Hilbert–Pólya conjecture

E259757

mathematical conjecture unproven idea in number theory

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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All labels observed (1)

Label	Occurrences
Hilbert–Pólya conjecture canonical	2

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 ⓘ surface form: 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 ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Riemann hypothesis → relatedTo → Hilbert–Pólya conjecture ⓘ

Über die Anzahl der Primzahlen unter einer gegebenen Grösse → influenceOn → Hilbert–Pólya conjecture ⓘ