## Electronic Communications in Probability

### The Barnes $G$ function and its relations with sums and products of generalized Gamma convolution variables

#### 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. This leads us to prove some non standard type of limit theorems for the logarithmic mean of the so called generalized gamma convolutions.

#### Article information

Source
Electron. Commun. Probab., Volume 14 (2009), paper no. 39, 396-411.

Dates
Accepted: 23 September 2009
First available in Project Euclid: 6 June 2016

https://projecteuclid.org/euclid.ecp/1465234748

Digital Object Identifier
doi:10.1214/ECP.v14-1488

Mathematical Reviews number (MathSciNet)
MR2545290

Zentralblatt MATH identifier
1189.60073

Rights

#### Citation

Nikeghbali, Ashkan; Yor, Marc. The Barnes $G$ function and its relations with sums and products of generalized Gamma convolution variables. Electron. Commun. Probab. 14 (2009), paper no. 39, 396--411. doi:10.1214/ECP.v14-1488. https://projecteuclid.org/euclid.ecp/1465234748

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