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Computer-assisted construction of Ramanujan-Sato series for 1 over π.

Ralf Hemmecke¹, Peter Paule^1,2, Cristian-Silviu Radu¹

¹Research Institute for Symbolic Computation (RISC), Johannes Kepler University, 4040 Linz, Austria.

The Ramanujan Journal

|March 2, 2026

Summary

This study introduces the "Sato construction," an algorithmic method for generating Ramanujan-Sato series for 1/π. These series, including famous examples, are rigorously proven using advanced algorithms like MultiSamba.

Keywords:

Holonomic differential equations Modular forms and functions MultiSamba algorithm Ramanujan–Sato series for 1 over

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Area of Science:

Number Theory
Computational Mathematics
Algebraic Geometry

Background:

Sato and Yang (2004) introduced a method for constructing series for 1/π using modular forms.
Existing series, like those by Ramanujan and the Chudnovsky brothers, are crucial for π computation.
The need for algorithmic and rigorously proven methods for these series is recognized.

Purpose of the Study:

To present an algorithmic version of Sato and Yang's construction, termed the "Sato construction".
To introduce and analyze "Ramanujan-Sato" series generated by this algorithmic approach.
To demonstrate the rigorous, non-numerical proof of these series using computational tools.

Main Methods:

Development of the "Sato construction" algorithm based on modular forms.
Utilizing infinite families of Sato triples (N, γN, τN) for series generation.
Application of the "MultiSamba" algorithm (an extension of Samba) for evaluating modular functions at imaginary quadratic points.
Integration with holonomic toolbox procedures and the "ModFormDE" algorithm for proving series identities.

Main Results:

The successful implementation of the algorithmic "Sato construction" for generating Ramanujan-Sato series.
Demonstration that famous series for 1/π fit within this framework.
Rigorous, non-numerical proofs for all series generated using the MultiSamba algorithm.

Conclusions:

The Sato construction provides a systematic and algorithmic approach to generating series for 1/π.
The MultiSamba algorithm is a powerful tool for proving evaluations of modular functions, leading to proven series for 1/π.
This work offers a rigorous framework for discovering and verifying mathematical constants through modular forms and computational algorithms.