A note on the irrationality of $\zeta_2(5)$

TL;DR

This paper constructs explicit rational approximations to the 2-adic zeta value 62(5), providing a new proof of its irrationality and an upper bound on its irrationality measure, inspired by Ape9ry's approach.

Contribution

It introduces a novel method for approximating 62(5) using 2-adic techniques, leading to a new proof of irrationality and a bound on its irrationality measure.

Findings

Constructed explicit rational approximations to 62(5).

Proved the irrationality of 62(5) using 2-adic approximations.

Established an upper bound for the irrationality measure of 62(5).

Abstract

In a spirit of Ap\'ery'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 $\mu(\zeta_2(5)) \le (16\log2)/(8\log2-5) = 20.342\dots$.