Dedekind zeta functions

E262117

Dirichlet series L-function number-theoretic function

Dedekind zeta functions are number-theoretic functions attached to algebraic number fields that encode their arithmetic properties, such as the distribution of prime ideals and class numbers.

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All labels observed (3)

Label	Occurrences
Dedekind zeta function	4
Dedekind zeta functions canonical	3
Dedekind zeta function for number fields	1

How this entity was disambiguated

This entity first appeared as the object of triple T2394174 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Dedekind zeta functions
Context triple: [Riemann zeta function, generalization, Dedekind zeta functions]

A. Dirichlet L-functions
Dirichlet L-functions are complex analytic functions built from Dirichlet characters that generalize the Riemann zeta function and play a central role in number theory, particularly in the study of primes in arithmetic progressions.
B. Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
C. Selberg class
The Selberg class is a collection of Dirichlet series with specific analytic properties introduced to generalize and axiomatize L-functions in number theory.
D. Hilbert’s twelfth problem
Hilbert’s twelfth problem is one of David Hilbert’s famous list of 23 problems, asking for a general explicit class field theory that would generate all abelian extensions of a given number field using special values of analytic functions.
E. Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Dedekind zeta functions
Target entity description: Dedekind zeta functions are number-theoretic functions attached to algebraic number fields that encode their arithmetic properties, such as the distribution of prime ideals and class numbers.

A. Dirichlet L-functions
Dirichlet L-functions are complex analytic functions built from Dirichlet characters that generalize the Riemann zeta function and play a central role in number theory, particularly in the study of primes in arithmetic progressions.
B. Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
C. Selberg class
The Selberg class is a collection of Dirichlet series with specific analytic properties introduced to generalize and axiomatize L-functions in number theory.
D. Hilbert’s twelfth problem
Hilbert’s twelfth problem is one of David Hilbert’s famous list of 23 problems, asking for a general explicit class field theory that would generate all abelian extensions of a given number field using special values of analytic functions.
E. Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
F. None of above. chosen

Statements (49)

Predicate	Object
instanceOf	Dirichlet series ⓘ L-function ⓘ number-theoretic function ⓘ
appearsIn	analytic class number formula ⓘ proofs of Dirichlet unit theorem ⓘ proofs of finiteness of class number ⓘ
associatedWith	ring of integers of a number field ⓘ
conjecturallySatisfies	generalized Riemann hypothesis for number fields ⓘ
definedBy	Dirichlet series over nonzero ideals of the ring of integers of a number field ⓘ
definedOn	algebraic number field ⓘ
dependsOn	degree of the number field ⓘ discriminant of the number field ⓘ signature of the number field ⓘ
domain	complex plane ⓘ
encodes	arithmetic properties of number fields ⓘ class numbers ⓘ discriminant of a number field ⓘ distribution of prime ideals ⓘ unit group information ⓘ
extendedBy	analytic continuation to all complex s except a pole ⓘ
fieldOfStudy	algebraic number theory ⓘ analytic number theory ⓘ
generalizes	Riemann zeta function ⓘ
hasConvergenceRegion	Re(s) > 1 ⓘ
hasEulerProduct	product over prime ideals ⓘ
hasInvariant	residue at s = 1 ⓘ
hasPoleAt	s = 1 ⓘ
hasProperty	analytic continuation ⓘ functional equation ⓘ meromorphic function of s ⓘ
hasVariable	complex variable s ⓘ
hasZeroType	nontrivial zeros ⓘ trivial zeros ⓘ
namedAfter	Richard Dedekind ⓘ
orderOfPoleAt	1 at s = 1 ⓘ
relatedTo	Artin L-functions ⓘ Chebotarev density theorem ⓘ Hecke L-functions ⓘ linked to: L-functions class number formula ⓘ prime ideal theorem ⓘ
satisfies	Euler product over prime ideals of the ring of integers ⓘ
specialCase	Riemann zeta function for the rational field ⓘ
usedIn	algebraic number theory ⓘ class field theory ⓘ distribution of splitting of primes in extensions ⓘ study of discriminants ⓘ
usedToDefine	class number of a number field ⓘ regulator of a number field ⓘ residue at s = 1 expressing class number and regulator ⓘ

How these facts were elicited