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- Montgomery's_pair_correlation_conjecture abstract "In mathematics, Montgomery's pair correlation conjecture is a conjecture made by Hugh Montgomery (1973) that the pair correlation between pairs of zeros of the Riemann zeta function (normalized to have unit average spacing) iswhich, as Freeman Dyson pointed out to him, is the same as the pair correlation function of random Hermitian matrices. Informally, this means that the chance of finding a zero in a very short interval of length 2πL/log(T) at a distance 2πu/log(T) from a zero 1/2+iT is about L times the expression above. (The factor 2π/log(T) is a normalization factor that can be thought of informally as the average spacing between zeros with imaginary part about T.) Andrew Odlyzko (1987) showed that the conjecture was supported by large-scale computer calculations of the zeros. The conjecture has been extended to correlations of more than 2 zeros, and also to zeta functions of automorphic representations (Rudnick & Sarnak 1996). In 1982 a student of Montgomery's, Ali Erhan Özlük, proved the pair correlation conjecture for some of Dirichlet's L-functions.A.E. Ozluk (1982)The connection with random unitary matrices could lead to a proof of the Riemann hypothesis. The Hilbert–Pólya conjecture asserts that the zeros of the Riemann Zeta function correspond to the eigenvalues of a linear operator, and implies RH. Some people think this is a promising approach (Andrew Odlyzko (1987)).Montgomery was studying the Fourier transform F(x) of the pair correlation function, and showed (assuming the Riemann hypothesis) that it wasequal to |x| for |x|<1. His methods were unable to determine it for |x|≥1, but he conjectured that it was equal to 1 for these x, which implies that the pair correlation function is as above.".
- Montgomery's_pair_correlation_conjecture thumbnail Montgomery-Odlyzko_law.png?width=300.
- Montgomery's_pair_correlation_conjecture wikiPageExternalLink 1077245671.
- Montgomery's_pair_correlation_conjecture wikiPageExternalLink home.html.
- Montgomery's_pair_correlation_conjecture wikiPageID "21627970".
- Montgomery's_pair_correlation_conjecture wikiPageRevisionID "599747635".
- Montgomery's_pair_correlation_conjecture authorlink "A.E.Ozluk".
- Montgomery's_pair_correlation_conjecture authorlink "Andrew Odlyzko".
- Montgomery's_pair_correlation_conjecture authorlink "Hugh Montgomery".
- Montgomery's_pair_correlation_conjecture first "A.E.".
- Montgomery's_pair_correlation_conjecture first "Andrew".
- Montgomery's_pair_correlation_conjecture first "Hugh".
- Montgomery's_pair_correlation_conjecture hasPhotoCollection Montgomery's_pair_correlation_conjecture.
- Montgomery's_pair_correlation_conjecture last "Montgomery".
- Montgomery's_pair_correlation_conjecture last "Odlyzko".
- Montgomery's_pair_correlation_conjecture last "Ozluk".
- Montgomery's_pair_correlation_conjecture year "1973".
- Montgomery's_pair_correlation_conjecture year "1982".
- Montgomery's_pair_correlation_conjecture year "1987".
- Montgomery's_pair_correlation_conjecture subject Category:Analytic_number_theory.
- Montgomery's_pair_correlation_conjecture subject Category:Conjectures.
- Montgomery's_pair_correlation_conjecture subject Category:Zeta_and_L-functions.
- Montgomery's_pair_correlation_conjecture type Abstraction100002137.
- Montgomery's_pair_correlation_conjecture type Cognition100023271.
- Montgomery's_pair_correlation_conjecture type Concept105835747.
- Montgomery's_pair_correlation_conjecture type Conjectures.
- Montgomery's_pair_correlation_conjecture type Content105809192.
- Montgomery's_pair_correlation_conjecture type Hypothesis105888929.
- Montgomery's_pair_correlation_conjecture type Idea105833840.
- Montgomery's_pair_correlation_conjecture type PsychologicalFeature100023100.
- Montgomery's_pair_correlation_conjecture type Speculation105891783.
- Montgomery's_pair_correlation_conjecture comment "In mathematics, Montgomery's pair correlation conjecture is a conjecture made by Hugh Montgomery (1973) that the pair correlation between pairs of zeros of the Riemann zeta function (normalized to have unit average spacing) iswhich, as Freeman Dyson pointed out to him, is the same as the pair correlation function of random Hermitian matrices.".
- Montgomery's_pair_correlation_conjecture label "Montgomery's pair correlation conjecture".
- Montgomery's_pair_correlation_conjecture label "モンゴメリー・オドリズコ予想".
- Montgomery's_pair_correlation_conjecture sameAs モンゴメリー・オドリズコ予想.
- Montgomery's_pair_correlation_conjecture sameAs m.05mxp1l.
- Montgomery's_pair_correlation_conjecture sameAs Q6905620.
- Montgomery's_pair_correlation_conjecture sameAs Q6905620.
- Montgomery's_pair_correlation_conjecture sameAs Montgomery's_pair_correlation_conjecture.
- Montgomery's_pair_correlation_conjecture wasDerivedFrom Montgomery's_pair_correlation_conjecture?oldid=599747635.
- Montgomery's_pair_correlation_conjecture depiction Montgomery-Odlyzko_law.png.
- Montgomery's_pair_correlation_conjecture isPrimaryTopicOf Montgomery's_pair_correlation_conjecture.