  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.
