Many $p$-adic odd zeta values are irrational

Li Lai; Johannes Sprang

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arXiv · 2306.10393

Many $p$-adic odd zeta values are irrational

Li Lai·Johannes Sprang

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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 $\zeta_p(3),\zeta_p(5),\dots,\zeta_p(s)$ 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.

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Li Lai, Johannes Sprang. 2023-06-17. Many $p$-adic odd zeta values are irrational. https://arxiv.org/abs/2306.10393

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