  In this paper we present a new mathematical rigorous technique for computing
the average free energy of a disordered system with quenched randomness, using
the replicas. The basic tool of this technique is a distributional
zeta-function, 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. This result supports the
use of integer moments of the partition function, computed via replicas, for
expressing the average free energy of the system. One advantage of the proposed
formalism is that it does not require the understanding of the properties of
the permutation group when the number of replicas goes to zero. Moreover, the
symmetry is broken using the saddle-point equations of the model. As an
application for the distributional zeta-function technique, we obtain the
average free energy of the disordered $\lambda\varphi^{4}$ model defined in a
$d$-dimensional Euclidean space.
