Many $p$-adic odd zeta values are irrational
Li Lai, Johannes Sprang
TL;DR
The paper establishes a p-adic analogue of the Lai–Yu elimination method for odd zeta values, proving that for any prime $p$ and any $\varepsilon>0$, a positive proportion $(c_p-\varepsilon)\sqrt{s/\log s}$ of the odd p-adic zeta values $\zeta_p(3),\zeta_p(5),\dots,\zeta_p(s)$ are irrational for sufficiently large odd $s$, where $c_p>0$ is explicitly given. The authors introduce a new irrationality criterion based on the $\ell(n)$-adic valuations of a family of linear forms, circumventing the non-vanishing obstacle that plagued previous p-adic adaptations. Central to the method is Volkenborn integration and p-adic Hurwitz zeta functions, enabling construction of fast-converging linear forms with controlled $p$-adic and Archimedean growth. Through a delicate, quantitative elimination argument and careful analysis of denominators and valuations, the paper yields a precise lower bound on the number of irrational odd zeta-values in the p-adic setting, extending the scope of classical irrationality results to the $p$-adic realm. The result contributes to the understanding of the arithmetic nature of $p$-adic zeta-values and demonstrates the efficacy of a new valuation-based irrationality criterion in tandem with the elimination framework.
Abstract
For any prime $p$ and $\varepsilon>0$ we prove that for any sufficiently large positive odd integer $s$ at least $(c_p-\varepsilon) \sqrt{\frac{s}{\log s}}$ of the $p$-adic zeta values $ζ_p(3),ζ_p(5),\dots,ζ_p(s)$ are irrational. The constant $c_p$ is positive and does only depend on $p$. This result establishes a $p$-adic version of the elimination technique used by Fischler--Sprang--Zudilin and Lai--Yu to prove a similar result on classical zeta values. The main difficulty consists in proving the non-vanishing of the resulting linear forms. We overcome this problem by using a new irrationality criterion.