  We investigate the thermodynamics of a Fermi gas whose single-particle energy
levels are given by the complex zeros of the Riemann zeta function. This is a
model for a gas, and in particular for an atomic nucleus, with an underlying
fully chaotic classical dynamics. The probability distributions of the quantum
fluctuations of the grand potential and entropy of the gas are computed as a
function of temperature and compared, with good agreement, with general
predictions obtained from random matrix theory and periodic orbit theory (based
on prime numbers). In each case the universal and non--universal regimes are
identified.
