math.NT — Number Theory

On the irrationality of the $3$-adic zeta value $\zeta_3(5)$

Contributed by Antti Hippeläinen

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 integer valued on $3^{-1}\mathbb{Z}_3^\times$, a local estimate at primes of the order of $n$ in which a pole of order $3$ coming from harmonic sums cancels exactly against a zero of order $3$ of the rational function, and a non-vanishing theorem at an auxiliary prime, in which Wilson's theorem makes one block of a reduced Gram matrix invertible. Here $n$ is the parameter of the rational function. Two constants of the estimates are bounded by finite computations, and with these bounds the $3$-adic valuation exceeds the size and the denominators together by more than $4.5$ bits per $n^2$. An appendix gives a second proof, independent of the first, which adapts to $\zeta_3(5)$ and the modular curve $X_0(3)$ a determinant argument that a preprint of Yi gives for $\zeta_7(3)$, and which rests on published theorems on overconvergent modular forms.