Absolute convergence of secant zeta function

Theorem

The series $\psi_s(z)$ 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 $s=2$.

References

• Matilde Lalín: Secant zeta functions (2014): Theorem 1.