# Meromorphic extension of a Dirichlet series

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 $s=1$ ?

a counter-example is $a_n = \ln n + n^{-1/4} (-1)^n \ln n$ such that $$\sum_{n=1}^\infty a_n n^{-s} = -\zeta'(s) + \eta'(s+1/4),\qquad \underset{ \text{for } Re(s) > 1/2+\epsilon}{\sum_{p} a_{p} p^{-s} = -\frac{\zeta'(s)}{\zeta(s)} + \frac{\zeta'(s+1/4)}{\zeta(s+1/4)}+ \mathcal{O}(1) }$$