Limit[Log[Max[EulerPhi[3]/(3/Log[Log[3]]), EulerPhi[4]/(4/Log[Log[4]]), \[Ellipsis], EulerPhi[n]/(n/Log[Log[n]])]], n -> Infinity] == EulerGamma

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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> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> max </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, EulerPhi] </annotation> </semantics> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <mfrac> <mn> 3 </mn> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mfrac> </mfrac> <mo> , </mo> <mfrac> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, EulerPhi] </annotation> </semantics> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> <mfrac> <mn> 4 </mn> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mfrac> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, EulerPhi] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mfrac> <mi> n </mi> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mfrac> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <ln /> <apply> <max /> <apply> <times /> <apply> <ci> EulerPhi </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ln /> <apply> <ln /> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <ci> EulerPhi </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <ln /> <apply> <ln /> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> <apply> <times /> <apply> <ci> EulerPhi </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <ln /> <apply> <ln /> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <eulergamma /> </apply> </annotation-xml> </semantics> </math>

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2001-10-29