Find Dirichlet series of $2^n$

Question

How can I find the Dirichlet series of $2^n$?

The Dirichlet series of a sequence $\{a_n\}_{n=1}^\infty$ is defined as $f(s) = \sum_{i = 1}^\infty \frac{a_n}{n^s}$.

If $\{a_n\}_{n=1}^\infty$ is multiplicative, then we have the following formula for Dirichlet series: $\sum_{i = 1}^\infty \frac{a_n}{n^s} = \Pi_{p \text{ is prime }} g_p(p^{-s})$, where $g_p(x) := \sum_{i=0}^\infty a_{p^n}x^n$ is the ordinary generating function of $\{a_{p^n}\}_{n = 0}^\infty$.

However, I can not use this formula, because $2^n$ is not multiplicative.

Answer 1

$\forall s \in \mathbb{R}$, $\lim_{n \to \infty} \frac{2^n}{n^s} = \infty$

Thus the Dirichlet series $\sum_{n = 1}^\infty \frac{2^n}{n^s}$ diverges $\forall s \in \mathbb{R}$, according to Cauchy convergence test.