Difference between revisions of "Secant zeta function"
From specialfunctionswiki
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$$\psi_s(z) = \displaystyle\sum_{n=1}^{\infty} \dfrac{\sec(\pi n z)}{n^s}$$ | $$\psi_s(z) = \displaystyle\sum_{n=1}^{\infty} \dfrac{\sec(\pi n z)}{n^s}$$ | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <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"> | ||
| + | <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. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
=References= | =References= | ||
[http://arxiv.org/pdf/1304.3922.pdf Secant zeta functions] | [http://arxiv.org/pdf/1304.3922.pdf Secant zeta functions] | ||
Revision as of 01:49, 6 March 2015
$$\psi_s(z) = \displaystyle\sum_{n=1}^{\infty} \dfrac{\sec(\pi n z)}{n^s}$$
Properties
Theorem: The series $\psi_s(z)$ converges absolutely in the following cases:
- when $z=\dfrac{p}{q}$ with $q$ odd, $s>1$
- when $z$ algebraic irrational number and $s >2$
- when $z$ is algebraic irrational and $s=2$.
Proof: █
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: █
