Title: Random characters under the $L$-measure, I : Dirichlet characters,
Abstract: We define the $L$-measure on the set of Dirichlet characters as an analogue
of the Plancherel measure, once considered as a measure on the irreducible
characters of the symmetric group.
We compare the two measures and study the limit in distribution of characters
evaluations when the size of the underlying group grows. These evaluations are
proven to converge in law to imaginary exponentials of a Cauchy distribution in
the same way as the rescaled windings of the complex Brownian motion. This
contrasts with the case of the symmetric group where the renormalised
characters converge in law to Gaussians after rescaling (Kerov Central Limit
Theorem).