Hardy spaces of general Dirichlet series — a survey

The main purpose of this article is to survey on some key elements of a recent ${\cal{H}}_p$-theory of general Dirichlet series $\sum a_n e^{-\lambda_{n}s}$, which was mainly inspired by the work of Bayart and Helson on ordinary Dirichlet series $\sum a_n n^{-s}$. In view of an ingenious identification of Bohr, the ${\cal H}_p$-theory of ordinary Dirichlet series can be seen as a sub-theory of Fourier analysis on the infinite dimensional torus ${\mathbb T}^\infty$. Extending these ideas, the ${\cal H}_p$-theory of $\lambda$-Dirichlet series is built as a sub-theory of Fourier analysis on so-called $\lambda$-Dirichlet groups. A number of problems are added.