There is a wonderful series of articles by Flajolet et. al. about Mellin Transforms and the asymptotic analysis of generating functions. In particular, on page 45 of the article Mellin Transforms and Asymptotics: Harmonic Sums, they state the following result:

Proposition 6 (Growth of Special Dirichlet Series): Let $\lambda_{k}$ and $\mu_{k}$ admit asymptotic expansions in descending powers of $k$ as $$\lambda_{k}\sim\sum_{r=0}^{\infty}\frac{a_{r}}{k^{\alpha_{r}}}$$ $$\mu_{k}\sim k^{w}\left(1+\sum_{r=1}^{\infty}\frac{b_{r}}{k^{\beta_{r}}}\right)$$ Then the Dirichlet series $\sum_{k}\lambda_{k}\mu_{k}^{-s}$ can be continued to a meromorphic function $\Lambda\left(s\right)$ in the whole of the complex plane.

Now, let $V$ be an arbitrary set of infinitely many positive integers, and let: $$\zeta_{V}\left(s\right)\overset{\textrm{def}}{=}\sum_{v\in V}\frac{1}{v^{s}}$$

Enumerating the elements of $V$ in increasing order as $v_{1},v_{2},\ldots$, recall that one way of defining the natural density of $V$ (denoted $d\left(V\right))$ is: $$d\left(V\right)=\lim_{n\rightarrow\infty}\frac{n}{v_{n}}$$ In the case where $d\left(V\right)$ exists and is positive, this gives the asymptotic $d\left(V\right)v_{n}\sim n$. As such, for: $$\frac{\zeta_{V}\left(s\right)}{\left(d\left(V\right)\right)^{s}}=\sum_{n=1}^{\infty}\frac{1}{\left(d\left(V\right)v_{n}\right)^{s}}$$ I have that $$\lambda_{k}=1$$ and $$\mu_{k}=d\left(V\right)v_{k}\sim k$$ which is obtained by taking:$$\mu_{k}\sim k^{w}\left(1+\sum_{r=1}^{\infty}\frac{b_{r}}{k^{\beta_{r}}}\right)$$ setting $w=1$, and letting all the $b_{r}$s be $0$. Hence, unless I am mistaken, Proposition 6 implies that $\zeta_{V}\left(s\right)$ extends to a meromorphic function on $\mathbb{C}$ whenever $V$ has positive natural density.

However, consider the following. Let $\mathbb{P}$ denote the set of prime numbers. Then, $\zeta_{\mathbb{P}}\left(s\right)$ is the so-called Prime Zeta Function, which is known to have a natural boundary on the imaginary axis. On the other hand, since $d\left(\mathbb{P}\right)=0$, it follows that $\mathbb{N}/\mathbb{P}$ has a well-defined natural density of $1$, and thus, by Proposition 6, that $\zeta_{\mathbb{N}/\mathbb{P}}\left(s\right)$ is meromorphic on $\mathbb{C}$. However, since I can write:$$\zeta_{\mathbb{P}}\left(s\right)=\zeta\left(s\right)-\zeta_{\mathbb{N}\backslash\mathbb{P}}\left(s\right)$$ it follows that $\zeta_{\mathbb{P}}\left(s\right)$ is the difference of two meromorphic functions, which forces $\zeta_{\mathbb{P}}\left(s\right)$ to be meromorphic on $\mathbb{C}$, which is obviously not correct.

So, where's the error, and how (if at all) can it be rectified? In particular, when, if ever, does the existence of $d\left(V\right)$ imply that $\zeta_{V}\left(s\right)$ is meromorphic on $\mathbb{C}$?