Translation of Dirichlet polynomial

Question

Assume $P(s)$ is a Dirichlet polynomial, say \begin{equation} P(s)=\sum_{n=1}^N\frac{a(n)}{n^s}. \end{equation}

Is $P(1-s)$ a Dirichlet polynomial too? I.e., is there a way to write \begin{equation} \sum_{n=1}^N\frac{a(n)}{n^{1-s}}=\sum_{n=1}^M\frac{b(n)}{n^s} \end{equation} with certain $b(n)$?

Answer 1

Only if $P$ is constant ($a(n) = 0$ for $n > 1$). If $P$ is a Dirichlet polynomial (or a Dirichlet series that converges somewhere), then we have

$$\lim_{\operatorname{Re} s \to +\infty} c^{-s}P(s) = 0$$

for every $c > 1$.

But if $P$ is a non-constant Dirichlet polynomial,

$$P(s) = \sum_{n = 1}^N \frac{a(n)}{n^s}$$

with $N > 1$ and $a(N) \neq 0$, then

$$\lim_{\operatorname{Re} s \to +\infty} N^{-s}P(1-s) = \frac{a(N)}{N} \neq 0.$$