Perhaps the most challenging open problem in classical mathematics is the Riemann hypothesis. Simply put, the Riemann hypothesis states that all of the infinite number of complex zeros of the zeta function lie symmetrically on the critical line 1/2 in the complex plane. In 1915, Hilbert and Polya independently suggested that the Riemann hypothesis could be proved if the nontrivial (complex) zeros of zeta could be shown to correspond directly to the spectrum of eigenvalues of some hermitian operator on a Hilbert space. This so-called spectral interpretation of the Riemann hypothesis has led many physicists to consider the possible links between number theory and quantum physics. (Of course, this interest could be related to the fact that there is a one-million dollar ($US) prize for a proof of the Riemann hypothesis!!).
In this talk, I will review several deep and interesting connections between number theory, statistical physics and quantum chaos. I will also present some of our recent work dealing with the prime numbers, Riemann zeros and quantum chaos.