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Moments of the derivative of the Riemann zeta-function and of characteristic polynomials

J. Brian Conrey·Michael O. Rubinstein·Nina C. Snaith
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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 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.

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J. Brian Conrey, Michael O. Rubinstein, Nina C. Snaith. 2006-03-17. Moments of the derivative of the Riemann zeta-function and of characteristic polynomials. https://arxiv.org/abs/math/0508378

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