On the irrationality of certain 2-adic zeta values

2023/04/03 by Lai, Li · 1 citation
#11J72 (Primary) 11F85 #11M06 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2304.00816

Abstract

Let ζ₂(⋅) 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 ζ₂(j) is irrational. In particular, at least one of ζ₂(7),ζ₂(9),ζ₂(11),ζ₂(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.

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