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"Gutzwiller gave a trace formula in the setting of quantum chaos which
relates the classical and quantum mechanical pictures. Given a chaotic
(classical) dynamical system, there will exist a dense set of periodic orbits,
and one side of the trace formula will be a sum over the lengths of these
orbits. On the other side will be a sum over the eigenvalues of the
Hamiltonian in the quantum-mechanical analog of the given classical dynamical
system.
So there is a resemblance between the GTF and an explicit formula
which relates directly to the RZF. Did Gutzwiller or Berry notice this first?
There is no known "reason" why there should be such a resemblance, of which I am
aware - it is a mystery. Burnol has written something to the effect that the QC material coming from Gutzwiller, then Berry and Keating is just "the modern version of the Hilbert-Pólya idea". Is he claiming that they've just reformulated the idea - that there is not something fundamentally new here? Remember Odlyzko's GUE hypothesis has been backed up by vast swathes of numerical evidence, and everyone's happy that the zeta zeros are eigenvalues of something. But it's more than just evidence that "the zeta zeros are eigenvalues of something". They have the statistical fingerprint of GUE eigenvalues in particular. The GUE fingerprint also shows up in a big way in quantum chaology.
So there are two reasons to think that QC is an important key to
understanding the zeta function:
/ \ GUE GTF \ / RZF In Keating's "Physics and the queen of mathematics" he tells us that
"Riemann's connection between the nontrivial zeros and the primes has a
particularly interesting form: it bears a striking resemblance to the
Gutzwiller formula, with the zeros behaving like energy levels and the
primes labelling the periodic orbits of some chaotic classical system.
Indeed from the general form of the relationship, the actions, periods
and stabilities of all these orbits may be obtained explicitly. In
this case the 'collective property' of the periodic orbits discovered
by Hannay and Ozorio de Almeida is just the prime number theorem!"
Yet more historical confusion: Gutzwiller, in his book, almost seems
to be suggesting that Selberg first recognised the GTF-RZF connection,
and was thus led to study the flow of geodesics on surfaces of constant
negative curvature. Somehow I don't think this is accurate - he wasn't
entirely clear.
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