"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)
 

There are some excellent introductory notes regarding the GTF in N. Snaith's thesis.
 

general notes on the number theoretic content of Gutzwiller's papers
 

quantum chaology resources