Referring to ideas of Sato and Yang in (Math Z 246:1–19, 2004) described a construction of series for 1 over <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation> starting with a pair (g, h), where g is a modular form of weight 2 and h is a modular function; i.e., a modular form of weight zero. In this article we present an algorithmic version, called “Sato construction”. Series for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$1/\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> obtained this way will be called “Ramanujan–Sato” series. Famous series fit into this definition, for instance, Ramanujan’s series used by Gosper and the series used by the Chudnovsky brothers for computing millions of digits of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation>. We show that these series are induced by members of infinite families of Sato triples <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(N, \gamma _N, \tau _N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <msub> <mi>γ</mi> <mi>N</mi> </msub> <mo>,</mo> <msub> <mi>τ</mi> <mi>N</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$N>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is an integer and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\gamma _N$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>N</mi> </msub> </math></EquationSource> </InlineEquation> a <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$2\times 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> matrix satisfying <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\gamma _N \tau _N=N \tau _N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mi>N</mi> </msub> <msub> <mi>τ</mi> <mi>N</mi> </msub> <mo>=</mo> <mi>N</mi> <msub> <mi>τ</mi> <mi>N</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\tau _N$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>τ</mi> <mi>N</mi> </msub> </math></EquationSource> </InlineEquation> being an element from the upper half of the complex plane. In addition to procedures for guessing and proving from the holonomic toolbox together with the algorithm “ModFormDE”, as described in Paule and Radu in Int J Number Theory (17:713–759, 2021), a central role is played by the algorithm “MultiSamba”, an extension of Samba (“subalgebra module basis algorithm”) originating from Radu in (J Symb Comput 68:225–253, 2015) and Hemmecke in (J Symb Comput 84:14–24, 2018). With the help of MultiSamba one can find and prove evaluations of modular functions, at imaginary quadratic points, in terms of nested algebraic expressions. As a consequence, all the series for <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$1/\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> constructed with the help of MultiSamba are proven completely in a rigorous non-numerical manner.

Computer-assisted construction of Ramanujan–Sato series for 1 over $\pi $

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2026年2月26日

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Ralf Hemmecke,
Peter Paule,
Cristian-Silviu Radu

摘要

Referring to ideas of Sato and Yang in (Math Z 246:1–19, 2004) described a construction of series for 1 over $\pi $ $π$ starting with a pair (g, h), where g is a modular form of weight 2 and h is a modular function; i.e., a modular form of weight zero. In this article we present an algorithmic version, called “Sato construction”. Series for $1/\pi $ $1 / π$ obtained this way will be called “Ramanujan–Sato” series. Famous series fit into this definition, for instance, Ramanujan’s series used by Gosper and the series used by the Chudnovsky brothers for computing millions of digits of $\pi $ $π$ . We show that these series are induced by members of infinite families of Sato triples $(N, \gamma _N, \tau _N)$ $(N, γ_{N}, τ_{N})$ where $N>1$ $N > 1$ is an integer and $\gamma _N$ $γ_{N}$ a $2\times 2$ $2 \times 2$ matrix satisfying $\gamma _N \tau _N=N \tau _N$ $γ_{N} τ_{N} = N τ_{N}$ for $\tau _N$ $τ_{N}$ being an element from the upper half of the complex plane. In addition to procedures for guessing and proving from the holonomic toolbox together with the algorithm “ModFormDE”, as described in Paule and Radu in Int J Number Theory (17:713–759, 2021), a central role is played by the algorithm “MultiSamba”, an extension of Samba (“subalgebra module basis algorithm”) originating from Radu in (J Symb Comput 68:225–253, 2015) and Hemmecke in (J Symb Comput 84:14–24, 2018). With the help of MultiSamba one can find and prove evaluations of modular functions, at imaginary quadratic points, in terms of nested algebraic expressions. As a consequence, all the series for $1/\pi $ $1 / π$ constructed with the help of MultiSamba are proven completely in a rigorous non-numerical manner.

Computer-assisted construction of Ramanujan–Sato series for 1 over \(\pi \)

摘要