A note on the irrationality of $ζ_2(5)$
Li Lai, Johannes Sprang, Wadim Zudilin
TL;DR
The paper constructs explicit Apéry-style rational approximations to the 2-adic zeta value $\zeta_2(5)$ and proves its irrationality via a new three-term recurrence that yields integer-valued components. It formulates and analyzes the linear forms $S_n=-\int_{\mathbb{Z}_2}R_n'(t+1/2)\,dt$ with $S_n=\rho_{n,0}+\rho_{n,3}\,\zeta_2(5)$, establishes precise arithmetic properties and asymptotics, and derives a strong $2$-adic decay $|S_n|_2\le 2^{-16n+o(n)}$. By scaling to $\widehat S_n=a_n+b_n\,\zeta_2(5)$ with integers $a_n,b_n$ and using a nonzero determinant along with Bel's lemma-based bounds, it obtains $\zeta_2(5)$ is irrational and a bound on its irrationality measure: $\mu(\zeta_2(5))\le \frac{16\log 2}{8\log 2-5}$. The work situates the result alongside recent irrationality breakthroughs for p-adic zeta values and highlights potential generalizations via hypergeometric methods and Calabi–Yau connections.
Abstract
In a spirit of Apéry's proof of the irrationality of $ζ(3)$, we construct a sequence $p_n/q_n$ of rational approximations to the $2$-adic zeta value $ζ_2(5)$ which satisfy $0 < |ζ_2(5)-p_n/q_n|_2 < \max\{|p_n|,|q_n|\}^{-1-δ}$ for an explicit constant $δ>0$. This leads to a new proof of the irrationality of $ζ_2(5)$, the result established recently by Calegari, Dimitrov and Tang using a different method. Furthermore, our approximations allow us to obtain an upper bound for the irrationality measure of this $2$-adic quantity; namely, we show that $μ(ζ_2(5)) \le (16\log2)/(8\log2-5) = 20.342\dots$.