Hilbert–Pólya conjecture
E259757
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hilbert–Pólya conjecture canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T2364376 — 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: Hilbert–Pólya conjecture Context triple: [Riemann hypothesis, relatedTo, Hilbert–Pólya conjecture]
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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–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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C.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
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D.
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.
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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: Hilbert–Pólya conjecture Target entity description: 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.
-
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–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.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
-
D.
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.
-
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 (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical conjecture
ⓘ
unproven idea in number theory ⓘ |
| aimsToExplain | distribution of nontrivial zeros of the Riemann zeta function ⓘ |
| approachType |
operator-theoretic approach to the Riemann hypothesis
ⓘ
spectral approach to the Riemann hypothesis ⓘ |
| associatedWith |
David Hilbert
ⓘ
George Pólya ⓘ |
| claimsAbout | location of nontrivial zeros of the Riemann zeta function ⓘ |
| conceptualBasis | analogy between spectra of operators and zeros of L-functions ⓘ |
| discussedIn |
literature on the Riemann hypothesis
ⓘ
surveys on spectral approaches to number theory ⓘ |
| field |
analytic number theory
ⓘ
mathematical physics ⓘ number theory ⓘ spectral theory ⓘ |
| generalizationTarget | spectral interpretations for other L-functions ⓘ |
| hasConsequence | would prove the Riemann hypothesis if true ⓘ |
| hasNo | explicitly known self-adjoint operator realizing the conjecture ⓘ |
| implies | Riemann hypothesis ⓘ |
| influenced |
Montgomery’s pair correlation conjecture
ⓘ
connections between Riemann zeros and random matrix theory ⓘ research on quantum chaos and the Riemann zeros ⓘ spectral interpretations of the Riemann zeta function ⓘ |
| involves |
critical line of the complex plane
ⓘ
self-adjoint operator on a Hilbert space ⓘ spectral interpretation of zeta zeros ⓘ |
| motivatedBy | search for a proof of the Riemann hypothesis ⓘ |
| namedAfter |
David Hilbert
ⓘ
George Pólya ⓘ |
| openProblemIn |
mathematical physics
ⓘ
number theory ⓘ |
| philosophicalNature | heuristic guiding principle rather than a precisely formulated theorem ⓘ |
| proposesThat | nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator ⓘ |
| relatedTo |
Gaussian Unitary Ensemble
ⓘ
linked to:
Gaussian unitary ensemble
Hermitian operator ⓘ Riemann hypothesis ⓘ Riemann zeta function ⓘ eigenvalues ⓘ nontrivial zeros of the Riemann zeta function ⓘ quantum chaos ⓘ random matrix theory ⓘ self-adjoint operator ⓘ spectral theory of operators ⓘ spectrum of an operator ⓘ |
| status |
conjectural
ⓘ
unproven ⓘ |
| timePeriod | early 20th century ⓘ |
| wouldImply | all nontrivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2 ⓘ |
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: Hilbert–Pólya conjecture Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
Über die Anzahl der Primzahlen unter einer gegebenen Grösse
→
influenceOn
→
Hilbert–Pólya conjecture
ⓘ