The barnes G function and its relations with sums
and products of generalized Gamma convolution variables
Ashkan Nikeghbali, University of Zurich
Marc Yor, Universite Paris 6
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.
Full text: PDF | PostScript
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona.
The preferred way is to send a scanned (jpeg or pdf) copy of the signed copyright form to the managing editor Philippe Carmona at ejpecpme@math.univ-nantes.fr.
If a paper has several authors, the corresponding author signs the copyright form
on behalf of all the authors.