Dedekind zeta functions
E262117
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Dedekind zeta functions canonical | 2 |
| Dedekind zeta function | 1 |
| 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.
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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.
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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.
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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.
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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 ⓘ L-functions ⓘ
surface form:
Hecke 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
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Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Dedekind zeta functions Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Dedekind zeta function for number fields
subject surface form:
L-function
this entity surface form:
Dedekind zeta function