Montgomery's pair correlation conjecture

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) is

1-\left(\frac{\sin(\pi u)}{\pi u}\right)^2 %2B\delta(u),

which, 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).

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.

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