Sum[KroneckerDelta[1, GCD[b, d]]/(b d (b + d))^2, {b, 1, Infinity}, {d, 1, Infinity}] == 1/3

Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["b", "=", "1"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["d", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["KroneckerDelta", "[", RowBox[List["1", ",", " ", RowBox[List["GCD", "[", RowBox[List["b", ",", "d"]], "]"]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", "d", " ", RowBox[List["(", RowBox[List["b", "+", "d"]], ")"]]]], ")"]], "2"]]]]]], "\[Equal]", FractionBox["1", "3"]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> b </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> d </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> , </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msub> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> d </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sum /> <bvar> <ci> b </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> KroneckerDelta </ci> <cn type='integer'> 1 </cn> <apply> <gcd /> <ci> b </ci> <ci> d </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <ci> b </ci> <ci> d </ci> <apply> <plus /> <ci> b </ci> <ci> d </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["b_", "=", "1"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["d_", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["KroneckerDelta", "[", RowBox[List["1", ",", RowBox[List["GCD", "[", RowBox[List["b_", ",", "d_"]], "]"]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["b_", " ", "d_", " ", RowBox[List["(", RowBox[List["b_", "+", "d_"]], ")"]]]], ")"]], "2"]]]]]], "]"]], "\[RuleDelayed]", FractionBox["1", "3"]]]]]

2001-10-29