Latest revision as of 06:10, 16 June 2016

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

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+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=
-<div class="toccolours mw-collapsible mw-collapsed">
+[[Absolute convergence of secant zeta function]]
-<strong>Theorem:</strong> The series $\psi_s(z)$ converges absolutely in the following cases:
-# when $z=\dfrac{p}{q}$ with $q$ odd, $s>1$
-# when $z$ [[algebraic number| algebraic]] [[irrational number]] and $s >2$
-# when $z$ is algebraic irrational and $s=2$.
-<div class="mw-collapsible-content">
-<strong>Proof:</strong> █
-</div>
-</div>
-<div class="toccolours mw-collapsible mw-collapsed">
+=References=
-<strong>Theorem:</strong> Let $z$ be irrational, $k \geq \dfrac{1}{2}$, and $\dfrac{p}{q}$ be a rational approximation to $z$ in reduced form for which
+* {{PaperReference|Secant zeta functions|2014|Matilde Lalín}}
-$$\left| z - \dfrac{p}{q} \right|< \dfrac{k}{q^2}.$$
-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.
-<div class="mw-collapsible-content">
-<strong>Proof:</strong> █
-</div>
-</div>
+[[Category:SpecialFunction]]
-=References=
-[http://arxiv.org/pdf/1304.3922.pdf Secant zeta functions]