Limit[(1/n^r) Sum[\[Ellipsis] Sum[GCD[Subscript[k, 1], Subscript[k, 2], \[Ellipsis], Subscript[k, r]]^k, {Subscript[k, r], 1, n}], {Subscript[k, 1], 1, n}, {Subscript[k, 2], 1, n}], n -> Infinity] == Zeta[r - k]/Zeta[r]

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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> <mfrac> <mn> 1 </mn> <msup> <mi> n </mi> <mi> r </mi> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mi> r </mi> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msup> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> k </mi> <mi> r </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> r </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[RowBox[List[&quot;r&quot;, &quot;-&quot;, &quot;k&quot;]], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;r&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mfrac> </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> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> n </ci> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <ci> &#8230; </ci> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> r </ci> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <power /> <apply> <gcd /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> r </ci> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> r </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Zeta </ci> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29