In the explicit formula for the Chebyshev function, one of the terms that appears is $$ \sum_{\rho} \frac{x^\rho}{\rho} $$ where the sum is taken over the nontrivial zeroes of the zeta function. I'm a bit stuck on how one shows that this series converges.

I'm trying to understand the proof of the explicit formula. I understand all the steps, but it's still not clear to me why this sum ought to converge.

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    $\begingroup$ You can't show it directly, you need to relate the partial sum with a contour integral of $\zeta'(s)/(s\zeta(s)) x^s$ with the residue theorem, see any text proving the explicit formula $\endgroup$ – reuns Apr 13 at 20:53
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    $\begingroup$ @Tuvasbien Do you understand my comment ? The explicit formula (its convergence and the limit function $\psi(x)$) follows from the residue theorem and non-trivial bounds on $\zeta'/\zeta$ among which the density of zeros $\endgroup$ – reuns Apr 13 at 20:59
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    $\begingroup$ @reuns - thanks for the tip. It should be $\frac1{2\pi i} \int_\gamma \frac{\zeta'(s)}{s\zeta(s)} x^s ds$ for an appropriate contour $\gamma$. I'll have to take a look at how to bound $\zeta'/\zeta$ then... I'm also a bit worried because I don't see how to ensure that I can always draw the contour sufficiently far away from the poles of $\zeta'/\zeta$. $\endgroup$ – Dark Malthorp Apr 13 at 21:02
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    $\begingroup$ You need the density of zeros and the approximation of $\zeta'/\zeta(s)$ as the sum of the poles near $s$ plus $O(\log |s|)$, this is for the "cross the critical strip" part, the $\Re(s)< -1$ part follows from the functional equation, it is recommended to follow one of the textbook proving it in details $\endgroup$ – reuns Apr 13 at 21:05
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    $\begingroup$ As @reuns comments, these estimates are not easy. The more-or-less conceptual approach I know goes by way of Hadamard's theorem about products for entire functions, and the result about asymptotics of zeros versus growth of the function. To apply this, one needs asymptotics of Gamma and Phragmen-Lindelof. Again, see remarks in the proofs of Explicit Formulas. $\endgroup$ – paul garrett Apr 13 at 21:26

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