  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.
