conjectures for many number fields.

Category:Zeta functions