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

TL;DR

This paper explores the Barnes G-function's probabilistic interpretation, linking it to sums and products of generalized gamma convolution variables, with implications for random matrix theory and number theory.

Contribution

It introduces a novel probabilistic perspective on the Barnes G-function and connects it to generalized gamma convolution variables, expanding understanding in random matrix and number theory.

Findings

Barnes G-function related to gamma, beta, and log-gamma variables

Modulus of characteristic polynomial expressed via gamma or beta products

Barnes G-function's reciprocal has a Lévy-Khintchin type representation

Abstract

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.