On the Irrationality of Certain 2-adic Zeta Values
Time: 2023-03-24
Published By: He Liu
Speaker(s): Li Lai(Tsinghua University)
Time: 15:30-16:30 April 5, 2023
Venue: Online
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 the (higher order) derivatives of these rational functions produce good linear combinations of $1$ and $2$-adic Hurwitz zeta values. The most difficult step is to prove that certain Volkenborn integrals are nonzero, which is resolved by careful manipulation of the binomial coefficients.
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