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On Barnes beta distributions, Selberg integral and Riemann xi

On Barnes beta distributions, Selberg integral and Riemann xi Abstract The theory of Barnes beta probability distributions is advanced and related to the Riemann xi function. The scaling invariance, multiplication formula, and Shintani factorization of Barnes multiple gamma functions are reviewed using the approach of Ruijsenaars and shown to imply novel properties of Barnes beta distributions. The applications are given to the meromorphic extension of the Selberg integral as a function of its dimension and the scaling invariance of the underlying probability distribution. This probability distribution in the critical case is described and conjectured to be the distribution of the derivative martingale. The Jacobi triple product is interpreted probabilistically resulting in an approximation of the Riemann xi function by the Mellin transform of the logarithm of a limit of Barnes beta distributions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

On Barnes beta distributions, Selberg integral and Riemann xi

Forum Mathematicum , Volume 28 (1) – Jan 1, 2016

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Publisher
de Gruyter
Copyright
Copyright © 2016 by the
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2013-0149
Publisher site
See Article on Publisher Site

Abstract

Abstract The theory of Barnes beta probability distributions is advanced and related to the Riemann xi function. The scaling invariance, multiplication formula, and Shintani factorization of Barnes multiple gamma functions are reviewed using the approach of Ruijsenaars and shown to imply novel properties of Barnes beta distributions. The applications are given to the meromorphic extension of the Selberg integral as a function of its dimension and the scaling invariance of the underlying probability distribution. This probability distribution in the critical case is described and conjectured to be the distribution of the derivative martingale. The Jacobi triple product is interpreted probabilistically resulting in an approximation of the Riemann xi function by the Mellin transform of the logarithm of a limit of Barnes beta distributions.

Journal

Forum Mathematicumde Gruyter

Published: Jan 1, 2016

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