I am new here, so I don't know how this works exactly. If I do something wrong, please let me know.
I'd like help to solve a problem I am studying: Let $A$ be finite set of positive integers and denote by $|A|$ the number of elements in $A$.
Let
$$D(s)=\sum_{n \in A}a(n)n^{-s}$$
be a Dirichlet polynomial with coefficients supported in $A$, e supose that $D(s)$ is not constant. For each complex number $w$, define
$$M(w)=|\{m \geq 0; D^{(m)}(w)=0\}|,$$
i.e, $M(w)$ is the number of derivatives of D such that $D^{(k)}(w)=0$. Show that $M(w)<|A| \forall w \in \mathbb{C}$ .
I know that $D^{(k)}(w)=(-1)^k\sum_{n \in A}a(n)\log^k(n)n^{-w}$. But I don't know how to find the zeroes of these derivatives and how to estimate then... Do someone has any idea of how to start this?