On the irrationality of the $3$-adic zeta value $\zeta_3(5)$
We give a proof that the $3$-adic zeta value $\zeta_3(5)$ is irrational. The proof adapts to $p=3$ a Hankel-determinant method that a recent preprint applies to $\zeta_2(7)$. A Volkenborn integral of an explicit rational function defines a linear functional with values in $\mathbb{Q}+\mathbb{Q}\,\zeta_3(5)$, and its Hankel determinants are the values at $\zeta_3(5)$ of polynomials with rational coefficients. We bound the $3$-adic valuation of these values and the size and the denominators of the coefficients. The new ingredients at $p=3$ are a triangular basis of the polynomials that are…
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