find the closed form of $\sum_{k=1}^\infty\left(\frac{\sec{kz}}{k^2}\right)^2$

How to evaluate $\displaystyle\sum_{k=1}^\infty\left(\frac{\sec{(k\pi\sqrt{5})}}{k^2}\right)^2$?

In general, how to find the closed form of infinite series $\displaystyle\sum_{k=1}^\infty\left(\frac{\sec{(kz)}}{k^2}\right)^2$ in term of variable $z$ ? Thanks in advance.

• what makes u believe that there is aclosed form? – tired Jan 10 '16 at 15:46
• Are you sure it converges? It probably does, but this is likely not a trivial result (and related to the irrationality measure of $\pi$). – Winther Jan 10 '16 at 15:53
• Disregard the last note in the comment above, I misread the sum. It has nothing to do with the irrationality measure of $\pi$ when $z = \pi \sqrt{5}$, but rather with that of $\sqrt{5}$. – Winther Jan 10 '16 at 16:12
• From wolfram alpha, the series is seem like converge. However, I can't prove it. I don't know is there a closed from. I'm just looking for possibilities. – Zoll Jan 10 '16 at 16:21