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The sum of reciprocal of zeroes of riemann zeta function converges conditionally that if they are paired as $\rho $ and $1-\rho$ My question is if the sum still converges if they are paired as $\rho$ and $\rho$ conjugate .

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Yes. $\textbf{}$ $\textbf{}$ $\textbf{}$

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Both convergence arguments rely on the fact that $\sum_\rho 1/(\Im\rho)^2$ converges, which is a consequence of the upper bound $cT\log T$ for the number of zeros with imaginary part bounded in absolute value by $T$. Then, note $1/\rho+1/\bar\rho = 2\Re\rho/((\Re\rho)^2+(\Im\rho)^2) \le 2/(\Im\rho)^2$. –  Greg Martin Nov 28 '13 at 7:28
    
I fixed the awkward spacing before the period. I hope you don't mind. –  Potato Nov 28 '13 at 7:29
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Not at all. The awkwardness should all come from the content! –  Greg Martin Nov 28 '13 at 7:29

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