grand Riemann hypothesis
E898461
The grand Riemann hypothesis is a far-reaching conjecture in number theory asserting that all nontrivial zeros of all automorphic L-functions lie on a critical line in the complex plane, generalizing the classical and generalized Riemann hypotheses.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Grand Riemann Hypothesis | 1 |
| grand Riemann hypothesis canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991104 — 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: grand Riemann hypothesis Context triple: [generalized Riemann hypothesis, relatedTo, grand 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.
generalized Riemann hypothesis
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.
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C.
Lindelöf hypothesis
The Lindelöf hypothesis is an unproven conjecture in analytic number theory about the growth rate of the Riemann zeta function along the critical line, with deep implications for the distribution of prime numbers.
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D.
Hilbert–Pólya conjecture
The Hilbert–Pólya conjecture is an unproven idea in number theory suggesting that the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a suitable self-adjoint operator, offering a potential spectral approach to proving the Riemann hypothesis.
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E.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
- 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: grand Riemann hypothesis Target entity description: The grand Riemann hypothesis is a far-reaching conjecture in number theory asserting that all nontrivial zeros of all automorphic L-functions lie on a critical line in the complex plane, generalizing the classical and generalized Riemann hypotheses.
-
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.
generalized Riemann hypothesis
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.
-
C.
Lindelöf hypothesis
The Lindelöf hypothesis is an unproven conjecture in analytic number theory about the growth rate of the Riemann zeta function along the critical line, with deep implications for the distribution of prime numbers.
-
D.
Hilbert–Pólya conjecture
The Hilbert–Pólya conjecture is an unproven idea in number theory suggesting that the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a suitable self-adjoint operator, offering a potential spectral approach to proving the Riemann hypothesis.
-
E.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical conjecture
ⓘ
unproven hypothesis in number theory ⓘ |
| asserts | all nontrivial zeros of all automorphic L-functions lie on a critical line ⓘ |
| concerns |
distribution of zeros in the complex plane
ⓘ
nontrivial zeros of automorphic L-functions ⓘ |
| criticalLine | Re(s)=1/2 ⓘ |
| criticalStrip | 0 < Re(s) < 1 ⓘ |
| difficulty | considered extremely difficult ⓘ |
| field |
L-functions
ⓘ
analytic number theory ⓘ automorphic forms ⓘ number theory ⓘ |
| generalizes |
Riemann hypothesis
ⓘ
generalized Riemann hypothesis ⓘ |
| hasConsequence |
refined estimates for distribution of primes in arithmetic progressions
ⓘ
results on distribution of automorphic spectra ⓘ results on distribution of eigenvalues of Hecke operators ⓘ strong bounds on error terms in prime counting functions ⓘ |
| implies |
Riemann hypothesis
ⓘ
generalized Riemann hypothesis ⓘ |
| importance | central problem in modern number theory ⓘ |
| involves |
algebraic number theory
ⓘ
complex analysis ⓘ harmonic analysis on groups ⓘ representation theory ⓘ |
| isStrongerThan |
extended Riemann hypothesis
ⓘ
linked to:
generalized Riemann hypothesis
generalized Riemann hypothesis ⓘ |
| motivatedBy |
properties of Dirichlet L-functions
ⓘ
properties of automorphic L-functions ⓘ properties of the Riemann zeta function ⓘ |
| nontrivialZeroDefinition | zeros in the critical strip excluding trivial zeros from functional equations ⓘ |
| openAsOf | 2024 ⓘ |
| relatedTo |
Langlands program
ⓘ
Selberg class ⓘ automorphic forms ⓘ prime number distribution ⓘ random matrix theory ⓘ spectral theory of automorphic forms ⓘ |
| status | open problem ⓘ |
| subjectOf | research in modern number theory ⓘ |
| typeOf | zero-free region conjecture ⓘ |
| usesConcept |
Dirichlet series
ⓘ
Euler product ⓘ automorphic L-function ⓘ
linked to:
L-functions
automorphic representation ⓘ critical line ⓘ functional equation of L-functions ⓘ |
| zeroLocationClaim | Re(s)=1/2 for all nontrivial zeros of automorphic 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: grand Riemann hypothesis Description of subject: The grand Riemann hypothesis is a far-reaching conjecture in number theory asserting that all nontrivial zeros of all automorphic L-functions lie on a critical line in the complex plane, generalizing the classical and generalized Riemann hypotheses.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: grand Riemann hypothesis