Vol. 2, No. 1, 2008

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Joint moments of derivatives of characteristic polynomials

Paul-Olivier Dehaye

Vol. 2 (2008), No. 1, 31–68

We investigate the joint moments of the 2k-th power of the characteristic polynomial of random unitary matrices with the 2h-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

discrete moment, random matrix theory, unitary characteristic polynomial, Riemann zeta function, Cauchy identity
Mathematical Subject Classification 2000
Primary: 11M26
Secondary: 60B15, 15A52, 33C80, 05E10
Received: 7 April 2007
Revised: 13 September 2007
Accepted: 15 September 2007
Published: 1 February 2008
Paul-Olivier Dehaye
University of Oxford
Merton College
Merton Street
OX1 4JD, Oxford
United Kingdom