A new approach to the characteristic polynomial of a random unitary matrix

Since the seminal work of Keating and Snaith, the characteristic polynomial of a random (Haar-distributed) unitary matrix has seen several of its functional studied in relation with the probabilistic study of the Riemann Zeta function. We will recall the history of the topic starting with the Montgommery-Dyson correspondance and will revisit the last twenty years of computations of integer moments of some functionals, with a particular focus on the mid-secular coefficients recently studied by Najnudel-PaquetteSimm. The new method here introduced will be compared with one of the classical ways to deal with such functionals, the Conrey-Farmer-Keating-Rubinstein-Snaith (CFKRS) formula.