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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
Article

Abstract

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.

Keywords

Unit Circle Conjugacy Class Characteristic Polynomial Critical Line Riemann Zeta Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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