Hypergeometric series and irrationality of the values of the Riemann zeta function (Q1424590)

The author presents a survey of the ideas around his striking discoveries about the values of the Riemann zeta-function at the odd integers. He shows how hypergeometric series and Padé approximants provide a common approach to Apéry's theorem on the irrationality of \(\zeta(3)\), the author's remarkable proof that infinitely many of the \(\zeta(2n+1)\) are irrational, and Zudilin's theorem, improving another result of the author, that at least one of the numbers \(\zeta(5),\zeta(7),\zeta(9),\zeta(11)\) is irrational. Some teasing questions are added in the conclusion.