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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]

Recommended reading

Nonstandard Mathematics