I asked this [question](https://math.stackexchange.com/questions/1901209/meromorphic-extension-of-a-dirichlet-series) in SEM but I got no answer, so I'm trying my luck here.


Let the Dirichlet series $\phi(s)=\sum\_{n\ge 1}\frac{a(n)}{n^s}$ be absolutely convergent for $\Re(s)>1$ and extend to a meromorphic function on the half plane $\Re(s)>1/2$ with only pole at $s=1.$ write 
$$\phi(s)=\sum\_{\text{prime}}\frac{a(p)}{p^s}+\sum\_{\text{non-prime}}\frac{a(n)}{n^s}$$
 If we suppose that $\sum\_{\text{prime}}\frac{a(p)}{p^s}$ converges absolutely for $\Re(s)>1$ and has a pole at $s=1.$ We can deduce that the series $\sum\_{\text{prime}}\frac{a(p)}{p^s}$ can be continued analytically to a meromorphic function in the half plane $\Re(s)>1/2$ with only pole at $s=1$ ?



Meromorphic continuation of a Dirichlet series
* Assuming the Riemann hypothesis is false, let $a\_n = \ln n$, and you have a counter-example.
* Assuming the Riemann hypothesis is true, let $a\_p = \ln p + p^{-1/4}\ln p, a\_{p+1} = \ln(p+1) - p^{-1/4}\ln p, a\_n = \ln n$ otherwise, and you get a counter-example.


