Joint moments of derivatives of characteristic polynomials

Dehaye, Paul-Olivier

doi:10.2140/ant.2008.2.31

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Joint moments of derivatives of characteristic polynomials

Paul-Olivier Dehaye

Vol. 2 (2008), No. 1, 31–68

DOI: 10.2140/ant.2008.2.31

Abstract

We investigate the joint moments of the $2 k$ -th power of the characteristic polynomial of random unitary matrices with the $2 h$ -th power of the derivative of this same polynomial. We prove that for a fixed $h$ , the moments are given by rational functions of $k$ , up to a well-known factor that already arises when $h = 0$ .

We fully describe the denominator in those rational functions (this had already been done by Hughes experimentally), and define the numerators through various formulas, mostly sums over partitions.

We also use this to formulate conjectures on joint moments of the zeta function and its derivatives, or even the same questions for the Hardy function, if we use a “real” version of characteristic polynomials.

Our methods should easily be applied to other similar problems, for instance with higher derivatives of characteristic polynomials.

More data and computer programs are available as expanded content.

Pour Annie & Jean-Paul

Keywords

discrete moment, random matrix theory, unitary characteristic polynomial, Riemann zeta function, Cauchy identity

Mathematical Subject Classification 2000

Primary: 11M26

Secondary: 60B15, 15A52, 33C80, 05E10

Milestones

Received: 7 April 2007

Revised: 13 September 2007

Accepted: 15 September 2007

Published: 1 February 2008

Authors

Paul-Olivier Dehaye
University of Oxford Merton College Merton Street OX1 4JD, Oxford United Kingdom
http://www.maths.ox.ac.uk/~pdehaye/

Vol. 2, No. 1, 2008