Probabilistic Methods for Dirichlet Series

The title of this chapter is a little emphatic, because the probabilistic methods will here concentrate essentially about one maximal inequality, which is fairly well-known in harmonic analysis, but will have a specific aspect, due to the Bohr point of view on Dirichlet series. We tried to keep the presentation as self-contained as possible, since the subject may be not completely familiar to some number-theoretists. Let us emphasize that those probabilistic methods have a great flexibility, and are nearly compulsory in some questions, even if the initial proof of the Bohnenblust-Hille theorem, to be proved in the last section, made no use of such methods.