6.3 Dedekind zeta functions

Dedekind zeta functions generalize the Riemann zeta function to number fields, encoding arithmetic information about rings of integers. They're crucial for studying ideal class groups and unit groups, extending key properties like analytic continuation and functional equations.

Initially defined for complex numbers with real part > 1, Dedekind zeta functions sum over nonzero ideals in the ring of integers. They have a simple pole at s = 1 and zeros in the critical strip, connecting to important arithmetic invariants like class numbers and discriminants.

Definition of Dedekind zeta functions

• Fundamental tool in algebraic number theory generalizes the Riemann zeta function to number fields
• Encodes arithmetic information about the ring of integers in a number field
• Plays crucial role in studying ideal class groups and unit groups of number fields

Relation to Riemann zeta function

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• Extends Riemann zeta function from rational numbers to arbitrary algebraic number fields
• Reduces to Riemann zeta function when applied to the field of rational numbers
• Incorporates additional factors related to the structure of the number field's ring of integers
• Shares many properties with Riemann zeta function (analytic continuation, functional equation)

Domain of definition

• Initially defined for complex numbers with real part greater than 1
• Expressed as a sum over nonzero ideals of the ring of integers in the number field
• Formula: $\zeta_K(s) = \sum_{I \neq 0} \frac{1}{N(I)^s}$, where I ranges over nonzero ideals and N(I) denotes the norm of the ideal
• Converges absolutely in the half-plane Re(s) > 1

Analytic continuation

• Extends to a meromorphic function on the entire complex plane
• Uses techniques similar to those employed for the Riemann zeta function (contour integration, Mellin transforms)
• Reveals important arithmetic properties of the number field through its behavior in the extended domain
• Allows study of special values at points where the original series diverges

Properties of Dedekind zeta functions

• Encode deep arithmetic information about number fields and their rings of integers
• Share many characteristics with the Riemann zeta function while incorporating field-specific data
• Serve as a bridge between analytic and algebraic aspects of number theory

Functional equation

• Relates values of ζK(s) to values of ζK(1-s)
• Involves factors dependent on the discriminant and degree of the number field
• General form: $\zeta_K(1-s) = \frac{|d_K|^{s-1/2}}{(2\pi)^{n_Ks}} \Gamma(s)^{r_1+r_2} \Gamma(s/2)^{r_2} \zeta_K(s)$
• Crucial for understanding behavior in the critical strip 0 < Re(s) < 1

Meromorphic continuation

• Extends ζK(s) to a meromorphic function on the entire complex plane
• Achieved through various methods (Hecke's proof, integral representations)
• Reveals additional poles and zeros beyond the original domain of definition
• Allows study of special values and residues at important points

Location of poles

• Simple pole at s = 1, analogous to the Riemann zeta function
• Residue at s = 1 related to important arithmetic invariants of the number field
• No other poles in the complex plane due to meromorphic continuation
• Zeros in the critical strip conjectured to lie on the critical line Re(s) = 1/2 (Generalized Riemann Hypothesis)

Arithmetic significance

• Dedekind zeta functions encode fundamental arithmetic properties of number fields
• Provide a powerful tool for studying ideal class groups, units, and other algebraic structures
• Connect analytic properties to algebraic invariants of number fields

Connection to ideal class groups

• Class number appears in the residue of ζK(s) at s = 1
• Reflects the extent to which unique factorization fails in the ring of integers
• Formula: $\lim_{s \to 1} (s-1)\zeta_K(s) = \frac{2^{r_1}(2\pi)^{r_2}hR}{w\sqrt{|d_K|}}$, where h denotes the class number
• Allows indirect computation of class numbers through analytic means

Relation to discriminants

• Discriminant of the number field appears in the functional equation
• Affects the growth rate of the Dedekind zeta function in certain regions
• Larger discriminants generally correspond to more complex arithmetic in the number field
• Impacts the distribution of zeros and the behavior near s = 1

Role in class number formula

• Expresses the residue at s = 1 in terms of various arithmetic invariants
• Involves class number, regulator, number of roots of unity, and discriminant
• Provides a deep connection between analytic and algebraic properties of the number field
• Used to derive bounds and estimates for class numbers and related quantities

Factorization of Dedekind zeta functions

• Dedekind zeta functions admit factorizations that reveal their arithmetic structure
• Decompositions relate to prime ideals and local properties of the number field
• Connect Dedekind zeta functions to other important L-functions in number theory

Euler product representation

• Expresses ζK(s) as an infinite product over prime ideals of the ring of integers
• Formula: $\zeta_K(s) = \prod_{\mathfrak{p}} \frac{1}{1 - N(\mathfrak{p})^{-s}}$, where p ranges over prime ideals
• Reflects the unique factorization of ideals in the ring of integers
• Converges absolutely for Re(s) > 1, mirroring the series representation

Local factors

• Each prime ideal contributes a factor to the Euler product
• Local factors encode information about how rational primes split in the number field
• Form: $(1 - N(\mathfrak{p})^{-s})^{-1} = (1 - p^{-fs})^{-1}$, where f denotes the inertia degree
• Determine the behavior of the zeta function at each prime

Artin L-functions vs Dedekind zeta functions

• Dedekind zeta functions factor into products of Artin L-functions
• Reflects decomposition of the regular representation of the Galois group
• Formula: $\zeta_K(s) = \zeta(s) \prod_{\chi \neq 1} L(s, \chi)$, where χ ranges over non-trivial irreducible characters
• Connects Dedekind zeta functions to representation theory of Galois groups