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The most general thing I've gotten is that the absolute convergence of $$\sum_{n=1}^{\infty} \frac{\csc^a (n + x)}{n^c}$$ implies that of $$\sum_{n=1}^{\infty} \frac{\csc^a \left(\frac{m}{2} n + \frac{x}{2} + \frac{\pi}{4} \pm \frac{\pi}{4}\right)}{n^c}$$ for integral $m$ and real $x$, but even this isn't enough to prove the theorem. It can easily be proven in the other direction, but I can't seem to prove that the absolute convergence of the secant series implies that of the cosecant series. Any ideas?

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