Difference between revisions of "Secant zeta function"

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The secant zeta functions $\psi_s$ are defined by
 
$$\psi_s(z) = \displaystyle\sum_{n=1}^{\infty} \dfrac{\sec(\pi n z)}{n^s}$$
 
$$\psi_s(z) = \displaystyle\sum_{n=1}^{\infty} \dfrac{\sec(\pi n z)}{n^s}$$
  
 
=Properties=
 
=Properties=
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[[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 $s=2$.
 
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<strong>Proof:</strong> █
 
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<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
 
$$\left| z - \dfrac{p}{q} \right|< \dfrac{k}{q^2}.$$
 
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.
 
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<strong>Proof:</strong> █
 
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=References=
 
=References=

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

Absolute convergence of secant zeta function

References