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For every prime number $p$ fix a natural number $n_p$ with $$ C\frac{p+1}2\le n_p\le C(p+1) $$ for some constant $C>0$, independent of $p$. The product $$ F(s)=\prod_p\frac1{(1-p^{-s})^{n_p}} $$ converges for $\mathrm{Re}(s)>2$. Is it true that such an Euler product cannot be meromorphically continued to the entire complex plane?

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    $\begingroup$ Have you tried special cases? E.g. $C = 1$ and $n_p = p$? $\endgroup$
    – Dirk
    Commented Jul 28, 2017 at 8:19
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    $\begingroup$ yes, I did. The case $ n_p=p$ satisfies the claim. $\endgroup$
    – user1688
    Commented Jul 28, 2017 at 8:38
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    $\begingroup$ What do you get when applying the argument there ? $\endgroup$
    – reuns
    Commented Jul 28, 2017 at 9:32

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