Montgomery's pair correlation conjecture is a conjecture made by the mathematician Hugh Montgomery concerning the pair correlation between pairs of zeros of the Riemann zeta function. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementThe pair correlation between pairs of zeros of the Riemann zeta function, normalized to have unit average spacing, is 1 - (sin(pi u) / (pi u))^2. 1 Proposed Year Progress Toward ResolutionNot proved in full. Montgomery showed, assuming the Riemann hypothesis, that the Fourier transform of the pair correlation function equals |x| for |x| less than 1. Odlyzko's 1987 large-scale computer calculations of the zeros support the conjecture, and it has been extended to correlations of more than two zeros and to zeta functions of automorphic representations (Rudnick and Sarnak 1996). 1 Classification
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1. Montgomery's Pair Correlation Conjecture (Wikipedia)
Wikimedia FoundationLead section
a conjecture made by Hugh Montgomery (1973)
Lead section, first sentence
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
Explanation section, Odlyzko sentence
Andrew Odlyzko (1987) showed that the conjecture was supported by large-scale computer calculations of the zeros.
Explanation section, Rudnick and Sarnak sentence
The conjecture has been extended to correlations of more than two zeros, and also to zeta functions of automorphic representations (Rudnick & Sarnak 1996).
Explanation section, Fourier transform sentence
showed (assuming the Riemann hypothesis) that it was equal to
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