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<p>In a previous paper (Sun, in: <a href="http://arxiv.org/abs/2406.06294">arXiv:2406.06294</a> [math.NT], 2024), the author proved the exact formulae for ranks of partitions modulo each prime <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1011_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\ge 5\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1011_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(p=5\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> and 7, we prove special vanishing properties of the Kloosterman sums appearing in the exact formulae. These vanishing properties imply a new proof of Dyson’s rank conjectures. Specifically, we give a new proof of Ramanujan’s congruences <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1011_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="162" /> </InlineMediaObject> <EquationSource Format="TEX">\(p(5n+4)\equiv 0\ (\textrm{mod}\ 5)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>≡</mo> <mn>0</mn> <mspace width="4pt" /> <mo stretchy="false">(</mo> <mtext>mod</mtext> <mspace width="4pt" /> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1011_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="162" /> </InlineMediaObject> <EquationSource Format="TEX">\(p(7n+5)\equiv 0\ (\textrm{mod}\ 7)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mn>7</mn> <mi>n</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>≡</mo> <mn>0</mn> <mspace width="4pt" /> <mo stretchy="false">(</mo> <mtext>mod</mtext> <mspace width="4pt" /> <mn>7</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Vanishing properties of Kloosterman sums and Dyson’s conjectures

  • 期刊论文

  • 发表日期：

    2025年1月31日

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  • Qihang Sun

摘要

In a previous paper (Sun, in: arXiv:2406.06294 [math.NT], 2024), the author proved the exact formulae for ranks of partitions modulo each prime \(p\ge 5\) p ≥ 5 . In this paper, for \(p=5\) p = 5 and 7, we prove special vanishing properties of the Kloosterman sums appearing in the exact formulae. These vanishing properties imply a new proof of Dyson’s rank conjectures. Specifically, we give a new proof of Ramanujan’s congruences \(p(5n+4)\equiv 0\ (\textrm{mod}\ 5)\) p ( 5 n + 4 ) ≡ 0 ( mod 5 ) and \(p(7n+5)\equiv 0\ (\textrm{mod}\ 7)\) p ( 7 n + 5 ) ≡ 0 ( mod 7 ) .