Eta-products, Eichler integrals, and the level-8 Apery limit
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908990680596480 |
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| author | Shvets, Alex |
| author_facet | Shvets, Alex |
| contents | We give an independent eta-product derivation of the level-8 Apery limit lim B_n^{(8)}/s_n = (7/32) zeta(3), where s_n = sum_{k=0}^n C(n,k)^2 C(2k,n)^2 and B_n^{(8)} is the rational companion sequence satisfying the same cubic recurrence with initial values B_0^{(8)}=0, B_1^{(8)}=1. This value was identified numerically by Almkvist-van Straten-Zudilin and was proved by Golyshev via Beukers's Atkin-Lehner modular method; it was later recomputed by Golyshev-Kerr-Sasaki in the motivic/normal-function framework. The continued fraction PCF((2n+1)(3n^2+3n+1),-n^6) = 8/(7 zeta(3)) already appears in Batut-Olivier and was later rediscovered by the Ramanujan Machine as conjecture Z1. The contribution of the present paper is an explicit rederivation, in the eta-product normalization, of the already-known level-8 Apery limit. We spell out the eta-product verification of the Wronskian identity, the normalization of the Eichler integral, the residue computation of the Fricke period polynomial, and the elementary continuant conversion. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_14219 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Eta-products, Eichler integrals, and the level-8 Apery limit Shvets, Alex Number Theory 11F67, 11F11, 11B37, 11M06 We give an independent eta-product derivation of the level-8 Apery limit lim B_n^{(8)}/s_n = (7/32) zeta(3), where s_n = sum_{k=0}^n C(n,k)^2 C(2k,n)^2 and B_n^{(8)} is the rational companion sequence satisfying the same cubic recurrence with initial values B_0^{(8)}=0, B_1^{(8)}=1. This value was identified numerically by Almkvist-van Straten-Zudilin and was proved by Golyshev via Beukers's Atkin-Lehner modular method; it was later recomputed by Golyshev-Kerr-Sasaki in the motivic/normal-function framework. The continued fraction PCF((2n+1)(3n^2+3n+1),-n^6) = 8/(7 zeta(3)) already appears in Batut-Olivier and was later rediscovered by the Ramanujan Machine as conjecture Z1. The contribution of the present paper is an explicit rederivation, in the eta-product normalization, of the already-known level-8 Apery limit. We spell out the eta-product verification of the Wronskian identity, the normalization of the Eichler integral, the residue computation of the Fricke period polynomial, and the elementary continuant conversion. |
| title | Eta-products, Eichler integrals, and the level-8 Apery limit |
| topic | Number Theory 11F67, 11F11, 11B37, 11M06 |
| url | https://arxiv.org/abs/2604.14219 |