Dirichlet series and complex analysis for functions in infinitely many variables

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In recent years we have seen a remarkable growth of interest in certain functional analytic aspects of the theory of ordinary Dirichlet series $\sum_n a_n n^{-s}$. It became more and more clear that the theory of natural classes of ordinary Dirichlet series through Bohr's fundamental work is intimately connected with infinite dimensional holomorphy and Fourier analysis for functions in infinitely many variables. The most important and most intensively studied classes of ordinary Dirichlet series are given by the Hardy spaces $\mathcal{H}_p$. Contemporary research almost exclusively seems to deal with ordinary Dirichlet series, although the founding fathers of the theory like Bohr, Landau, Hardy and Riesz (among others) started with deep work on so-called general Dirichlet series $\sum_n a_n e^{-\lambda_n s}$ where $\lambda= (\lambda_n)$ is a strictly increasing sequence of positive real numbers tending to $\infty$ (called frequency). The main goal of our project is to reproduce some of the key results of the ordinary theory of $\mathcal{H}_p$-theory for general Dirichlet spaces (fixing a frequency). This seems a difficult task (at least for some topics) since the setting of general Dirichlet series is of course much broader than that of ordinary ones. In fact, the big challenge is to find reasonable replacements for the dramatic loss of tools from complex analysis and Fourier analysis on the infinite dimensional poly disc and torus.