On the irrationality of certain 2-adic zeta values

Abstract

Let [Formula: see text] denote the Kubota–Leopoldt [Formula: see text]-adic zeta function. We prove that, for every nonnegative integer [Formula: see text], there exists an odd integer [Formula: see text] in the interval [Formula: see text] such that [Formula: see text] is irrational. In particular, at least one of [Formula: see text] is irrational. Our approach is inspired by the recent work of Sprang. We construct explicit rational functions. The Volkenborn integrals of these rational functions’ (higher-order) derivatives produce good linear combinations of [Formula: see text] and [Formula: see text]-adic Hurwitz zeta values. The most difficult step is proving that certain Volkenborn integrals are nonzero, which is resolved by carefully manipulating the binomial coefficients.