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http://functions.wolfram.com/04.08.25.0002.01
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Limit[(1/n^2) Sum[KroneckerDelta[1, GCD[k, l]], {k, 1, n}, {l, 1, n}],
n -> Infinity] == 6/Pi^2
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Cell[BoxData[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List[FractionBox["1", SuperscriptBox["n", "2"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l", "=", "1"]], "n"], RowBox[List["KroneckerDelta", "[", RowBox[List["1", ",", RowBox[List["GCD", "[", RowBox[List["k", ",", "l"]], "]"]]]], "]"]]]]]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]], "\[Equal]", FractionBox["6", SuperscriptBox["\[Pi]", "2"]]]]]]
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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> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mfrac> <mn> 1 </mn> <msup> <mi> n </mi> <mn> 2 </mn> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> l </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> l </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msub> </mrow> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mn> 6 </mn> <msup> <mi> π </mi> <mn> 2 </mn> </msup> </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> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> l </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> KroneckerDelta </ci> <cn type='integer'> 1 </cn> <apply> <gcd /> <ci> k </ci> <ci> l </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "n_"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l_", "=", "1"]], "n_"], RowBox[List["KroneckerDelta", "[", RowBox[List["1", ",", RowBox[List["GCD", "[", RowBox[List["k_", ",", "l_"]], "]"]]]], "]"]]]]]], SuperscriptBox["n_", "2"]], ",", RowBox[List["n_", "\[Rule]", "\[Infinity]"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox["6", SuperscriptBox["\[Pi]", "2"]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
