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= Welcome to MWiki = | = Welcome to MWiki = | ||
== Theorem of the month == | == Theorem of the month == | ||
− | === | + | === Finite representation for odd <math>\zeta</math>-arguments === |
− | + | Using the digamma function <math>\psi</math>, it holds for <math>n \in {}^{\omega}2\mathbb{N}^{*}</math>, small <math>\varepsilon \in ]0, 1]</math> and <math>{{d}_{\varepsilon k n}}:={{\varepsilon}^{{\hat{n}}}}{e}^{\hat{n}2k\pi i}</math> that<div style="text-align:center;"><math>\zeta(\grave{n}) = \underset{\varepsilon \to 0}{\mathop{\lim }}\,\widehat{-\varepsilon n}\sum\limits_{k=1}^{n}{\left( \gamma +\psi ({{d}_{\varepsilon k n}}) \right)}+\mathcal{O}(\varepsilon )</math></div>and<div style="text-align:center;"><math>\zeta(\grave{n}) = \underset{\varepsilon \to 0}{\mathop{\lim }}\,\widehat{2\varepsilon n}\sum\limits_{k=1}^{n}{\left( \psi ({{d}_{\varepsilon k n}}{{i}^{\hat{n}2}})-\psi ({{d}_{\varepsilon k n}}) \right)}+\mathcal{O}({{\varepsilon }^{2}}).</math></div> | |
==== Proof: ==== | ==== Proof: ==== | ||
− | <div style="text-align:center;"><math>\ | + | The claim results easily via the geometric series from <div style="text-align:center;"><math>\psi (z)+\gamma +\hat{z}=\sum\limits_{m=1}^{\omega }{\left( \hat{m}-\widehat{m+z} \right)}=-\sum\limits_{m=1}^{\omega }{\zeta(\grave{m}){{(-z)}^{m}}}=z\sum\limits_{m=1}^{\omega }{\hat{m}\widehat{m+z}}.\square</math></div> |
== Recommended reading == | == Recommended reading == |
Revision as of 17:59, 3 March 2020
Welcome to MWiki
Theorem of the month
Finite representation for odd [math]\displaystyle{ \zeta }[/math]-arguments
Using the digamma function [math]\displaystyle{ \psi }[/math], it holds for [math]\displaystyle{ n \in {}^{\omega}2\mathbb{N}^{*} }[/math], small [math]\displaystyle{ \varepsilon \in ]0, 1] }[/math] and [math]\displaystyle{ {{d}_{\varepsilon k n}}:={{\varepsilon}^{{\hat{n}}}}{e}^{\hat{n}2k\pi i} }[/math] that
[math]\displaystyle{ \zeta(\grave{n}) = \underset{\varepsilon \to 0}{\mathop{\lim }}\,\widehat{-\varepsilon n}\sum\limits_{k=1}^{n}{\left( \gamma +\psi ({{d}_{\varepsilon k n}}) \right)}+\mathcal{O}(\varepsilon ) }[/math]
and
[math]\displaystyle{ \zeta(\grave{n}) = \underset{\varepsilon \to 0}{\mathop{\lim }}\,\widehat{2\varepsilon n}\sum\limits_{k=1}^{n}{\left( \psi ({{d}_{\varepsilon k n}}{{i}^{\hat{n}2}})-\psi ({{d}_{\varepsilon k n}}) \right)}+\mathcal{O}({{\varepsilon }^{2}}). }[/math]
Proof:
The claim results easily via the geometric series from
[math]\displaystyle{ \psi (z)+\gamma +\hat{z}=\sum\limits_{m=1}^{\omega }{\left( \hat{m}-\widehat{m+z} \right)}=-\sum\limits_{m=1}^{\omega }{\zeta(\grave{m}){{(-z)}^{m}}}=z\sum\limits_{m=1}^{\omega }{\hat{m}\widehat{m+z}}.\square }[/math]