Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











DivisorSigma






Mathematica Notation

Traditional Notation









Number Theory Functions > DivisorSigma[k,n] > Identities > Functional identities





http://functions.wolfram.com/13.05.17.0005.01









  


  










Input Form





DivisorSum[Subscript[d, j]^\[Mu] DivisorSigma[\[Mu], Subscript[d, j]], {Subscript[d, j], n}] == DivisorSum[(n/Subscript[d, j])^(2 \[Mu]) DivisorSigma[\[Mu], Subscript[d, j]], {Subscript[d, j], n}] /; Element[Subscript[d, j], Divisors[n]] && Element[\[Mu], Integers]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List[SubsuperscriptBox["d", "j", "\[Mu]"], RowBox[List["DivisorSigma", "[", RowBox[List["\[Mu]", ",", SubscriptBox["d", "j"]]], "]"]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["d", "j"], ",", "n"]], "}"]]]], "]"]], "\[Equal]", RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["n", SubscriptBox["d", "j"]], ")"]], RowBox[List["2", "\[Mu]"]]], " ", RowBox[List["DivisorSigma", "[", RowBox[List["\[Mu]", ",", 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"]]]]]]]]










MathML Form







<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> &#8721; </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> <mi> &#956; </mi> </msubsup> <mo> &#8290; </mo> <mrow> <msub> <mi> &#963; </mi> <mi> &#956; </mi> </msub> <mo> ( </mo> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <munder> <mo> &#8721; </mo> <mrow> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> | </mo> <mi> n </mi> </mrow> </munder> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> n </mi> <msub> <mi> d </mi> <mi> j </mi> </msub> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> &#963; </mi> <mi> &#956; </mi> </msub> <mo> ( </mo> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> &#8712; </mo> <mrow> <mi> divisors </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#956; </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, 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> &#8721; </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> <ms> &#956; </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#963; </ms> <ms> &#956; </ms> </apply> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <ms> &#8721; </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> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <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> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> &#956; </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#963; </ms> <ms> &#956; </ms> </apply> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </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> &#8712; </ms> <apply> <ci> RowBox </ci> <list> <ms> divisors </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <ms> n </ms> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> &#8712; </ms> <apply> <ci> TagBox </ci> <ms> &#8484; </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List[SubsuperscriptBox["d_", "j", "\[Mu]_"], " ", RowBox[List["DivisorSigma", "[", RowBox[List["\[Mu]_", ",", SubscriptBox["d_", "j"]]], "]"]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["d_", "j"], ",", "n_"]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["n", SubscriptBox["d", "j"]], ")"]], RowBox[List["2", " ", "\[Mu]"]]], " ", RowBox[List["DivisorSigma", "[", RowBox[List["\[Mu]", ",", 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"]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2001-10-29