On the irrationality of certain 2-adic zeta values

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.