  Keating and Snaith showed that the $2k^{th}$ absolute moment of the
characteristic polynomial of a random unitary matrix evaluated on the unit
circle is given by a polynomial of degree $k^2$. In this article, uniform
asymptotics for the coefficients of that polynomial are derived, and a maximal
coefficient is located. Some of the asymptotics are given in explicit form.
Numerical data to support these calculations are presented. Some apparent
connections between random matrix theory and the Riemann zeta function are
discussed.
