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

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Mathematical proofs demonstrate the relationships of Apéry limits in number theory, suggesting new connections in continued fractions.

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)

Alex Shvets (2026) studied this question.

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