The Domb Ap'ery-limit and a proof of the Ramanujan Machine conjecture Z2
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910109999824896 |
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| author | Shvets, Alex |
| author_facet | Shvets, Alex |
| contents | We prove that the ratio $B_n/D_n$ of the Apéry-like sequence $B_n$ to the Domb numbers $D_n$ converges to $(7/24)ζ(3)$, and that $\sum_{n=1}^{\infty} 64^n/(n^3 D_n D_{n-1}) = (56/3)ζ(3)$. As a corollary we establish the value $Z_2 = 12/(7ζ(3))$ conjectured by the Ramanujan Machine project. The proof uses level-6 eta products, Atkin--Lehner involutions, and Eichler integrals of weight-4 modular forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06239 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Domb Ap'ery-limit and a proof of the Ramanujan Machine conjecture Z2 Shvets, Alex Number Theory 11F11, 11B83, 33C20 We prove that the ratio $B_n/D_n$ of the Apéry-like sequence $B_n$ to the Domb numbers $D_n$ converges to $(7/24)ζ(3)$, and that $\sum_{n=1}^{\infty} 64^n/(n^3 D_n D_{n-1}) = (56/3)ζ(3)$. As a corollary we establish the value $Z_2 = 12/(7ζ(3))$ conjectured by the Ramanujan Machine project. The proof uses level-6 eta products, Atkin--Lehner involutions, and Eichler integrals of weight-4 modular forms. |
| title | The Domb Ap'ery-limit and a proof of the Ramanujan Machine conjecture Z2 |
| topic | Number Theory 11F11, 11B83, 33C20 |
| url | https://arxiv.org/abs/2604.06239 |