Open Access
2008 Joint moments of derivatives of characteristic polynomials
Paul-Olivier Dehaye
Algebra Number Theory 2(1): 31-68 (2008). DOI: 10.2140/ant.2008.2.31


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.


Download Citation

Paul-Olivier Dehaye. "Joint moments of derivatives of characteristic polynomials." Algebra Number Theory 2 (1) 31 - 68, 2008.


Received: 7 April 2007; Revised: 13 September 2007; Accepted: 15 September 2007; Published: 2008
First available in Project Euclid: 20 December 2017

zbMATH: 1204.11143
MathSciNet: MR2377362
Digital Object Identifier: 10.2140/ant.2008.2.31

Primary: 11M26
Secondary: 05E10 , 15A52 , 33C80 , 60B15

Keywords: Cauchy identity , discrete moment , Random matrix theory , Riemann zeta function , unitary characteristic polynomial

Rights: Copyright © 2008 Mathematical Sciences Publishers

Vol.2 • No. 1 • 2008
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