Secant zeta function
From specialfunctionswiki
The secant zeta functions $\psi_s$ are defined by $$\psi_s(z) = \displaystyle\sum_{n=1}^{\infty} \dfrac{\sec(\pi n z)}{n^s}$$
Properties
Absolute convergence of secant zeta function
Theorem: Let $z$ be irrational, $k \geq \dfrac{1}{2}$, and $\dfrac{p}{q}$ be a rational approximation to $z$ in reduced form for which $$\left| z - \dfrac{p}{q} \right|< \dfrac{k}{q^2}.$$ Then either $\dfrac{p}{q}$ is a convergent $\dfrac{p_{\ell}}{q_{\ell}}$ to $z$, or $$\dfrac{p}{q} = \dfrac{ap_{\ell}+bp_{\ell-1}}{aq_{\ell}+bq_{\ell-1}}; |a|,|b|<2k,$$ where $a$ and $b$ are integers.
Proof: █
