On the irrationality of certain $2$-adic zeta values

Abstract

Let ${\zeta }_2(\cdot )$ be the Kubota-Leopoldt $2$-adic zeta function. We prove that, for every nonnegative integer $s$, there exists an odd integer $j$ in the interval $[s+3,3s+5]$ such that ${\zeta }_2(j)$ is irrational. In particular, at least one of ${\zeta }_2(7),{\zeta }_2(9),{\zeta }_2(11),{\zeta }_2(13)$ 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 $1$ and $2$-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.

21 pages