  We give a probabilistic interpretation for the Barnes G-function which
appears in random matrix theory and in analytic number theory in the important
moments conjecture due to Keating-Snaith for the Riemann zeta function, via the
analogy with the characteristic polynomial of random unitary matrices. We show
that the Mellin transform of the characteristic polynomial of random unitary
matrices and the Barnes G-function are intimately related with products and
sums of gamma, beta and log-gamma variables. In particular, we show that the
law of the modulus of the characteristic polynomial of random unitary matrices
can be expressed with the help of products of gamma or beta variables, and that
the reciprocal of the Barnes G-function has a L\'{e}vy-Khintchin type
representation. These results lead us to introduce the so called generalized
gamma convolution variables.

Probability