On the irrationality of certain $p$-adic zeta values

Abstract

A famous theorem of Zudilin states that at least one of the Riemann zeta values $\zeta (5),\zeta (7),\zeta (9),\zeta (11)$ is irrational. In this paper, we establish the $p$-adic analogue of Zudilin's theorem. As a weaker form of our result, it is proved that for any prime number $p⩾5$ there exists an odd integer $i$ in the interval $[3,p+p/\log p+5]$ such that the $p$-adic zeta value ${\zeta }_p(i)$ is irrational.

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