What question did this study set out to answer?

The aim is to prove two conjectures related to continued fractions from the Ramanujan Machine project.

April 12, 2026Open Access

Apéry limits via Eichler integrals: proofs of Ramanujan Machine conjectures Z1 and Z2

Key Points

The aim is to prove two conjectures related to continued fractions from the Ramanujan Machine project.
Utilized the Domb Apéry limit on level 6
Applied the A290575 Apéry limit on level 8
Employed eta-product parametrization
Used Eichler integrals in the proofs
Applied Atkin–Lehner extraction of the ζ(3) period
Proved PCF((2n+1)(5n²+5n+2), −16n⁶) = 12/(7ζ(3))
Established PCF((2n+1)(3n²+3n+1), −n⁶) = 8/(7ζ(3))
Both proofs confirm relationships in number theory regarding ζ(3)

Abstract

We prove two conjectures from the Ramanujan Machine project (Nature 2021): the continued fraction PCF((2n+1)(5n²+5n+2), −16n⁶) = 12/(7ζ(3)) via the Domb Apéry limit on level 6, and PCF((2n+1)(3n²+3n+1), −n⁶) = 8/(7ζ(3)) via the A290575 Apéry limit on level 8. Both proofs use the same method: eta-product parametrization, Eichler integrals, and Atkin–Lehner extraction of the ζ(3) period.

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Cite This Study

Alex Shvets (Fri,) studied this question.

synapsesocial.com/papers/69db37df4fe01fead37c5ebc https://doi.org/https://doi.org/10.5281/zenodo.19491686