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http://functions.wolfram.com/13.05.17.0004.01
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DivisorSum[Subscript[d, j]^(\[Mu] - \[Nu]) DivisorSigma[\[Nu] + \[Tau],
Subscript[d, j]] DivisorSigma[\[Mu] + \[Rho], n/Subscript[d, j]],
{Subscript[d, j], n}] == DivisorSum[Subscript[d, j]^(\[Mu] - \[Nu])
DivisorSigma[\[Nu] + \[Rho], Subscript[d, j]]
DivisorSigma[\[Mu] + \[Tau], n/Subscript[d, j]], {Subscript[d, j], n}] /;
Element[Subscript[d, j], Divisors[n]] && Element[\[Mu], Integers] &&
Element[\[Nu], Integers] && Element[\[Rho], Integers] &&
Element[\[Tau], Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List[SubsuperscriptBox["d", "j", RowBox[List["\[Mu]", "-", "\[Nu]"]]], RowBox[List["DivisorSigma", "[", RowBox[List[RowBox[List["\[Nu]", "+", "\[Tau]"]], ",", SubscriptBox["d", "j"]]], "]"]], RowBox[List["DivisorSigma", "[", RowBox[List[RowBox[List["\[Mu]", "+", "\[Rho]"]], ",", FractionBox["n", SubscriptBox["d", "j"]]]], "]"]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["d", "j"], ",", "n"]], "}"]]]], "]"]], "\[Equal]", RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List[SubsuperscriptBox["d", "j", RowBox[List["\[Mu]", "-", "\[Nu]"]]], RowBox[List["DivisorSigma", "[", RowBox[List[RowBox[List["\[Nu]", "+", "\[Rho]"]], ",", SubscriptBox["d", "j"]]], "]"]], " ", RowBox[List["DivisorSigma", "[", RowBox[List[RowBox[List["\[Mu]", "+", "\[Tau]"]], ",", FractionBox["n", SubscriptBox["d", "j"]]]], "]"]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["d", "j"], ",", "n"]], "}"]]]], "]"]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["d", "j"], "\[Element]", RowBox[List["Divisors", "[", "n", "]"]]]], "\[And]", RowBox[List["\[Mu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Rho]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Tau]", "\[Element]", "Integers"]]]]]]]]
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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> <munder> <mo> ∑ </mo> <mrow> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> | </mo> <mi> n </mi> </mrow> </munder> <mrow> <msubsup> <mi> d </mi> <mi> j </mi> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> </mrow> </msubsup> <mo> ⁢ </mo> <mrow> <msub> <mi> σ </mi> <mrow> <mi> ν </mi> <mo> + </mo> <mi> τ </mi> </mrow> </msub> <mo> ( </mo> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> σ </mi> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ρ </mi> </mrow> </msub> <mo> ( </mo> <mfrac> <mi> n </mi> <msub> <mi> d </mi> <mi> j </mi> </msub> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <munder> <mo> ∑ </mo> <mrow> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> | </mo> <mi> n </mi> </mrow> </munder> <mrow> <msubsup> <mi> d </mi> <mi> j </mi> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> </mrow> </msubsup> <mo> ⁢ </mo> <mrow> <msub> <mi> σ </mi> <mrow> <mi> ν </mi> <mo> + </mo> <mi> ρ </mi> </mrow> </msub> <mo> ( </mo> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> σ </mi> <mrow> <mi> μ </mi> <mo> + </mo> <mi> τ </mi> </mrow> </msub> <mo> ( </mo> <mfrac> <mi> n </mi> <msub> <mi> d </mi> <mi> j </mi> </msub> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> ∈ </mo> <mrow> <mi> divisors </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> μ </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> ν </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> ρ </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> τ </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> ∑ </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> | </ms> <ms> n </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> d </ms> <ms> j </ms> <apply> <ci> RowBox </ci> <list> <ms> μ </ms> <ms> - </ms> <ms> ν </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> σ </ms> <apply> <ci> RowBox </ci> <list> <ms> ν </ms> <ms> + </ms> <ms> τ </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> σ </ms> <apply> <ci> RowBox </ci> <list> <ms> μ </ms> <ms> + </ms> <ms> ρ </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> n </ms> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <ms> ∑ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> | </ms> <ms> n </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> d </ms> <ms> j </ms> <apply> <ci> RowBox </ci> <list> <ms> μ </ms> <ms> - </ms> <ms> ν </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> σ </ms> <apply> <ci> RowBox </ci> <list> <ms> ν </ms> <ms> + </ms> <ms> ρ </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> σ </ms> <apply> <ci> RowBox </ci> <list> <ms> μ </ms> <ms> + </ms> <ms> τ </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> n </ms> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> ∈ </ms> <apply> <ci> RowBox </ci> <list> <ms> divisors </ms> <ms> ( </ms> <ms> n </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> μ </ms> <ms> ∈ </ms> <apply> <ci> TagBox </ci> <ms> ℤ </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> ν </ms> <ms> ∈ </ms> <apply> <ci> TagBox </ci> <ms> ℤ </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> ρ </ms> <ms> ∈ </ms> <apply> <ci> TagBox </ci> <ms> ℤ </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> τ </ms> <ms> ∈ </ms> <apply> <ci> TagBox </ci> <ms> ℤ </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List[SubsuperscriptBox["d_", "j", RowBox[List["\[Mu]_", "-", "\[Nu]_"]]], " ", RowBox[List["DivisorSigma", "[", RowBox[List[RowBox[List["\[Nu]_", "+", "\[Tau]_"]], ",", SubscriptBox["d_", "j"]]], "]"]], " ", RowBox[List["DivisorSigma", "[", RowBox[List[RowBox[List["\[Mu]_", "+", "\[Rho]_"]], ",", FractionBox["n_", SubscriptBox["d_", "j"]]]], "]"]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["d_", "j"], ",", "n_"]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List[SubsuperscriptBox["d", "j", RowBox[List["\[Mu]", "-", "\[Nu]"]]], " ", RowBox[List["DivisorSigma", "[", RowBox[List[RowBox[List["\[Nu]", "+", "\[Rho]"]], ",", SubscriptBox["d", "j"]]], "]"]], " ", RowBox[List["DivisorSigma", "[", RowBox[List[RowBox[List["\[Mu]", "+", "\[Tau]"]], ",", FractionBox["n", SubscriptBox["d", "j"]]]], "]"]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["d", "j"], ",", "n"]], "}"]]]], "]"]], "/;", RowBox[List[RowBox[List[SubscriptBox["d", "j"], "\[Element]", RowBox[List["Divisors", "[", "n", "]"]]]], "&&", RowBox[List["\[Mu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Rho]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Tau]", "\[Element]", "Integers"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
