Many $p$-adic odd zeta values are irrational

Abstract

For any prime $p$ and $\epsilon >0$ we prove that for any sufficiently large positive odd integer $s$ at least $(c_p-\epsilon )\sqrt{\frac{s}{\log s}}$ of the $p$-adic zeta values $\hat{I}{\P }_p(3),\hat{I}{\P }_p(5),\dots ,\hat{I}{\P }_p(s)$ 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.

36 pages, to appear in Michigan Math. J