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 zeta 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.
|Translated title of the contribution||Moments of the derivative of characteristic polynomials wirh an application to the Riemann zeta function|
|Pages (from-to)||611 - 629|
|Journal||Communications in Mathematical Physics|
|Publication status||Published - Nov 2006|
Bibliographical notePublisher: Springer
Other identifier: IDS number 082OJ