The Benford phenomenon for random variables. Discussion of Feller's way

Abstract

This is a detailed survey which mainly presents the Pinkham-Feller way. I added some new points to the first version [V2] and I suppressed "Examples" devoted to Gamma, Fr\'echet and Weibull laws. Theorem 2 is a bit more general (no assumption of density: this answers a question of T. Hill). Section 10 is new and devoted to an argument (Poincar\'e, Fewster) about the effect of high frequencies oscillations. Maybe many works, many efforts, have been devoted to the study of a sufficient condition of poor value: see Sections 7 and 11. The final Section gives some suggestions.