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)