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](http://algo.inria.fr/flajolet/Publications/mellin-harm.pdf), 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 $\mathbb{C}$?



On the relation between the asymptotics of a Dirichlet series' coefficients and the series' analytic continuability
Proposition 6 does not imply that if $V$ has positive natural density then $\zeta\_V(s)$ extends to a meromorphic function. This is because the density assumption is much weaker than the assumptions in Proposition 6. Indeed, if the elements of $V$ are $v\_1<v\_2<\dotsb$, then the density assumption says that $v\_k$ is asymptotically a constant times $k$, while Proposition 6 requires a full asymptotic expansion of $v\_k$ in descending powers of $k$. The latter means that, for any nonnegative integer $R$, there are exponents $w\_1>w\_2>\dotsb>w\_R$ and coefficients $c\_1,c\_2,\dotsc,c\_R\in\mathbb{R}$ such that
$$v\_k=c\_1 k^{w\_1}+c\_2 k^{w\_2}+\dotsb+c\_{R-1}k^{w\_{R-1}}+(c\_R+o(1))k^{w\_R}.$$
The density assumption implies that we have such a relation for $R=1$, namely $w\_1=1$ and $c\_1=1/d(V)$ work, but it does not imply the relation for $R=2$. And in fact, for the sequence of non-primes $\mathbb{N}\setminus\mathbb{P}$, the above approximation only holds for $R=1$.


