Abs[RamanujanTau[n]] <= (2 \[Chi]^2 + \[Chi]) n^5 Log[Log[n]]^2 /; \[Chi] == E^EulerGamma + 6482/Log[Log[3]]^2 && n > 0

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <semantics> <mi> &#964; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Tau]&quot;, RamanujanTau] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8804; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#967; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> &#967; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> n </mi> <mn> 5 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> &#967; </mi> <mo> &#63449; </mo> <mrow> <msup> <mi> &#8519; </mi> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[List[], EulerGamma]] </annotation> </semantics> </msup> <mo> + </mo> <mfrac> <mn> 6482 </mn> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <leq /> <apply> <abs /> <apply> <ci> RamanujanTau </ci> <ci> n </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> &#967; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#967; </ci> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <apply> <ln /> <apply> <ln /> <ci> n </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> &#967; </ci> <apply> <plus /> <apply> <power /> <exponentiale /> <eulergamma /> </apply> <apply> <times /> <cn type='integer'> 6482 </cn> <apply> <power /> <apply> <power /> <apply> <ln /> <apply> <ln /> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <gt /> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02