Response of a canonical ensemble of quantum oscillators to a random metric

We calculate the susceptibility of a canonical ensemble of quantum oscillators to the singular random metric. If the covariance of the metric is $\mathrm{\mid }\mathbf{x}-{\mathbf{x}}^{\mathrm{\prime }}{\mathrm{\mid }}^{-4\alpha }$ $0<\alpha <\frac{1}{2}$ then the expansion of the partition function in powers of the temperature involves non-integer indices.