Absolute convergence of secant zeta function

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Theorem

The series defining the secant zeta function $\psi_s(z)$ converges absolutely in the following cases:

  1. when $z=\dfrac{p}{q}$ with $q$ odd, $s>1$
  2. when $z$ algebraic irrational number and $s >2$
  3. when $z$ is algebraic irrational and $s=2$.

Proof

References