# A formula for ζ(2n + 1) and a proof of their irrationality

Abstract : Using a polylogarithmic identity, we express the values of $\zeta$ at odd integers $2n+1$ as integrals over unit $n-$dimensional hypercubes of simple functions involving products of logarithms. We also prove a useful property of those functions as some of their variables are raised to a power. In the case $n=2$, we prove two closed-form expressions concerning related integrals. Finally, another family of related integrals is introduced which, combined with Beukers's method, allows to show that all $\zeta(2n+1)$ (in fact all $\zeta(n)$) are irrational numbers.
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https://hal.archives-ouvertes.fr/hal-01352764
Contributor : Thomas Sauvaget <>
Submitted on : Saturday, December 10, 2016 - 8:31:08 PM
Last modification on : Monday, April 9, 2018 - 12:20:05 PM
Document(s) archivé(s) le : Monday, March 27, 2017 - 3:33:48 PM

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### Identifiers

• HAL Id : hal-01352764, version 3
• ARXIV : 1608.03174

### Citation

Thomas Sauvaget. A formula for ζ(2n + 1) and a proof of their irrationality. 2016. ⟨hal-01352764v3⟩

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