What question did this study set out to answer?

The aim is to establish the Domb Apéry-limit and its relationship to the Ramanujan Machine conjecture.

April 5, 2026Open Access

The Domb Apéry-limit and a proof of the Ramanujan Machine conjecture Z2

Puntos clave

The aim is to establish the Domb Apéry-limit and its relationship to the Ramanujan Machine conjecture.
Deriving an Atkin–Lehner transformation law for the Eichler integral.
Extracting the connection constant from the derivative of the period polynomial at the CM fixed point.
Establishing a continued fraction identity for the Apéry constant.
The Domb Apéry-limit is proven, resolving an open question since 2008.
A continued fraction identity for the Apéry constant is verified, linked to the Ramanujan Machine project.

Resumen

We prove the Domb Apéry-limit, listed without proof in the literature since 2008, by deriving an Atkin–Lehner transformation law for the Eichler integral of an Eisenstein form on the genus-zero modular curve of level six, then extracting the connection constant from the derivative of the period polynomial at the CM fixed point. As a corollary we establish a continued fraction identity for the Apéry constant conjectured by the Ramanujan Machine project and open since 2019.

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Cite This Study

Alex Shvets (Fri,) studied this question.

synapsesocial.com/papers/69d1fd4ea79560c99a0a350b https://doi.org/https://doi.org/10.5281/zenodo.19394212

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