A note on the irrationality of ${\zeta }_2(5)$

Abstract

In a spirit of Apéry's proof of the irrationality of $\zeta (3)$, we construct a sequence $p_n/q_n$ of rational approximations to the $2$-adic zeta value ${\zeta }_2(5)$ which satisfy $0<|{\zeta }_2(5)-p_n/q_n{|}_2<\max \{|p_n|,|q_n|{\}}^{-1-\delta }$ for an explicit constant $\delta >0$. This leads to a new proof of the irrationality of ${\zeta }_2(5)$, 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 $\mu ({\zeta }_2(5))\le (16\log 2)/(8\log 2-5)=20.342\dots$.

2^2 x 5 pages