Montgomery's pair correlation conjecture
is the imaginary part of the non-trivial zeros of Riemann zeta function, that is
(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 two 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 (RH).
The Hilbert–Pólya conjecture asserts that the zeros of the Riemann Zeta function correspond to the eigenvalues of a linear operator, and implies RH.
Montgomery was studying the Fourier transform F(x) of the pair correlation function, and showed (assuming the Riemann hypothesis) that it was equal 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.
He was also motivated by the notion that the Riemann hypothesis is not a brick wall, and one should feel free to make stronger conjectures.
stand for non-trivial zeros of the Riemann zeta function.
In the 1980s, motivated by Montgomery's conjecture, Odlyzko began an intensive numerical study of the statistics of the zeros of ζ(s).
For non-trivial zero, 1/2 + iγn, let the normalized spacings be Then we would expect the following formula as the limit for
: Based on a new algorithm developed by Odlyzko and Arnold Schönhage that allowed them to compute a value of ζ(1/2 + it) in an average time of tε steps, Odlyzko computed millions of zeros at heights around 1020 and gave some evidence for the GUE conjecture.
As more zeros are sampled, the more closely their distribution approximates the shape of the GUE random matrix.
