  We investigate the joint moments of the 2k-th power of the characteristic
polynomial of random unitary matrices with the 2h-th power of the derivative of
this same polynomial. We prove that for a fixed h, the moments are given by
rational functions of k, up to a well-known factor that already arises when
h=0. We fully describe the denominator in those rational functions (this had
already been done by Hughes experimentally), and define the numerators through
various formulas, mostly sums over partitions. We also use this to formulate
conjectures on joint moments of the zeta function and its derivatives, or even
the same questions for the Hardy function, if we use a ``real'' version of
characteristic polynomials. Our methods should easily be applicable to other
similar problems, for instance with higher derivatives of characteristic
polynomials.
  NOTE: More data is available on the author's website or attached to the LaTeX
source of this arXiv submission.
