On Bohr's theorem for general Dirichlet series

We present quantitative versions of Bohr's theorem on general Dirichlet series $D=\sum a_{n} e^{-λ_{n}s}$ assuming different assumptions on the frequency $λ:=(λ_{n})$, 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 $S_{N}(D):=\sum_{n=1}^{N} a_{n}(D)e^{-λ_{n}s}$ of length $N$ on the space $\mathcal{D}_{\infty}^{ext}(λ)$ of all somewhere convergent $λ$-Dirichlet series allowing a holomorphic and bounded extension to the open right half plane $[Re>0]$. As a consequence for some classes of $λ$'s we obtain a Montel theorem in $\mathcal{D}_{\infty}(λ)$; the space of all $D \in \mathcal{D}_{\infty}^{ext}(λ)$ which converge on $[Re>0]$. Moreover following the ideas of Neder we give a construction of frequencies $λ$ for which $\mathcal{D}_{\infty}(λ)$ fails to be complete.

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