# Hardy spaces of general Dirichlet series - a survey

## TL;DR

This survey reviews recent developments in the $_p$-theory of general Dirichlet series, extending classical Fourier analysis techniques to broader settings involving $$-Dirichlet groups.

## Contribution

It synthesizes recent advances in $_p$-theory for Dirichlet series and introduces the framework of $$-Dirichlet groups for analysis.

## Key findings

- Extension of $_p$-theory to general Dirichlet series
- Identification of $_p$-theory as a sub-theory of Fourier analysis on $$-Dirichlet groups
- Discussion of open problems in the field

## Abstract

The main purpose of this article is to survey on some key elements of a recent $\mathcal{H}_p$-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 $\mathcal{H}_p$-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 $\mathcal{H}_p$-theory of $\lambda$-Dirichlet series is build as a sub-theory of Fourier analysis on what we call $\lambda$-Dirichlet groups. A number of problems is added.