Thermodynamics of the Bosonic Randomized Riemann Gas

The partition function of a bosonic Riemann gas is given by the Riemann zeta function. We assume that the hamiltonian of this gas at a given temperature $\beta^{-1}$ has a random variable $\omega$ with a given probability distribution over an ensemble of hamiltonians. We study the average free energy density and average mean energy density of this arithmetic gas in the complex $\beta$-plane. Assuming that the ensemble is made by an enumerable infinite set of copies, there is a critical temperature where the average free energy density diverges due to the pole of the Riemann zeta function. Considering an ensemble of non-enumerable set of copies, the average free energy density is non-singular for all temperatures, but acquires complex values in the critical region. Next, we study the mean energy density of the system which depends strongly on the distribution of the non-trivial zeros of the Riemann zeta function. Using a regularization procedure we prove that the this quantity is continuous and bounded for finite temperatures.