On Bohr's theorem for general Dirichlet series

Abstract

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