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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?



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The mathworld article on the Riemann Zeta Function zeroes seems to discuss a little bit about your "Zetor" function, at least in the case of integer values of it. Look in particular at equation (3) in the article where they consider "Zetor" for the first time. Although the article does not seem to discuss this in exactly the generality you want, it has some references which may be useful to you. – Adrián Barquero Dec 29 '10 at 20:49
@ Adrian: Thank you, looks useful indeed. – Max Muller Dec 29 '10 at 20:55
This article by D.H. Lehmer seems relevant, it is listed in the mathworld entry I referred to in my other comment. I don't have access to it right now so I can't pass from the first page. – Adrián Barquero Dec 29 '10 at 20:57
Luckily, the AMS was generous enough to allow free access to old Mathematics of Computation issues... so, here is the Lehmer article @Adrián was referring to. – J. M. Dec 30 '10 at 10:58
@Max Muller, The book Zeta Functions Over Zeros of Zeta Functions by André Voros discusses this "Zetor function". – Neves Dec 30 '10 at 15:28

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 $\sigma>1$.

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