Irrationality of zeta values (following Apéry, Rivoal, \(\dots\)).
(Irrationalité de valeurs de zêta (d’après Apéry, Rivoal, \(\dots\)).)

*(French)*Zbl 1101.11024
Bourbaki seminar. Volume 2002/2003. Exposes 909–923. Paris: Société Mathématique de France (ISBN 2-85629-156-2/pbk). Astérisque 294, 27-62, Exp. No. 910 (2004).

This Bourbaki lecture is a survey of irrationality results for the values taken by the Riemann zeta function at integers \(\geq 2\). Of course, as is well-known, Euler proved that for any integer \(n\geq 1\), the number \(\zeta(2n)\) is a rational multiple of \(\pi^{2n}\). But no such result has ever been proved for the numbers \(\zeta(2n+1)\). Conjecturally, the numbers \(\pi, \zeta(3), \zeta(5), \zeta(7)\), etc, are algebraically independent over \(\mathbb{Q}\). Those results that we have supporting this conjecture are much weaker and were all proved in the last thirty years.

The article begins with the famous theorem that Apéry proved in 1978: \(\zeta(3)\) is irrational. Many proofs have been given of this result and the author presents a sketch of each proof known at the time of writing. An interesting feature is that these proofs all seemed at first sight to be based on different ideas (Padé approximants, modular forms, continued fractions, hypergeometric series and integrals, complex Barnes integrals, etc) but end up with the sequences originally used by Apéry. Consider the integers \(u_n=\sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2\); there exists a sequence \((v_n)_{n\geq 0}\) of rational numbers such that \(| u_n\zeta(3) - v_n | = (\sqrt{2}-1)^{4n+o(n)}\) and \(\{1, 2, \ldots, n\}^3v_n\in\mathbb{Z}\). These properties imply the irrationality of \(\zeta(3)\) and it is quite interesting that, except from trivial perturbations of the sequences \(u_n\) and \(v_n\), nobody has found so far a really different proof.

The other parts of the survey give more or less detailed presentations of some of the quantitative and qualitative results that followed Apéry’s work. First, the constructive nature of Apéry’s proof yielded an upper bound for the irrationality exponent \(\mu\) of \(\zeta(3)\). After successive improvements by Hata and, independently, Rhin and Viola, the latter proved in 2001 the best known result to date that \(2\leq \mu\leq 5.5139.\) Similar bounds are known for \(\log(2)\) (Rukhadze) and \(\pi^2\) (Rhin & Viola). Secondly, some completely new results for \(\zeta(2n+1)\) (for \(2n+1\geq 5\)) were proved in 2000 and the following years. It was for example proved that infinitely many of the numbers \(\zeta(2n+1)\) are linearly independent over \(\mathbb{Q}\) (Rivoal, Ball-Rivoal) and that at least one amongst the numbers \(\zeta(5), \zeta(7), \zeta(9), \zeta(11)\) is irrational (Zudilin). The main new idea used to prove these results is the introduction in this context of the classical notion of very-well-poised hypergeometric series, whose symmetries enable to construct sequences of rational linear forms in the \(\zeta(2n+1)\)’s only, i.e, the linear forms do not contain the \(\zeta(2n)\)’s which used to appear in other hypergeometric constructions. The proofs of these two results are sketched.

The author provides some interesting side remarks: optimality of Nesterenko’s criteria, the connection with Padé approximants and the linear forms generated by the generalized Beukers’ integrals introduced by Vasil’ev. He also mentions the denominators conjecture formulated by the reviewer (and proved by Krattenthaler and the reviewer jointly). In fact, a proof of Zudilin’s more general denominators conjecture could lead to an improvement of his above mentioned result. Finally, let us mention that the problem raised in the last sentences before section 3 on page 52 (i.e proving directly that Vasil’ev’s integrals are equal to linear forms in the \(\zeta(2n+1)\)’s) has now been solved by Zlobin in a remarkable tour de force.

For the entire collection see [Zbl 1052.00010].

The article begins with the famous theorem that Apéry proved in 1978: \(\zeta(3)\) is irrational. Many proofs have been given of this result and the author presents a sketch of each proof known at the time of writing. An interesting feature is that these proofs all seemed at first sight to be based on different ideas (Padé approximants, modular forms, continued fractions, hypergeometric series and integrals, complex Barnes integrals, etc) but end up with the sequences originally used by Apéry. Consider the integers \(u_n=\sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2\); there exists a sequence \((v_n)_{n\geq 0}\) of rational numbers such that \(| u_n\zeta(3) - v_n | = (\sqrt{2}-1)^{4n+o(n)}\) and \(\{1, 2, \ldots, n\}^3v_n\in\mathbb{Z}\). These properties imply the irrationality of \(\zeta(3)\) and it is quite interesting that, except from trivial perturbations of the sequences \(u_n\) and \(v_n\), nobody has found so far a really different proof.

The other parts of the survey give more or less detailed presentations of some of the quantitative and qualitative results that followed Apéry’s work. First, the constructive nature of Apéry’s proof yielded an upper bound for the irrationality exponent \(\mu\) of \(\zeta(3)\). After successive improvements by Hata and, independently, Rhin and Viola, the latter proved in 2001 the best known result to date that \(2\leq \mu\leq 5.5139.\) Similar bounds are known for \(\log(2)\) (Rukhadze) and \(\pi^2\) (Rhin & Viola). Secondly, some completely new results for \(\zeta(2n+1)\) (for \(2n+1\geq 5\)) were proved in 2000 and the following years. It was for example proved that infinitely many of the numbers \(\zeta(2n+1)\) are linearly independent over \(\mathbb{Q}\) (Rivoal, Ball-Rivoal) and that at least one amongst the numbers \(\zeta(5), \zeta(7), \zeta(9), \zeta(11)\) is irrational (Zudilin). The main new idea used to prove these results is the introduction in this context of the classical notion of very-well-poised hypergeometric series, whose symmetries enable to construct sequences of rational linear forms in the \(\zeta(2n+1)\)’s only, i.e, the linear forms do not contain the \(\zeta(2n)\)’s which used to appear in other hypergeometric constructions. The proofs of these two results are sketched.

The author provides some interesting side remarks: optimality of Nesterenko’s criteria, the connection with Padé approximants and the linear forms generated by the generalized Beukers’ integrals introduced by Vasil’ev. He also mentions the denominators conjecture formulated by the reviewer (and proved by Krattenthaler and the reviewer jointly). In fact, a proof of Zudilin’s more general denominators conjecture could lead to an improvement of his above mentioned result. Finally, let us mention that the problem raised in the last sentences before section 3 on page 52 (i.e proving directly that Vasil’ev’s integrals are equal to linear forms in the \(\zeta(2n+1)\)’s) has now been solved by Zlobin in a remarkable tour de force.

For the entire collection see [Zbl 1052.00010].

Reviewer: Tanguy Rivoal (Grenoble)

##### MSC:

11J72 | Irrationality; linear independence over a field |

11J82 | Measures of irrationality and of transcendence |

11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |