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I asked this question 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 $s=1$ ?

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  • 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.

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  • $\begingroup$ Thanks for your answer !!! I am really interested to the case where the $a(n)$ are Fourier coefficients of a modular forms... $\endgroup$
    – Adam
    Aug 24, 2016 at 15:09
  • $\begingroup$ @Adam the simplest counter-example is probably $a_n = \ln n + (-1)^n n^{-1/4} \ln n$. And if you meant $a(n)/\ln n$ are coefs of a modular form (otherwise $\sum_n a_n n^{-s}$ has no pole at $s=1$, or $\sum_p a_p p^{-s}$ is not meromorphic at $s=1$) by the Hecke operator theory it means $\phi(s) = \sum_{m=0}^M \sum_n b_n^m n^{-s} \ln n$ where $ b_n^m$ are multiplicative (and there is a Riemann hypothesis for each $\sum_n b_n^m n^{-s}$), so it should really be equivalent with GRH $\endgroup$
    – reuns
    Aug 24, 2016 at 15:22
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    $\begingroup$ @Adam, I think you must add it to your question, and you should specify if you're interested to modular forms of integral wieght or half-ntegral weight or Maass wave forms ...! $\endgroup$
    – Med
    Aug 24, 2016 at 15:28
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    $\begingroup$ @Adam it changes completely the question, and I can't answer if you meant non integral weight modular forms $\endgroup$
    – reuns
    Aug 24, 2016 at 15:43
  • $\begingroup$ @Med what would be the answer in the case of modular forms ? $\endgroup$
    – reuns
    Aug 24, 2016 at 15:47

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