A note on the irrationality of ζ₂(5)

2025/05/08 by Lai, Li, Sprang, Johannes, Zudilin, Wadim
#11J72 #11J82 #11M06 #33C20 #Algebraic Geometry (math.AG) #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2505.05005

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 ζ₂(5) which satisfy 0 < |ζ₂(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 ζ₂(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(ζ₂(5)) ≤ (16log2)/(8log2-5) = 20.342….

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