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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.