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The famous Riemann Hypothesis (RH) is one of the most important open problem in Number Theory. As many other outstanding problems, RH has many equivalent statements. Ten years ago the speaker reformulated the Riemann Hypothesis as statements about the eigenvalues of certain Hankel matrices, entries of which are defined via the Taylor series coefficients of Riemann's zeta function. Numerical calculations revealed some very interesting visual patterns in the behaviour of the eigenvalues and allowed the speaker to state a number of new conjectures related to the RH.
Recently computations have been performed on supercomputers. This led to new conjectures about the finer structure of the eigenvalues and eigenvectors and to conjectures that are (formally) stronger than RH. Further refinement of these
conjectures would require extensive computations on more powerful computers than those that were available to the speaker.
More information can be found at http://logic.pdmi.ras.ru/~yumat/personaljournal/zetahiddenlife
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