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http://functions.wolfram.com/05.11.06.0001.01
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Cyclotomic[n, z] == Sum[a[n, j] z^(EulerPhi[n] - j), {j, 0, EulerPhi[n]}] /;
a[n, j] == (-(1/j)) (MoebiusMu[n] Sum[a[n, m] MoebiusMu[GCD[n, j - m]]
EulerPhi[GCD[n, j - m]], {m, 0, j - 1}]) && a[n, 0] == 1 &&
!(Element[Sqrt[n], Integers] && n >= 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Cyclotomic", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["EulerPhi", "[", "n", "]"]]], RowBox[List[RowBox[List["a", "[", RowBox[List["n", ",", "j"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["EulerPhi", "[", "n", "]"]], "-", "j"]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["a", "[", RowBox[List["n", ",", "j"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "j"]]], RowBox[List["(", RowBox[List[RowBox[List["MoebiusMu", "[", "n", "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["a", "[", RowBox[List["n", ",", "m"]], "]"]], " ", RowBox[List["MoebiusMu", "[", RowBox[List["GCD", "[", RowBox[List["n", ",", RowBox[List["j", "-", "m"]]]], "]"]], "]"]], " ", RowBox[List["EulerPhi", "[", RowBox[List["GCD", "[", RowBox[List["n", ",", RowBox[List["j", "-", "m"]]]], "]"]], "]"]]]]]]]], ")"]]]]]], "\[And]", RowBox[List[RowBox[List["a", "[", RowBox[List["n", ",", "0"]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List["Element", "[", RowBox[List[RowBox[List["Sqrt", "[", "n", "]"]], ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]], "]"]]]]]]]]
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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> <msub> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", Cyclotomic] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", EulerPhi] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </munderover> <mrow> <mrow> <mi> a </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mrow> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", EulerPhi] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mi> j </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> a </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <semantics> <mi> μ </mi> <annotation encoding='Mathematica'> TagBox["\[Mu]", MoebiusMu] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mi> j </mi> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <mi> a </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> μ </mi> <annotation encoding='Mathematica'> TagBox["\[Mu]", MoebiusMu] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mrow> <mi> j </mi> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", EulerPhi] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mrow> <mi> j </mi> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <msqrt> <mi> n </mi> </msqrt> <mo> ∉ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Cyclotomic </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <ci> EulerPhi </ci> <ci> n </ci> </apply> </uplimit> <apply> <times /> <apply> <ci> a </ci> <ci> n </ci> <ci> j </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <ci> EulerPhi </ci> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> a </ci> <ci> n </ci> <ci> j </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> MoebiusMu </ci> <ci> n </ci> </apply> <apply> <power /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> a </ci> <ci> n </ci> <ci> m </ci> </apply> <apply> <ci> MoebiusMu </ci> <apply> <gcd /> <ci> n </ci> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <ci> EulerPhi </ci> <apply> <gcd /> <ci> n </ci> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> a </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <notin /> <apply> <power /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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