  We reconsider the problem of calculating arbitrary negative integer moments
of the (regularized) characteristic polynomial for $N\times N$ random matrices
taken from the Gaussian Unitary Ensemble (GUE). A very compact and convenient
integral representation is found via the use of a matrix integral close to that
considered by Ingham and Siegel. We find the asymptotic expression for the
discussed moments in the limit of large $N$. The latter is of interest because
of a conjectured relation to properties of the Riemann $\zeta-$ function
zeroes. Our method reveals a striking similarity between the structure of the
negative and positive integer moments which is usually obscured by the use of
the Hubbard-Stratonovich transformation. This sheds a new light on "bosonic"
versus "fermionic" replica trick and has some implications for the
supersymmetry method. We briefly discuss the case of the chiral GUE model from
that perspective.
