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

We prove that the dimension of the $\mathbb{Q}$-linear span of $1,\zeta (3),\zeta (5),\dots ,\zeta (s-1)$ is at least $(1.119\cdot \log s)\mathrm{/}(1+\log 2)$ 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 $1,{\zeta }_p(3),{\zeta }_p(5),\dots ,{\zeta }_p(s-1)$ is at least $(1.119\cdot \log s)\mathrm{/}(1+\log 2)$ for any sufficiently large even integer $s$. This is new, it slightly refines a result of Sprang (2020).