On the irrationality of 2-adic zeta values in short intervals

Abstract

We prove that, for every odd integer s ≥ 7, there exists an odd integer j in the interval [s, 1.7s) such that the Kubota–Leopoldt 2-adic zeta value \(\zeta_2(j)\) is irrational. This shortens the interval [s+3, 3s+5] of Lai. In particular, ζ₂(7) is irrational (a new proof), at least one of ζ₂(9) and ζ₂(11) is irrational, and each of {ζ₂(11), ζ₂(13), ζ₂(15)} and {ζ₂(13), ζ₂(15), ζ₂(17)} contains an irrational number. For ζ₂(9) and ζ₂(11) the best previously published result was that at least one of the six values ζ₂(9), ζ₂(11), …, ζ₂(19) is irrational. The proof constructs Hankel determinants from moments of Volkenborn integrals with poles of order q, zeros of order q and d derivatives, whose values lie in \(\mathbb{Q}+\mathbb{Q}\zeta_2(d+4)+\dots+\mathbb{Q}\zeta_2(d+q)\). After a relaxation, the denominator estimate is exactly homogeneous in q, so a single exact rational certificate covers all intervals, while nonvanishing follows from strict \(p\)-adic dominance at the auxiliary prime 2n+1.