Montgomery's pair correlation conjecture

E259758

mathematical conjecture number theory conjecture

Montgomery's pair correlation conjecture is a deep number-theoretic prediction about the statistical spacing of the nontrivial zeros of the Riemann zeta function, linking them to eigenvalues of random matrices and suggesting profound connections between number theory and quantum physics.

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All labels observed (5)

Label	Occurrences
GUE conjecture for zeta zeros	1
Montgomery pair correlation conjecture	1
Montgomery's pair correlation conjecture canonical	1
Montgomery–Odlyzko law	1
pair correlation conjecture	1

Statements (45)

Predicate	Object
instanceOf	mathematical conjecture ⓘ number theory conjecture ⓘ
appliesTo	nontrivial zeros of the Riemann zeta function on the critical line ⓘ
assumes	Riemann hypothesis for the formulation of zeros on the critical line ⓘ
author	Hugh L. Montgomery ⓘ
concerns	asymptotic behavior of pair correlation as the height on the critical line tends to infinity ⓘ
conjectures	that the normalized gaps between high zeros of the Riemann zeta function follow the GUE pair correlation function ⓘ that the pair correlation function of zeros is given by 1 - (sin(πx)/(πx))^2 for the scaled spacing variable x ⓘ that the pair correlation of high Riemann zeros matches that of eigenvalues of large random Hermitian matrices from the Gaussian unitary ensemble ⓘ
connectedTo	Hilbert–Pólya idea relating zeros of zeta to eigenvalues of a self-adjoint operator ⓘ spectral interpretation of zeros of the Riemann zeta function ⓘ
describes	two-point correlation function of zeros of the Riemann zeta function ⓘ
discoveredDuring	a discussion between Hugh Montgomery and Freeman Dyson at the Institute for Advanced Study ⓘ
field	analytic number theory ⓘ mathematical physics ⓘ random matrix theory ⓘ
generalizedBy	pair correlation conjectures for zeros of general L-functions ⓘ
hasConsequence	prediction of level repulsion between zeros of the Riemann zeta function ⓘ prediction that small gaps between zeros are less frequent than for a Poisson process ⓘ prediction that zeros of the Riemann zeta function behave like eigenvalues of large random Hermitian matrices ⓘ
hasMathematicalExpression	pair correlation function R_2(x) = 1 - (sin(πx)/(πx))^2 for the scaled zeros ⓘ
hasType	Montgomery's pair correlation conjecture self-linksurface differs ⓘ surface form: pair correlation conjecture
inception	1973 ⓘ
influenced	Katz–Sarnak philosophy on statistics of zeros of L-functions ⓘ
inspired	connections between zeros of L-functions and eigenvalues of random matrices ⓘ
mainSubject	Riemann zeta function ⓘ nontrivial zeros of the Riemann zeta function ⓘ pair correlation of zeros ⓘ statistical distribution of zeros ⓘ
motivated	development of random matrix models for L-functions ⓘ
namedAfter	Hugh L. Montgomery ⓘ
predicts	universal local statistics for zeros of the Riemann zeta function matching GUE statistics ⓘ
relatedTo	Montgomery's pair correlation conjecture self-linksurface differs ⓘ surface form: GUE conjecture for zeta zeros Gaussian unitary ensemble ⓘ Montgomery's pair correlation conjecture self-linksurface differs ⓘ surface form: Montgomery–Odlyzko law Riemann hypothesis ⓘ quantum chaos ⓘ random matrix theory model of zeta zeros ⓘ
statedIn	Hugh Montgomery's 1973 paper on the pair correlation of zeros of the zeta function ⓘ
status	unproven ⓘ
supportedBy	extensive numerical computations of Riemann zeros by Andrew Odlyzko ⓘ
topic	spacing of zeros of the Riemann zeta function ⓘ statistical properties of zeros of L-functions ⓘ
usedIn	heuristics for gaps between primes ⓘ heuristics for the distribution of primes ⓘ

Referenced by (5)

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

Riemann hypothesis → relatedTo → Montgomery's pair correlation conjecture ⓘ

random matrix theory → relatedTo → Montgomery's pair correlation conjecture ⓘ

this entity surface form: Montgomery pair correlation conjecture

Montgomery's pair correlation conjecture → relatedTo → Montgomery's pair correlation conjecture self-linksurface differs ⓘ

this entity surface form: GUE conjecture for zeta zeros

Montgomery's pair correlation conjecture → relatedTo → Montgomery's pair correlation conjecture self-linksurface differs ⓘ

this entity surface form: Montgomery–Odlyzko law

Montgomery's pair correlation conjecture → hasType → Montgomery's pair correlation conjecture self-linksurface differs ⓘ

this entity surface form: pair correlation conjecture