What question did this study set out to answer?

The study aims to prove the limit of a specific Apéry-like sequence and establish the Ramanujan Machine conjecture Z1.

April 27, 2026Open Access

The level-8 Apéry-limit and a proof of the Ramanujan Machine conjecture Z1

Key Points

The study aims to prove the limit of a specific Apéry-like sequence and establish the Ramanujan Machine conjecture Z1.
Parametrization of eta products
Use of Eichler integrals
Application of Mellin-Barnes period polynomials and singularity transfer
Proved lim B_n^{(8)}/s_n = (7/32)ζ(3) for level-8 Apéry-like sequences.
Established Ramanujan Machine conjecture Z1: PCF((2n+1)(3n²+3n+1), −n⁶) = 8/(7ζ(3)).
First complete proof of the conjecture following prior identification.

Abstract

We prove lim Bₙ^ (8) /sₙ = (7/32) ζ (3) for the level-8 Apéry-like sequence sₙ = Σ C (n, k) ²C (2k, n) ² and its rational companion Bₙ^ (8), and deduce the Ramanujan Machine conjecture Z1: PCF ( (2n+1) (3n²+3n+1), −n⁶) = 8/ (7ζ (3) ). The value was identified by Almkvist–van Straten–Zudilin (2008) and stated by Golyshev (arXiv: 0908. 1458) ; this is the first complete proof. Method: eta-product parametrization, Eichler integral, Mellin–Barnes period polynomial, singularity transfer. The companion result Z2 was proved in arXiv: 2604. 06239. https: //arxiv. org/abs/2604. 14219

The level-8 Apéry-limit and a proof of the Ramanujan Machine conjecture Z1

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