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

TL;DR

This paper proves a $p$-adic analogue of Zudilin's theorem, showing that for primes $p \\geq 5$, at least one $p$-adic zeta value at an odd integer in a specific range is irrational, extending classical results to the $p$-adic setting.

Contribution

It establishes the first $p$-adic analogue of Zudilin's theorem, demonstrating irrationality of certain $p$-adic zeta values within a specified range for primes $p \\geq 5.

Findings

For any prime $p \\geq 5$, there exists an odd integer $i$ in $[3,p+p/\\log p+5]$ with $\\zeta_p(i)$ irrational.

The result extends classical irrationality theorems to the $p$-adic context.

Provides a new perspective on the irrationality of special values in $p$-adic number theory.

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.