On the Hilbert-Pólya and Pair Correlation Conjectures Hudoba de Badyn, Mathias
The Hilbert-Pólya Conjecture supposes that there exists an operator in a Hilbert space whose eigenvalues are the zeroes of the Riemann Zeta function ζ(s). This conjecture, if true, would very likely expedite the proof of the Riemann Hypothesis, namely that the non-trivial zeroes of ζ(s) have real part 1/2. In this thesis we summarize work by Berry, Keating and others in constructing such an operator. Although the work so far has not yet yielded such an operator, some have been found that have properties very close to what is desired. We also summarize a (partially proven) conjecture by Montgomery that motivates the search for this operator. He conjectures that the pair correlation function for the spacing between the imaginary parts of the Riemann zeroes is the same as the correlation function for the spacing between eigenvalues of random Gaussian unitary matrices.
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