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.