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It is known that the infinite sum of the non-trivial zeros $\rho_n =\beta + \gamma_ni$ of $\zeta(z)$, when taken in pairs that are either conjugated or reflexive (they give the same outcome), converges:

$$Z_1(s):=\displaystyle \sum_{n=1}^\infty \frac{1}{\rho_n^s} + \sum_{n=1}^\infty \frac{1}{\overline{\rho_n^s}}$$

For $s \ge 1, s \in \mathbb{R}$ the sum can be described analytically for integer values e.g. $Z(1)= 1 + \frac{\gamma}{2}-\frac{\ln(4\pi)}{2}$. For $s<1$ the sum diverges.

However the individual sums $\dfrac{1}{\rho_n^s}$ and $\dfrac{1}{\overline{\rho_n^s}}$ do not converge. Even though the real parts do, their imaginary parts diverge and are annihilated by adding both sums together.

I believe there exists an alternative 'alternating' way to split up the total sum into subtotals that each do converge:

Take $\mu_n =\beta + (-1)^n \gamma_ni$ and: $$Z_2(s):= \sum_{n=1}^\infty \frac{1}{\mu_n^s}+\sum_{n=1}^\infty \frac{1}{\overline{\mu_n^s}}$$

and the individual sums seem to nicely converge for all $\beta \in \mathbb{R}$ and $s \ge 1$. Obviously $Z_1(s)=Z_2(s)$, however I found that contrary to $Z_1(s)$ the domain $Z_2(s)$ can be expanded towards $0 \lt s \lt 1$ by taking:

$$Z_{2-}(s):= \sum_{n=1}^\infty \frac{1}{\mu_n^s}-\sum_{n=1}^\infty \frac{1}{\overline{\mu_n^s}}$$

The difference converges towards an imaginary value and the diverging real parts are annihilated.


1) Can it be proven that $\displaystyle \sum_{n=1}^\infty \frac{1}{\mu_n^s}$ converges for all $s\ge1$ ?

2) Is there anything known about analytic (closed form) values of $\displaystyle \sum_{n=1}^\infty \frac{1}{\mu_n^s}$ ?

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Agno, sage/mpmath can compute numerically your first sum for complex $s$ too. – joro Apr 1 '13 at 15:40
...or compute something quite similar to it, not quite sure right now. – joro Apr 1 '13 at 15:41
Hi Joro. Complex $s$ do not seem to converge in my Maple program. What does work though is to make $\beta \in \mathbb{C}$ as long as $\beta \pm \gamma_n \ne 0$ since that would induce a pole. – Agno Apr 1 '13 at 16:31
@Agno the functions is secondzeta. The documentation is here:… – joro Apr 2 '13 at 5:51
@Joro, I don't believe these sums are the same. The secondzeta-function in mpmath sums $\displaystyle \sum_{n=1}^\infty \frac{1}{\gamma_n^s}$ and can indeed be analytically continued (via four components). However, $Z_1(s)$ sums over all (paired) $\rho_n$'s. The secondzeta-function for instance has a pole at $s=1$ whereas $Z_1(1)$ doesn't. – Agno Apr 2 '13 at 15:21

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