Modular Forms and Numerical Explorations of Rational Approximations to $ζ(3)$
For number theorists, this provides a broader framework for irrationality proofs, but the results are incremental as they do not improve upon known bounds or establish new irrationality results.
The paper extends Beukers' modular-form proof of the irrationality of ζ(3) by showing his choice is part of a one-parameter family with the same exponential decay and denominator-growth estimate, and applies the construction to other genus-zero Fricke groups.
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.