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<p>The expectation that Ramanujan’s tau function does not vanish, commonly known as Lehmer’s Conjecture, has inspired several extensions to broader settings. In this paper, we focus on one such direction, proposed by Rouse, concerning the non-vanishing of traces of Hecke operators <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1235_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(T_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. We refine an algorithm originally introduced by Rouse to resolve the case <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1235_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(n=2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and, together with tools from our earlier work on the case <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1235_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(n=3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> in level one, we settle the conjecture for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1235_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(T_3\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation> in full generality. We also discuss an implication of our result for the non-vanishing of all coefficients of the characteristic polynomial of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1235_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(T_3\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>.</p>

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A generalized Lehmer conjecture for the trace of \(T_3\)

  • 期刊论文

  • 发表日期：

    2025年11月5日

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  • Liubomir Chiriac,
  • Erin Williams

摘要

The expectation that Ramanujan’s tau function does not vanish, commonly known as Lehmer’s Conjecture, has inspired several extensions to broader settings. In this paper, we focus on one such direction, proposed by Rouse, concerning the non-vanishing of traces of Hecke operators \(T_n\) T n . We refine an algorithm originally introduced by Rouse to resolve the case \(n=2\) n = 2 , and, together with tools from our earlier work on the case \(n=3\) n = 3 in level one, we settle the conjecture for \(T_3\) T 3 in full generality. We also discuss an implication of our result for the non-vanishing of all coefficients of the characteristic polynomial of \(T_3\) T 3 .