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

We prove that the dimension of the $\mathbb{Q}$-linear span of the Riemann zeta values $ζ(3),ζ(5),\ldots,ζ(s-1)$ is at least $(1.119 \cdot \log s)/(1+\log 2)$ for any sufficiently large even integer $s \geqslant s_0$. This result slightly refines the theorem of Rivoal (2000) or Ball--Rivoal (2001).