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http://functions.wolfram.com/13.06.23.0009.01
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Sum[1/EulerPhi[k], {k, 1, n}] \[Proportional]
((315 Zeta[3])/(2 Pi^4)) Log[n] + (315 EulerGamma Zeta[3])/(2 Pi^4) -
Sum[(Abs[MoebiusMu[k]]^2 Log[k])/(k EulerPhi[k]), {k, 1, Infinity}] +
O[Log[n]/n] /; (n -> Infinity)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox["1", RowBox[List["EulerPhi", "[", "k", "]"]]]]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox[RowBox[List["315", " ", RowBox[List["Zeta", "[", "3", "]"]]]], RowBox[List["2", " ", SuperscriptBox["\[Pi]", "4"]]]], " ", RowBox[List["Log", "[", "n", "]"]]]], "+", FractionBox[RowBox[List["315", " ", "EulerGamma", " ", RowBox[List["Zeta", "[", "3", "]"]]]], RowBox[List["2", " ", SuperscriptBox["\[Pi]", "4"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Abs", "[", RowBox[List["MoebiusMu", "[", "k", "]"]], "]"]], "2"], " ", RowBox[List["Log", "[", "k", "]"]]]], RowBox[List["k", " ", RowBox[List["EulerPhi", "[", "k", "]"]]]]]]], "+", RowBox[List["O", "[", FractionBox[RowBox[List["Log", "[", "n", "]"]], "n"], "]"]]]]]], "/;", RowBox[List["(", RowBox[List["n", "\[Rule]", "\[Infinity]"]], ")"]]]]]]
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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> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mfrac> <mn> 1 </mn> <mrow> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", EulerPhi] </annotation> </semantics> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ∝ </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 315 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["3", Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 315 </mn> <mo> ⁢ </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["3", Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <semantics> <mi> μ </mi> <annotation encoding='Mathematica'> TagBox["\[Mu]", MoebiusMu] </annotation> </semantics> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mrow> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", EulerPhi] </annotation> </semantics> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> EulerPhi </ci> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 315 </cn> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 315 </cn> <eulergamma /> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <apply> <abs /> <apply> <ci> MoebiusMu </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ln /> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <ci> k </ci> <apply> <ci> EulerPhi </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> O </ci> <apply> <times /> <apply> <ln /> <ci> n </ci> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> n </ci> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n_"], FractionBox["1", RowBox[List["EulerPhi", "[", "k", "]"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["315", " ", RowBox[List["Zeta", "[", "3", "]"]]]], ")"]], " ", RowBox[List["Log", "[", "n", "]"]]]], RowBox[List["2", " ", SuperscriptBox["\[Pi]", "4"]]]], "+", FractionBox[RowBox[List["315", " ", "EulerGamma", " ", RowBox[List["Zeta", "[", "3", "]"]]]], RowBox[List["2", " ", SuperscriptBox["\[Pi]", "4"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Abs", "[", RowBox[List["MoebiusMu", "[", "k", "]"]], "]"]], "2"], " ", RowBox[List["Log", "[", "k", "]"]]]], RowBox[List["k", " ", RowBox[List["EulerPhi", "[", "k", "]"]]]]]]], "+", RowBox[List["O", "[", FractionBox[RowBox[List["Log", "[", "n", "]"]], "n"], "]"]]]], "/;", RowBox[List["(", RowBox[List["n", "\[Rule]", "\[Infinity]"]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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