Small improvements on the Ball-Rivoal theorem and its $p$-adic variant

TL;DR

This paper improves lower bounds on the dimension of the span of 1 and odd zeta values, both real and p-adic, using a simplified proof that offers explicit non-vanishing linear forms and refines previous results.

Contribution

It provides a simpler proof that refines existing bounds on the linear independence of zeta values and introduces a new p-adic variant with improved estimates.

Findings

Lower bound on the dimension of the span of 1 and odd zeta values improved

Explicit non-vanishing small linear forms constructed

p-adic variant established with refined bounds

Abstract

We prove that the dimension of the $\mathbb{Q}$-linear span of $1,\zeta(3),\zeta(5),\ldots,\zeta(s-1)$ is at least $(1.119 \cdot \log s)/(1+\log 2)$ for any sufficiently large even integer $s$. This slightly refines a well-known result of Rivoal (2000) or Ball-Rivoal (2001). Quite unexpectedly, the proof only involves inserting the arithmetic observation of Zudilin (2001) into the original proof of Ball-Rivoal. Although this result is covered by a recent development of Fischler (2021+), our proof has the advantages of being simple and providing explicit non-vanishing small linear forms in $1$ and odd zeta values. Moreover, we establish the $p$-adic variant: for any prime number $p$, the dimension of the $\mathbb{Q}$-linear span of $1,\zeta_p(3),\zeta_p(5),\ldots,\zeta_p(s-1)$ is at least $(1.119 \cdot \log s)/(1+\log 2)$ for any sufficiently large even integer $s$. This is new, it slightly refines a result of Sprang (2020).