A new formula for ζ(2n + 1) (and how not to prove that ζ(5) is irrational)

Thomas Sauvaget (1)

1 Chercheur indépendant

Using a new polylogarithmic identity, we express the values of ζ at odd integers 2n + 1 as integrals over unit n−dimensional hypercubes of simple functions involving products of logarithms. We then make several conjectures, based on numerical evidence, on the behaviour of those functions as some of their variables are raised to a power. Finally, we discuss how one attempt to adapt Beukers' integral-based proof of the irrationality of ζ(2) and ζ(3) to the case of ζ(5) fails.

Mots clés: riemann zeta function; irrationality

Domaines: Théorie des nombres [math.NT]

Thomas Sauvaget. A new formula for ζ(2n + 1) (and how not to prove that ζ(5) is irrational). 2016. ⟨hal-01352764v2⟩