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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 28, Issue 1


On Barnes beta distributions, Selberg integral and Riemann xi

Dmitry Ostrovsky
Published Online: 2014-09-09 | DOI: https://doi.org/10.1515/forum-2013-0149


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.

Keywords: Multiple gamma function; Infinite divisibility; Selberg integral; Mellin transform; Shintani factorization; Riemann xi function

MSC: 33B15; 60E07; 60E10; 11M06; 40A20; 60G57

About the article

Received: 2013-09-16

Revised: 2014-02-10

Published Online: 2014-09-09

Published in Print: 2016-01-01

Citation Information: Forum Mathematicum, Volume 28, Issue 1, Pages 1–23, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2013-0149.

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