  In this paper, we propose a probabilistic approach to the study of the
characteristic polynomial of a random unitary matrix. We recover the Mellin
Fourier transform of such a random polynomial, first obtained by Keating and
Snaith, using a simple recursion formula, and from there we are able to obtain
the joint law of its radial and angular parts in the complex plane. In
particular, we show that the real and imaginary parts of the logarithm of the
characteristic polynomial of a random unitary matrix can be represented in law
as the sum of independent random variables. From such representations, the
celebrated limit theorem obtained by Keating and Snaith is now obtained from
the classical central limit theorems of Probability Theory, as well as some new
estimates for the rate of convergence and law of the iterated logarithm type
results.
