<p>First we give general formulas for proving real or complex Ramanujan series for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1178_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <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>. Then, as an example, we apply them for providing complete proofs of the fastest series for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1178_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <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> due to Ramanujan using Russell and Weber modular polynomials. We recommend the reader to use the Maple code in Guillera (Maple program: automatic proofs of Ramanujan series for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1178_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <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> using modular equations, <a href="https://anamat.unizar.es/jguillera/Maple.html">https://anamat.unizar.es/jguillera/Maple.html</a>) for automatically proving any Ramanujan-type series for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1178_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <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>.</p>

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The fastest series for \(1/\pi \) due to Ramanujan: proofs from modular polynomials

  • Jesús Guillera

摘要

First we give general formulas for proving real or complex Ramanujan series for \(1/\pi \) 1 / π . Then, as an example, we apply them for providing complete proofs of the fastest series for \(1/\pi \) 1 / π due to Ramanujan using Russell and Weber modular polynomials. We recommend the reader to use the Maple code in Guillera (Maple program: automatic proofs of Ramanujan series for \(1/\pi \) 1 / π using modular equations, https://anamat.unizar.es/jguillera/Maple.html) for automatically proving any Ramanujan-type series for \(1/\pi \) 1 / π .