Modular Forms and Numerical Explorations of Rational Approximations to $\zeta(3)$
Number Theory
2026-05-04 v1 Numerical Analysis
Numerical Analysis
Abstract
We revisit Beukers' modular-form proof of the irrationality of from the point of view of the auxiliary weight two modular form. For the Fricke group , 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.
Cite
@article{arxiv.2605.00673,
title = {Modular Forms and Numerical Explorations of Rational Approximations to $\zeta(3)$},
author = {Cynthia Bortolotto and Lucas Oliveira},
journal= {arXiv preprint arXiv:2605.00673},
year = {2026}
}