"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.

This setup resembles the explicit formulas of prime number theory. In this analogy, the lengths of the prime periodic orbits play the role of the rational primes, while the eigenvalues of the Hamiltonian play the role of the zeros of the zeta function. Based on this analogy and pearls mined from Odlyzko's numerical evidence, Sir Michael Berry proposes that there exists a classical dynamical system, asymmetric with respect to time reversal, the lengths of whose periodic orbits correspond to the rational primes, and whose quantum-mechanical analog has a Hamiltonian with zeros equal to the imaginary parts of the nontrivial zeros of the zeta function. The search for such a dynamical system is one approach to proving the Riemann hypothesis."   (Daniel Bump)
 

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:

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.