Selberg class
E246701
The Selberg class is a collection of Dirichlet series with specific analytic properties introduced to generalize and axiomatize L-functions in number theory.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Selberg class canonical | 2 |
| Selberg orthonormality conjecture | 1 |
| grand Riemann hypothesis | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2252109 — 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 class Context triple: [Atle Selberg, knownFor, Selberg class]
-
A.
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.
-
B.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
C.
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.
-
D.
Multiplicative Number Theory
Multiplicative Number Theory is a branch of analytic number theory that studies arithmetic functions and prime number distributions through their multiplicative properties and associated Dirichlet series.
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E.
A. Ivić, The Riemann Zeta-Function
"A. Ivić, The Riemann Zeta-Function" is a comprehensive monograph on the analytic theory of the Riemann zeta function, widely regarded as a standard modern reference in analytic number theory.
- 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 class Target entity description: The Selberg class is a collection of Dirichlet series with specific analytic properties introduced to generalize and axiomatize L-functions in number theory.
-
A.
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.
-
B.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
C.
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.
-
D.
Multiplicative Number Theory
Multiplicative Number Theory is a branch of analytic number theory that studies arithmetic functions and prime number distributions through their multiplicative properties and associated Dirichlet series.
-
E.
A. Ivić, The Riemann Zeta-Function
"A. Ivić, The Riemann Zeta-Function" is a comprehensive monograph on the analytic theory of the Riemann zeta function, widely regarded as a standard modern reference in analytic number theory.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
class of Dirichlet series
ⓘ
concept in analytic number theory ⓘ |
| appearsIn |
literature on Selberg’s conjectures on L-functions
ⓘ
literature on generalized Riemann hypothesis ⓘ |
| contains |
Dirichlet L-functions
ⓘ
L-functions ⓘ
surface form:
Hecke L-functions
Riemann zeta function ⓘ automorphic L-functions under suitable conditions ⓘ |
| context | axiomatic theory of L-functions ⓘ |
| definedAs | collection of Dirichlet series satisfying specific axioms ⓘ |
| field | number theory ⓘ |
| generalizes |
Dedekind zeta functions
ⓘ
classical Dirichlet L-functions ⓘ |
| hasAxiom |
Dirichlet series representation
ⓘ
Euler product formula for the Riemann zeta function ⓘ
surface form:
Euler product
Ramanujan hypothesis type growth condition ⓘ analytic continuation ⓘ functional equation ⓘ |
| hasConjecture |
Selberg class
self-linksurface differs
ⓘ
surface form:
Selberg orthonormality conjecture
degree conjecture for elements of the Selberg class ⓘ |
| hasInvariant |
conductor of an L-function
ⓘ
degree of an L-function ⓘ |
| hasProperty |
Euler product has local factors of polynomial type in p^{-s}
ⓘ
closed under multiplication of L-functions ⓘ closed under taking Dirichlet series quotients in some formulations ⓘ coefficients satisfy polynomial growth conditions ⓘ each element admits meromorphic continuation to the complex plane ⓘ each element has an Euler product expansion ⓘ each element is a Dirichlet series absolutely convergent in some right half-plane ⓘ each element satisfies a functional equation relating s and 1−s ⓘ elements satisfy certain growth bounds in vertical strips ⓘ |
| introducedBy | Atle Selberg ⓘ |
| introducedInContextOf | L-functions ⓘ |
| namedAfter | Atle Selberg ⓘ |
| purpose |
to axiomatize L-functions
ⓘ
to generalize classical L-functions ⓘ |
| relatedTo |
Dirichlet characters
ⓘ
Euler products ⓘ automorphic representations ⓘ functional equations of L-functions ⓘ modular forms ⓘ |
| studiedFor |
distribution of zeros of L-functions
ⓘ
value distribution of L-functions ⓘ |
| subfield | analytic number theory ⓘ |
| topicOf | research in analytic number theory ⓘ |
| usedFor |
formulating generalized Riemann hypothesis
ⓘ
studying zeros of L-functions ⓘ unifying different types of L-functions ⓘ |
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 class Description of subject: The Selberg class is a collection of Dirichlet series with specific analytic properties introduced to generalize and axiomatize L-functions in number theory.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
grand Riemann hypothesis
this entity surface form:
Selberg orthonormality conjecture