Derivative moments of CUE characteristic polynomials and the Riemann zeta function

I will discuss recent work on the derivative of the characteristic polynomial from the Circular Unitary Ensemble. The main focus is on the calculation of moments with values of the spectral parameter z inside the unit disc. We investigate three asymptotic regimes depending on the distance of z to the unit circle, as the size of the matrices tends to infinity. I will also discuss some corresponding results for the derivative of the Riemann zeta function. This is joint work with Fei Wei (Sussex).