Section: New Results

A formal proof of the irrationality of $\zeta \left(3\right)$

We have obtained a formal proof, machine-checked by the Coq proof assistant, of the irrationality of the constant $\zeta \left(3\right)$, that is, the evaluation at 3 of the Riemann zeta function of number theory. The result has been known in mathematics since the French mathematician Apéry's work in 1978, and several alternative proofs have been given since then. Our formalized result is the first complete proof by the computer (under the single assumption of the asymptotic behavior of the least common multiple of the first $n$ natural numbers). The core of this formal proof is based on (untrusted) computer-algebra calculations performed outside the proof assistant with the Mgfun Maple library developed by members of the team in the past. Then, we verify formally and a posteriori the desired properties of the objects computed by Maple and complete the proof of irrationality. This work [11] was formally presented at the conference on interactive theorem proving, ITP'14, and also as talks at MSC 2014 (Mathematical Structures of Computation) (http://smc2014.univ-lyon1.fr/ ), at the meeting MAP 2014 of the community on mathematics, algorithms and proofs (http://perso.crans.org/cohen/map2014/ ), and at JNCF'14, the meeting of the French computer-algebra community (http://www.lifl.fr/jncf2014/ ).