Hi there, I am struggling with a theorem about truncated Dirichlet series. I am trying to prove the following theorem:

Let $(a_n)_n \subset \mathbb{C}$ and $N \in \mathbb{N}$. Then $\sup_{t \in \mathbb{R}} \vert \sum_{n=1}^N a_n n^{-it} \vert = \sup_{\Re s \ge 0} \vert \sum_{n=1}^N a_n n^{-s} \vert$.

I think the trick is to use the maximum modulus principle for holomorphic functions (a holomorphic function, continuous up to the boundary attains its maximum on the boundary). Using this for the domain $-a < \Im s < a, 0 < \Re s < b$ could lead to the answer ($a,b \to \infty$), but how do I show that such a truncated Dirichlet series attains its maximum on the *left* border?

I would appreciate any hint on the proof or reference to any literature.