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.

**Question:**

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 $\displaystyle \sum_{n=1}^\infty \frac{1}{\mu_n^s}$ ?

`secondzeta`

. The documentation is here: mpmath.googlecode.com/svn/trunk/doc/build/functions/… $\endgroup$ – joro Apr 2 '13 at 5:51