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We give a probabilistic interpretation for the Barnes G-function whichappears in random matrix theory and in analytic number theory in the importantmoments conjecture due to Keating-Snaith for the Riemann zeta function, via theanalogy with the characteristic polynomial of random unitary matrices. We showthat the Mellin transform of the characteristic polynomial of random unitarymatrices and the Barnes G-function are intimately related with products andsums of gamma, beta and log-gamma variables. In particular, we show that thelaw of the modulus of the characteristic polynomial of random unitary matricescan be expressed with the help of products of gamma or beta variables, and thatthe reciprocal of the Barnes G-function has a L\'{e}vy-Khintchin typerepresentation. These results lead us to introduce the so called generalizedgamma convolution variables.

The barnes G function and its relations with sums and products of generalized Gamma convolution variables