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Let $p_n$ be the n'th non-trivial zero of the Riemann zeta function. We define the Zetor function (acronym of 'zeta' and 'zero') as follows: $\zeta \rho (s) = \sum_{n=1}^{\infty} \frac{1}{(p_n)^s}. $

For which values of $s$ does $\zeta \rho (s)$ converge?

Thanks,

Max

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    @ Neves: wow a whole book! And I though I had thought of something new... thanks a lot.2010-12-31

1 Answers 1

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From Hadamard's theorem about products versus the summability of powers of zeros, from the functional equation (and Phragmen-Lindelof) we know that $\sum {1\over |\rho|^\sigma}$ is (absolutely) convergent for $\sigma>1$.