Hyperg{\'e}om{\'e}trie et fonction z{\^e}ta de Riemann

TL;DR

This paper proves a key conjecture about the denominators of coefficients in linear forms of zeta values, leading to new results on the irrationality and independence of specific zeta constants.

Contribution

It establishes the denominator conjecture for linear forms in zeta values, enabling new proofs of irrationality and independence of certain zeta constants.

Findings

Proved the denominator conjecture for coefficients of linear forms in zeta values.

Showed at least one of the eight zeta values between 5 and 19 is irrational.

Identified an odd integer j between 5 and 165 such that 1, ζ(3), and ζ(j) are linearly independent over Q.

Abstract

We prove the second author's "denominator conjecture" [40] concerning the common denominators of coefficients of certain linear forms in zeta values. These forms were recently constructed to obtain lower bounds for the dimension of the vector space over $\mathbb Q$ spanned by $1,\zeta(m),\zeta(m+2),...,\zeta(m+2h)$, where $m$ and $h$ are integers such that $m\ge2$ and $h\ge0$. In particular, we immediately get the following results as corollaries: at least one of the eight numbers $\zeta(5),\zeta(7),...,\zeta(19)$ is irrational, and there exists an odd integer $j$ between 5 and 165 such that 1, $\zeta(3)$ and $\zeta(j)$ are linearly independent over $\mathbb{Q}$. This strengthens some recent results in [41] and [8], respectively. We also prove a related conjecture, due to Vasilyev [49], and as well a conjecture, due to Zudilin [55], on certain rational approximations of $\zeta(4)$. The proofs are based on a hypergeometric identity between a single sum and a multiple sum due to Andrews [3]. We hope that it will be possible to apply our construction to the more general linear forms constructed by Zudilin [56], with the ultimate goal of strengthening his result that one of the numbers $\zeta(5),\zeta(7),\zeta(9),\zeta(11)$ is irrational.