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.

中文翻译：

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关于一般狄利克雷级数的玻尔定理

我们在一般狄利克雷级数 $D=\sum a_{n} e^{-\lambda_{n}s}$ 上呈现波尔定理的定量版本，假设对频率 $\lambda:=(\lambda_{n}) $，包括 Bohr 和 Landau 引入的条件。因此，使用 M. Riesz 发明的典型（第一）方法求和方法，在 $\lambda$ 上没有任何条件，我们给出部分和运算符 $S_{N}(D):=\sum_ 的范数的上限{n=1}^{N} a_{n}(D)e^{-\lambda_{n}s}$ 空间 $\mathcal{D}_{\infty}^{ext }(\lambda)$ 的所有某处收敛 $\lambda$-Dirichlet 级数允许全纯和有界扩展到开放的右半平面 $[Re>0]$。因此，对于某些类别的 $\lambda$，我们在 $\mathcal{D}_{\infty}(\lambda)$ 中获得了 Montel 定理；收敛于 $[Re>0]$ 的所有 $D \in \mathcal{D}_{\infty}^{ext}(\lambda)$ 的空间。此外，遵循 Neder 的思想，我们给出了频率 $\lambda$ 的构造，其中 $\mathcal{D}_{\infty}(\lambda)$ 不完整。