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

Thomas Sauvaget (1)

1 Chercheur indépendant

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.

Mots clés: irrationality; riemann zeta function

Domaines: Théorie des nombres [math.NT]

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