Modular Forms and Numerical Explorations of Rational Approximations to $ζ(3)$

We revisit Beukers' modular-form proof of the irrationality of $ζ(3)$ from the point of view of the auxiliary weight two modular form. For the Fricke group $Γ_0(6)^\star$, we show that Beukers' choice is not isolated: it belongs to a one-parameter affine family. These approximations have the same exponential decay as the classical Apéry approximations and satisfy the same denominator-growth estimate needed in Beukers' irrationality argument. We then apply the same construction to several other genus-zero Fricke groups.

Source: arXiv:2605.00673v1 - http://arxiv.org/abs/2605.00673v1 PDF: https://arxiv.org/pdf/2605.00673v1 Original Link: http://arxiv.org/abs/2605.00673v1