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MoebiusMu






Mathematica Notation

Traditional Notation









Number Theory Functions > MoebiusMu[n] > Summation > Finite summation





http://functions.wolfram.com/13.07.23.0003.01









  


  










Input Form





DivisorSum[MoebiusMu[Subscript[d, j]]^2, {Subscript[d, j], n}] == 2^k /; FactorInteger[n] == {{Subscript[p, 1], Subscript[n, 1]}, \[Ellipsis], {Subscript[p, k], Subscript[n, k]}} && Element[Subscript[n, j], Integers] && Subscript[n, j] > 0 && Element[Subscript[p, j], Primes] && Element[Subscript[d, j], Divisors[n]]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["DivisorSum", "[", RowBox[List[SuperscriptBox[RowBox[List["MoebiusMu", "[", SubscriptBox["d", "j"], "]"]], "2"], ",", RowBox[List["{", RowBox[List[SubscriptBox["d", "j"], ",", "n"]], "}"]]]], "]"]], "\[Equal]", SuperscriptBox["2", "k"]]], "/;", RowBox[List[RowBox[List[RowBox[List["FactorInteger", "[", "n", "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["p", "1"], ",", SubscriptBox["n", "1"]]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["p", "k"], ",", SubscriptBox["n", "k"]]], "}"]]]], "}"]]]], "\[And]", RowBox[List[SubscriptBox["n", "j"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["n", "j"], ">", "0"]], "\[And]", RowBox[List[SubscriptBox["p", "j"], "\[Element]", "Primes"]], "\[And]", RowBox[List[SubscriptBox["d", "j"], "\[Element]", RowBox[List["Divisors", "[", "n", "]"]]]]]]]]]]










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> <msup> <mrow> <semantics> <mi> &#956; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Mu]&quot;, MoebiusMu] </annotation> </semantics> <mo> ( </mo> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> &#10869; </mo> <msup> <mn> 2 </mn> <mi> k </mi> </msup> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> factors </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> p </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> p </mi> <mi> k </mi> </msub> <mo> , </mo> <msub> <mi> n </mi> <mi> k </mi> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> n </mi> <mi> j </mi> </msub> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> p </mi> <mi> j </mi> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8473; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalP]&quot;, Function[Primes]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> &#8712; </mo> <mrow> <mi> divisors </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </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> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#956; </ms> <ci> MoebiusMu </ci> </apply> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> ) </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> &#10869; </ms> <apply> <ci> SuperscriptBox </ci> <ms> 2 </ms> <ms> k </ms> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> factors </ms> <ms> ( </ms> <ms> n </ms> <ms> ) </ms> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> p </ms> <ms> 1 </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> } </ms> </list> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> p </ms> <ms> k </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> k </ms> </apply> </list> </apply> <ms> } </ms> </list> </apply> </list> </apply> <ms> } </ms> </list> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> j </ms> </apply> <ms> &#8712; </ms> <apply> <ci> SuperscriptBox </ci> <ms> &#8469; </ms> <ms> + </ms> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> p </ms> <ms> j </ms> </apply> <ms> &#8712; </ms> <apply> <ci> TagBox </ci> <ms> &#8473; </ms> <apply> <ci> Function </ci> <primes /> </apply> </apply> </list> </apply> <ms> &#8743; </ms> <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> <ms> ( </ms> <ms> n </ms> <ms> ) </ms> </list> </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[SuperscriptBox[RowBox[List["MoebiusMu", "[", SubscriptBox["d_", "j"], "]"]], "2"], ",", RowBox[List["{", RowBox[List[SubscriptBox["d_", "j"], ",", "n_"]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", "k"], "/;", RowBox[List[RowBox[List[RowBox[List["FactorInteger", "[", "n", "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["p", "1"], ",", SubscriptBox["n", "1"]]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["p", "k"], ",", SubscriptBox["n", "k"]]], "}"]]]], "}"]]]], "&&", RowBox[List[SubscriptBox["n", "j"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["n", "j"], ">", "0"]], "&&", RowBox[List[SubscriptBox["p", "j"], "\[Element]", "Primes"]], "&&", RowBox[List[SubscriptBox["d", "j"], "\[Element]", RowBox[List["Divisors", "[", "n", "]"]]]]]]]]]]]]










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





2001-10-29





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