generalized Riemann hypothesis
E259754
The generalized Riemann hypothesis is a major unproven conjecture in number theory asserting that the nontrivial zeros of all Dirichlet L-functions lie on a critical line in the complex plane, extending the classical Riemann hypothesis.
All labels observed (3)
| Label | Occurrences |
|---|---|
| generalized Riemann hypothesis canonical | 2 |
| Generalized Riemann Hypothesis for Dirichlet L-functions | 1 |
| extended Riemann hypothesis | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2364370 — 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: generalized Riemann hypothesis Context triple: [Riemann hypothesis, hasGeneralization, generalized Riemann hypothesis]
-
A.
Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
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B.
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.
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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.
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.
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E.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
- 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: generalized Riemann hypothesis Target entity description: The generalized Riemann hypothesis is a major unproven conjecture in number theory asserting that the nontrivial zeros of all Dirichlet L-functions lie on a critical line in the complex plane, extending the classical Riemann hypothesis.
-
A.
Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
-
B.
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.
-
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.
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.
-
E.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
conjecture in number theory
ⓘ
mathematical conjecture ⓘ unproven hypothesis ⓘ |
| alsoKnownAs | GRH ⓘ |
| appliesTo |
Dirichlet L-functions
ⓘ
surface form:
Dirichlet L-functions modulo q
Dirichlet characters ⓘ primitive Dirichlet characters ⓘ |
| asserts | all nontrivial zeros of Dirichlet L-functions lie on the critical line Re(s) = 1/2 ⓘ |
| concerns |
critical strip 0 < Re(s) < 1
ⓘ
location of zeros of L-functions ⓘ |
| criticalLine | Re(s) = 1/2 ⓘ |
| domain | complex plane ⓘ |
| extends | Riemann hypothesis ⓘ |
| field | number theory ⓘ |
| generalizes | Riemann zeta function case ⓘ |
| hasConsequence |
sharper bounds in many arithmetic counting functions
ⓘ
zero-free regions for Dirichlet L-functions off the critical line ⓘ |
| implies |
bounds on least prime in an arithmetic progression
ⓘ
bounds on least quadratic nonresidue modulo a prime ⓘ improved error terms in the prime number theorem for arithmetic progressions ⓘ results on distribution of primes in residue classes ⓘ strong bounds on prime numbers in arithmetic progressions ⓘ |
| importance |
central conjecture in analytic number theory
ⓘ
major unsolved problem in mathematics ⓘ |
| involves |
Dirichlet characters
ⓘ
surface form:
Dirichlet characters modulo q
Euler products ⓘ analytic continuation of L-functions ⓘ |
| namedAfter | Bernhard Riemann ⓘ |
| nontrivialZerosLieOn | critical line Re(s) = 1/2 ⓘ |
| openAsOf | 2024 ⓘ |
| relatedTo |
Lindelöf hypothesis
ⓘ
Riemann hypothesis ⓘ generalized Riemann hypothesis self-linksurface differs ⓘ
surface form:
extended Riemann hypothesis
generalized Lindelöf hypothesis ⓘ grand Riemann hypothesis ⓘ |
| statementAbout |
Dirichlet L-functions
ⓘ
zeros of L-functions ⓘ |
| status |
open problem
ⓘ
unproven ⓘ |
| subfield |
L-function theory
ⓘ
algebraic number theory ⓘ analytic number theory ⓘ |
| typeOfZero | nontrivial zeros ⓘ |
| usedIn |
algorithmic number theory
ⓘ
complexity theory conditional results ⓘ computational number theory ⓘ cryptography conditional analyses ⓘ |
| yearFormulatedApprox | 19th century ⓘ |
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: generalized Riemann hypothesis Description of subject: The generalized Riemann hypothesis is a major unproven conjecture in number theory asserting that the nontrivial zeros of all Dirichlet L-functions lie on a critical line in the complex plane, extending the classical Riemann hypothesis.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
generalized Riemann hypothesis
→
relatedTo
→
generalized Riemann hypothesis
self-linksurface differs
ⓘ
this entity surface form:
extended Riemann hypothesis
this entity surface form:
Generalized Riemann Hypothesis for Dirichlet L-functions