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

Abstract

We prove that the dimension of the $\mathbb{Q}$-linear span of the Riemann zeta values $\zeta (3),\zeta (5),\dots ,\zeta (s-1)$ is at least $(1.119\cdot \log s)/(1+\log 2)$ for any sufficiently large even integer $s⩾s_0$. This result slightly refines the theorem of Rivoal (2000) or Ball--Rivoal (2001). The proof incorporates the arithmetic factor ${\Phi }_n$ introduced by Zudilin (2001) into a slight generalization of the Ball--Rivoal hypergeometric construction. The asymptotic impact of the arithmetic factor ${\Phi }_n$ was not anticipated prior to this work. Although this result is subsumed by a much stronger recent development by Fischler (2026), our approach yields a new result in the $p$-adic setting, slightly refining a theorem of Sprang (2020). We prove that the dimension of the $\mathbb{Q}$-linear span of the $p$-adic zeta values ${\zeta }_p(3),{\zeta }_p(5),\dots ,{\zeta }_p(s-1)$ is at least $(1.119\cdot \log s)/(1+\log 2)$ for any prime $p$ and any sufficiently large even integer $s⩾s_0(p)$.

arXiv comment50 pages, 1 figure, 2 tables; v3 made minor corrections, to appear in FCNT