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arXiv · 2505.23088

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

Li Lai·Cezar Lupu·Johannes Sprang
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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 $p \geqslant 5$ 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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Li Lai, Cezar Lupu, Johannes Sprang. 2025-05-29. On the irrationality of certain $p$-adic zeta values. https://arxiv.org/abs/2505.23088

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