Modular Forms and Numerical Explorations of Rational Approximations to $ζ(3)$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909008132046848 |
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| author | Bortolotto, Cynthia Oliveira, Lucas |
| author_facet | Bortolotto, Cynthia Oliveira, Lucas |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_00673 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Modular Forms and Numerical Explorations of Rational Approximations to $ζ(3)$ Bortolotto, Cynthia Oliveira, Lucas Number Theory Numerical Analysis 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. |
| title | Modular Forms and Numerical Explorations of Rational Approximations to $ζ(3)$ |
| topic | Number Theory Numerical Analysis |
| url | https://arxiv.org/abs/2605.00673 |