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We present quantitative versions of Bohr's theorem on general Dirichlet series assuming different assumptions on the frequency , including the conditions introduced by Bohr and Landau. Therefore, using the summation method by typical (first) means invented by M. Riesz, without any condition on λ, we give upper bounds for the norm of the partial sum operator of length N on the space of all somewhere convergent λ‐Dirichlet series, which allow a holomorphic and bounded extension to the open right half plane . As a consequence for some classes of λ's we obtain a Montel theorem in ; the space of all which converge on . Moreover, following the ideas of Neder we give a construction of frequencies λ for which fails to be complete.

A result of Helson on general Dirichlet series a n e −λns states that, whenever (a n ) is 2-summable and λ = (λ n ) satisfies a certain condition introduced by Bohr, then for almost all homomorphism ω : (R, +) → T the Dirichlet series a n ω(λ n )e −λns converges on the open right half plane [Re > 0]. For ordinary Dirichlet series a n n −s Hedenmalm and Saksman related this result with the famous Carleson-Hunt theorem on pointwise convergence of Fourier series, and Bayart extended it within his theory of Hardy spaces H p of such series. The aim here is to prove variants of Helson's theorem within our recent theory of Hardy spaces H p (λ), 1 ≤ p ≤ ∞, of general Dirichlet series. To be more precise, in the reflexive case 1 < p < ∞ we extend Helson's result to Dirichlet series in H p (λ) without any further condition on the frequency λ, and in the non-reflexive case p = 1 to the wider class of frequencies satisfying the so-called Landau condition (more general than Bohr's condition). In both cases we add relevant maximal inequalities. Finally, we give several applications to the structure theory of Hardy spaces of general Dirichlet series.

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The main purpose of this article is to survey on some key elements of a recent H p -theory of general Dirichlet series a n e −λns , which was mainly inspired by the work of Bayart and Helson on ordinary Dirichlet series a n n −s . In view of an ingenious identification of Bohr, the H p -theory of ordinary Dirichlet series can be seen as a sub-theory of Fourier analysis on the infinite dimensional torus T ∞ . Extending these ideas, the H p -theory of λ-Dirichlet series is build as a sub-theory of Fourier analysis on what we call λ-Dirichlet groups. A number of problems is added.

A. For a general Dirichlet series ∑ a n e −λ n s with frequency λ = (λ n ) n , we study how horizontal translation (i.e. convolution with a Poisson kernel) improves its integrability properties. We characterize hypercontractive frequencies in terms of their additive structure answering some questions posed by Bayart. We also provide sharp bounds for the strips S p (λ) that encode the minimum translation necessary for series in the Hardy space H p (λ) to have absolutely convergent coefficients.

Inspired by a recent article on Fréchet spaces of ordinary Dirichlet series ∑ a n n −s due to J. Bonet, we study topological and geometrical properties of certain scales of Fréchet spaces of general Dirichlet spaces ∑ a n e −λ n s . More precisely, fixing a frequency λ = (λ n ), we focus on the Fréchet space of λ -Dirichlet series which have limit functions bounded on all half planes strictly smaller than the right half plane [Re > 0]. We develop an abstract setting of pre-Fréchet spaces of λ -Dirichlet series generated by certain admissible normed spaces of λ -Dirichlet series and the abscissas of convergence they generate, which allows also to define Fréchet spaces of λ -Dirichlet series for which a n e −λ n /k for each k equals the Fourier coefficients of a function on an appropriate λ -Dirichlet group.

Opened up by early contributions due to, among others, H. Bohr, Hardy-Riesz, Bohnenblust-Hille, Neder and Landau the last 20 years show a substantial revival of systematic research on ordinary Dirichlet series a n n −s , and more recently even on general Dirichlet series a n e −λns . This involves the intertwining of classical work with modern functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory. Motivated through this line of research the main goal of this article is to start a systematic study of a variety of fundamental aspects of vector-valued general Dirichlet series a n e −λns , so Dirichlet series, where the coefficients are not necessarily in C but in some arbitrary Banach space X.