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We present quantitative versions of Bohr's theorem on general Dirichletseries $D=\sum a_{n} e^{-\lambda_{n}s}$ assuming different assumptions on thefrequency $\lambda:=(\lambda_{n})$, including the conditions introduced by Bohrand Landau. Therefore using the summation method by typical (first) meansinvented by M. Riesz, without any condition on $\lambda$, we give upper boundsfor 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 theopen right half plane $[Re>0]$. As a consequence for some classes of$\lambda$'s we obtain a Montel theorem in $\mathcal{D}_{\infty}(\lambda)$; thespace of all $D \in \mathcal{D}_{\infty}^{ext}(\lambda)$ which converge on$[Re>0]$. Moreover following the ideas of Neder we give a construction offrequencies $\lambda$ for which $\mathcal{D}_{\infty}(\lambda)$ fails to becomplete.

On Bohr's theorem for general Dirichlet series