On the irrationality of certain $p$-adic zeta values
Li Lai, Cezar Lupu, Johannes Sprang
TL;DR
The paper proves a p-adic analogue of Zudilin's irrationality theorem for primes $p\ge 5$ by constructing a sequence of rational functions $R_n(t)$ and associated Volkenborn-analytic linear forms in $1$ and $p$-adic zeta values. By careful control of the $p$-adic norms and Archimedean growth, together with strong divisibility properties of the coefficients (mirroring Zudilin's approach), the authors show that for some odd $i$ in a computable interval $[3,c_p]$ the $p$-adic zeta value $\zeta_p(i)$ must be irrational. The argument hinges on combining an elementary $p$-adic irrationality criterion with explicit constructions of $R_n(t)$, their primitive $\widetilde{R}_n(t)$, and the resulting linear forms $S_n$, whose $p$-adic size decays faster than the coefficients grow in the Archimedean sense. This yields not only the existence of an irrational $\zeta_p(i)$ but also a concrete upper bound on the index interval, $c_p$, with $c_p\le p+p/\log p+5$ and asymptotics $c_p=p+(\gamma+o(1))\frac{p}{\log p}$ as $p\to\infty$. The result broadens the landscape of irrationality phenomena in $p$-adic zeta values and provides a framework for further refinements and extensions.
Abstract
A famous theorem of Zudilin states that at least one of the Riemann zeta values $ζ(5), ζ(7), ζ(9), ζ(11)$ 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 $ζ_p(i)$ is irrational.