An Introduction to Dirichlet Series

Sammanfattning: We establish the central convergence properties of ordinary Dirichlet series, including the classical result by Bohr, providing uniform convergence of the series where it has a bounded analytic continuation. We also derive a lower bound for the supremum of Dirichlet polynomials using Kronecker's theorem, of which we see one proof. With this knowledge and some probability theory we can follow the work of Queffélec and Boas proving the existence of random series, with terms ±n^-s, with certain convergence properties. In particular Boas work is a probabilistic version of what Bohnenblust and Hille did, namely showing that the estimate of the distance of the abscissae of absolute and uniform convergence - estimated from above by 1/2 - is sharp. An abscissa denotes the vertical lines Re(s) to the right of which the Dirichlet series converges and to the left of which it diverges (in some sense of convergence), and is throroughly introduced in Chapter 1.