Selberg zeta function
E865101
The Selberg zeta function is an analytic function associated with the lengths of closed geodesics on a Riemannian manifold, playing a central role in spectral theory and the study of automorphic forms.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Selberg zeta function canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T10462031 — 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: Selberg zeta function Context triple: [Selberg trace formula, relatedTo, Selberg zeta function]
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A.
Selberg trace formula
The Selberg trace formula is a fundamental result in analytic number theory and spectral theory that relates lengths of closed geodesics on a Riemannian manifold to the spectrum of its Laplace operator, serving as a non-abelian analogue of the Poisson summation formula.
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B.
Dedekind zeta functions
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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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.
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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E.
L-functions
L-functions are complex analytic functions, often arising from number theory and algebraic geometry, that encode deep arithmetic information and generalize the Riemann zeta function.
- 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: Selberg zeta function Target entity description: The Selberg zeta function is an analytic function associated with the lengths of closed geodesics on a Riemannian manifold, playing a central role in spectral theory and the study of automorphic forms.
-
A.
Selberg trace formula
The Selberg trace formula is a fundamental result in analytic number theory and spectral theory that relates lengths of closed geodesics on a Riemannian manifold to the spectrum of its Laplace operator, serving as a non-abelian analogue of the Poisson summation formula.
-
B.
Dedekind zeta functions
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.
-
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.
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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E.
L-functions
L-functions are complex analytic functions, often arising from number theory and algebraic geometry, that encode deep arithmetic information and generalize the Riemann zeta function.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
analytic function
ⓘ
object in analytic number theory ⓘ object of spectral theory ⓘ zeta function ⓘ |
| appearsIn |
Selberg’s work on harmonic analysis on locally symmetric spaces
ⓘ
theory of rank-one locally symmetric spaces ⓘ |
| associatedWith |
Fuchsian groups
ⓘ
linked to:
Fuchsian group
Laplace–Beltrami operator ⓘ
linked to:
Laplace operator
Riemannian manifold ⓘ
linked to:
Riemannian manifolds
closed geodesics ⓘ cocompact Fuchsian groups ⓘ cofinite Fuchsian groups ⓘ discrete groups of isometries ⓘ hyperbolic surfaces ⓘ spectrum of the Laplacian ⓘ |
| definedFor | Re(s) sufficiently large ⓘ |
| definedOn |
compact hyperbolic surfaces
ⓘ
finite-area hyperbolic surfaces ⓘ quotients of hyperbolic plane by Fuchsian groups ⓘ |
| encodes |
length spectrum of closed geodesics
ⓘ
multiplicities of closed geodesics ⓘ primitive closed geodesics ⓘ |
| field |
analytic number theory
ⓘ
automorphic forms ⓘ differential geometry ⓘ spectral theory ⓘ |
| generalizationOf | Riemann zeta function in geometric setting ⓘ |
| hasDomain | complex plane ⓘ |
| hasEulerProduct | over primitive closed geodesics ⓘ |
| hasVariable | complex variable s ⓘ |
| namedAfter | Atle Selberg ⓘ |
| property |
admits meromorphic continuation to the complex plane
ⓘ
logarithmic derivative appears in Selberg trace formula ⓘ satisfies functional equation ⓘ zeros encode spectral data of Laplacian ⓘ zeros related to eigenvalues of Laplace–Beltrami operator ⓘ |
| relatedTo |
Eisenstein series
ⓘ
Maass forms ⓘ Ruelle zeta function ⓘ Selberg trace formula ⓘ automorphic Laplacian ⓘ automorphic representations ⓘ dynamical zeta function ⓘ prime geodesic theorem ⓘ |
| usedIn |
inverse spectral problems
ⓘ
proofs of prime geodesic theorem ⓘ quantum chaos on hyperbolic surfaces ⓘ study of automorphic spectra ⓘ study of resonances on hyperbolic manifolds ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
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: Selberg zeta function Description of subject: The Selberg zeta function is an analytic function associated with the lengths of closed geodesics on a Riemannian manifold, playing a central role in spectral theory and the study of automorphic forms.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
L-functions