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http://functions.wolfram.com/13.06.23.0005.01
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Sum[EulerPhi[n]/n^s, {n, 1, Infinity}] == Zeta[s - 1]/Zeta[s] /; Re[s] > 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["EulerPhi", "[", "n", "]"]], SuperscriptBox["n", "s"]]]], "\[Equal]", FractionBox[RowBox[List["Zeta", "[", RowBox[List["s", "-", "1"]], "]"]], RowBox[List["Zeta", "[", "s", "]"]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "s", "]"]], ">", "1"]]]]]]
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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> n </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", EulerPhi] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <msup> <mi> n </mi> <mi> s </mi> </msup> </mfrac> </mrow> <mo> ⩵ </mo> <mfrac> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["s", "-", "1"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["s", Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> EulerPhi </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <power /> <ci> n </ci> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n_", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["EulerPhi", "[", "n_", "]"]], SuperscriptBox["n_", "s_"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["Zeta", "[", RowBox[List["s", "-", "1"]], "]"]], RowBox[List["Zeta", "[", "s", "]"]]], "/;", RowBox[List[RowBox[List["Re", "[", "s", "]"]], ">", "1"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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