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

TL;DR

This paper explores modular forms related to Beukers' proof of the irrationality of ζ(3), revealing a family of approximations with similar decay properties and extending the approach to other Fricke groups.

Contribution

It uncovers a one-parameter family of modular form-based approximations to ζ(3) and generalizes the method to multiple genus-zero Fricke groups.

Findings

Approximations exhibit exponential decay comparable to Apéry's.

The family of approximations shares the same denominator-growth estimates.

The approach applies to several other genus-zero Fricke groups.

Abstract

We revisit Beukers' modular-form proof of the irrationality of $\zeta(3)$ from the point of view of the auxiliary weight two modular form. For the Fricke group $\Gamma_0(6)^\star$, 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 Ap\'ery 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.