Disordered Field Theory in $d=0$ and Distributional Zeta-Function

Recently we introduced a new technique for computing the average free energy of a system with quenched randomness. The basic tool of this technique is a distributional zeta-function. The distributional zeta-function is a complex function whose derivative at the origin yields the average free energy of the system as the sum of two contributions: the first one is a series in which all the integer moments of the partition function of the model contribute; the second one, which can not be written as a series of the integer moments, can be made as small as desired. In this paper we present a mathematical rigorous proof that the average free energy of one disordered $\lambda\varphi^{4}$ model defined in a zero-dimensional space can be obtained using the distributional zeta-function technique. We obtain an analytic expression for the average free energy of the model.