Moments of the Derivative of Characteristic Polynomials with an Application to the Riemann Zeta Function
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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.
KeywordsUnit Circle Conjugacy Class Characteristic Polynomial Critical Line Riemann Zeta Function
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