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Let $\psi$ be the Chebyshev function. I would like to prove that the function $\sum_{n\ge0}(\sum_{k=0}^n\binom{n}{k}e^{\psi(k)})w^n$ can be analyticaly continued on an open set of $\mathbb C$ containing $\mathbb R^-$.

Does anyone know a way to prove that?

Thanks in advance

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  • $\begingroup$ What is the Chebyshev function? $\endgroup$ Mar 15, 2014 at 13:52
  • $\begingroup$ This the function defined from the Von Mangoldt function:$$\Psi(x)=\sum_{p\text{ prime}}\left[\log_p(x)\right]\log(p)$$ $\endgroup$
    – joaopa
    Mar 16, 2014 at 8:14
  • $\begingroup$ Yes this entire function is defined on the disk with radius $1/3$. But I would like to know whether it can be analytically expanded to an open se of $\mathbb C$ containing $\mathbb R^-$. $\endgroup$
    – joaopa
    Jun 5, 2014 at 5:17
  • $\begingroup$ Sorry, I didn't read properly. $\endgroup$ Jun 5, 2014 at 7:33

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