Communications in Mathematical Physics

, Volume 267, Issue 3, pp 611–629 | Cite as

Moments of the Derivative of Characteristic Polynomials with an Application to the Riemann Zeta Function

  • J. B. Conrey
  • M. O. RubinsteinEmail author
  • N. C. Snaith


We investigate the moments of the derivative, on the unit circle, of characteristic polynomials of random unitary matrices and use this to formulate a conjecture for the moments of the derivative of the Riemann ζ function on the critical line. We do the same for the analogue of Hardy’s Z-function, the characteristic polynomial multiplied by a suitable factor to make it real on the unit circle. Our formulae are expressed in terms of a determinant of a matrix whose entries involve the I-Bessel function and, alternately, by a combinatorial sum.


Unit Circle Conjugacy Class Characteristic Polynomial Critical Line Riemann Zeta Function 
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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • J. B. Conrey
    • 1
    • 2
  • M. O. Rubinstein
    • 3
    Email author
  • N. C. Snaith
    • 2
  1. 1.American Institute of MathematicsPalo AltoUSA
  2. 2.School of MathematicsUniversity of BristolBristolUnited Kingdom
  3. 3.Pure MathematicsUniversity of WaterlooWaterlooCanada

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