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| | + | The secant zeta functions $\psi_s$ are defined by |
| | $$\psi_s(z) = \displaystyle\sum_{n=1}^{\infty} \dfrac{\sec(\pi n z)}{n^s}$$ | | $$\psi_s(z) = \displaystyle\sum_{n=1}^{\infty} \dfrac{\sec(\pi n z)}{n^s}$$ |
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| | =Properties= | | =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:
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| − | # when $z=\dfrac{p}{q}$ with $q$ odd, $s>1$
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| − | # when $z$ [[algebraic number| algebraic]] [[irrational number]] and $s >2$
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| − | # when $z$ is algebraic irrational and $s=2$.
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − | <div class="toccolours mw-collapsible mw-collapsed">
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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
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| − | $$\left| z - \dfrac{p}{q} \right|< \dfrac{k}{q^2}.$$
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| − | Then either $\dfrac{p}{q}$ is a [[convergent]] $\dfrac{p_{\ell}}{q_{\ell}}$ to $z$, or
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| − | $$\dfrac{p}{q} = \dfrac{ap_{\ell}+bp_{\ell-1}}{aq_{\ell}+bq_{\ell-1}}; |a|,|b|<2k,$$
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| − | where $a$ and $b$ are integers.
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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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
