Sum of divisor powers: Summation (formula 13.05.23.0003)

DivisorSigma

$Mathematica Notation$

Number Theory Functions

DivisorSigma[k,n]

Summation

Finite summation

http://functions.wolfram.com/13.05.23.0003.01

Input Form

Sum[DivisorSigma[r, k] DivisorSigma[s, n - k], {k, 1, n - 1}] == (DivisorSigma[r + s + 1, n] Gamma[r + 1] Gamma[s + 1] Zeta[r + 1] Zeta[s + 1])/(Gamma[r + s + 2] Zeta[r + s + 2]) + (n DivisorSigma[r + s - 1, n] (Zeta[1 - r] + Zeta[1 - s]))/(r + s) - (1/2) DivisorSigma[s, n] Zeta[-r] - (1/2) DivisorSigma[r, n] Zeta[-s] /; (r == 1 && s == 3) || (r == 1 && s == 5) || (r == 1 && s == 7) || (r == 1 && s == 11) || (r == 3 && s == 3) || (r == 3 && s == 5) || (r == 3 && s == 9) || (r == 5 && s == 7)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["DivisorSigma", "[", RowBox[List["r", ",", "k"]], "]"]], " ", RowBox[List["DivisorSigma", "[", RowBox[List["s", ",", RowBox[List["n", "-", "k"]]]], "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["DivisorSigma", "[", RowBox[List[RowBox[List["r", "+", "s", "+", "1"]], ",", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["r", "+", "1"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["s", "+", "1"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["r", "+", "1"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["s", "+", "1"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["r", "+", "s", "+", "2"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["r", "+", "s", "+", "2"]], "]"]]]], ")"]]]], "+", FractionBox[RowBox[List["n", " ", RowBox[List["DivisorSigma", "[", RowBox[List[RowBox[List["r", "+", "s", "-", "1"]], ",", "n"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["1", "-", "r"]], "]"]], "+", RowBox[List["Zeta", "[", RowBox[List["1", "-", "s"]], "]"]]]], ")"]]]], RowBox[List["r", "+", "s"]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["DivisorSigma", "[", RowBox[List["s", ",", "n"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["-", "r"]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["DivisorSigma", "[", RowBox[List["r", ",", "n"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["-", "s"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["r", "\[Equal]", "1"]], "\[And]", RowBox[List["s", "\[Equal]", "3"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["r", "\[Equal]", "1"]], "\[And]", RowBox[List["s", "\[Equal]", "5"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["r", "\[Equal]", "1"]], "\[And]", RowBox[List["s", "\[Equal]", "7"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["r", "\[Equal]", "1"]], "\[And]", RowBox[List["s", "\[Equal]", "11"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["r", "\[Equal]", "3"]], "\[And]", RowBox[List["s", "\[Equal]", "3"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["r", "\[Equal]", "3"]], "\[And]", RowBox[List["s", "\[Equal]", "5"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["r", "\[Equal]", "3"]], "\[And]", RowBox[List["s", "\[Equal]", "9"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["r", "\[Equal]", "5"]], "\[And]", RowBox[List["s", "\[Equal]", "7"]]]], ")"]]]]]]]]

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> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <msub> <semantics> <mi> σ </mi> <annotation encoding='Mathematica'> TagBox["\[Sigma]", DivisorSigma] </annotation> </semantics> <mi> r </mi> </msub> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> σ </mi> <annotation encoding='Mathematica'> TagBox["\[Sigma]", DivisorSigma] </annotation> </semantics> <mi> s </mi> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["r", "+", "1"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mo> ⁢ </mo> <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> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> r </mi> <mo> + </mo> <mi> s </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> r </mi> <mo> + </mo> <mi> s </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["r", "+", "s", "+", "2"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> σ </mi> <annotation encoding='Mathematica'> TagBox["\[Sigma]", DivisorSigma] </annotation> </semantics> <mrow> <mi> r </mi> <mo> + </mo> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> σ </mi> <annotation encoding='Mathematica'> TagBox["\[Sigma]", DivisorSigma] </annotation> </semantics> <mi> s </mi> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> r </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["-", "r"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mfrac> <mi> n </mi> <mrow> <mi> r </mi> <mo> + </mo> <mi> s </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> σ </mi> <annotation encoding='Mathematica'> TagBox["\[Sigma]", DivisorSigma] </annotation> </semantics> <mrow> <mi> r </mi> <mo> + </mo> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> r </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["1", "-", "r"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> <mo> + </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["1", "-", "s"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> σ </mi> <annotation encoding='Mathematica'> TagBox["\[Sigma]", DivisorSigma] </annotation> </semantics> <mi> r </mi> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["-", "s"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> r </mi> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> s </mi> <mo> ⩵ </mo> <mn> 3 </mn> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> r </mi> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> s </mi> <mo> ⩵ </mo> <mn> 5 </mn> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> r </mi> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> s </mi> <mo> ⩵ </mo> <mn> 7 </mn> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> r </mi> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> s </mi> <mo> ⩵ </mo> <mn> 11 </mn> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> r </mi> <mo> ⩵ </mo> <mn> 3 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> s </mi> <mo> ⩵ </mo> <mn> 3 </mn> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> r </mi> <mo> ⩵ </mo> <mn> 3 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> s </mi> <mo> ⩵ </mo> <mn> 5 </mn> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> r </mi> <mo> ⩵ </mo> <mn> 3 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> s </mi> <mo> ⩵ </mo> <mn> 9 </mn> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> r </mi> <mo> ⩵ </mo> <mn> 5 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> s </mi> <mo> ⩵ </mo> <mn> 7 </mn> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> DivisorSigma </ci> <ci> r </ci> <ci> k </ci> </apply> <apply> <ci> DivisorSigma </ci> <ci> s </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> r </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> r </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> r </ci> <ci> s </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> r </ci> <ci> s </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> DivisorSigma </ci> <apply> <plus /> <ci> r </ci> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ci> DivisorSigma </ci> <ci> s </ci> <ci> n </ci> </apply> <apply> <ci> Zeta </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <plus /> <ci> r </ci> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> DivisorSigma </ci> <apply> <plus /> <ci> r </ci> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> <ci> n </ci> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ci> DivisorSigma </ci> <ci> r </ci> <ci> n </ci> </apply> <apply> <ci> Zeta </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <eq /> <ci> r </ci> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <ci> s </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> r </ci> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <ci> s </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> r </ci> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <ci> s </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> r </ci> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <ci> s </ci> <cn type='integer'> 11 </cn> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> r </ci> <cn type='integer'> 3 </cn> </apply> <apply> <eq /> <ci> s </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> r </ci> <cn type='integer'> 3 </cn> </apply> <apply> <eq /> <ci> s </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> r </ci> <cn type='integer'> 3 </cn> </apply> <apply> <eq /> <ci> s </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> r </ci> <cn type='integer'> 5 </cn> </apply> <apply> <eq /> <ci> s </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["n_", "-", "1"]]], RowBox[List[RowBox[List["DivisorSigma", "[", RowBox[List["r_", ",", "k"]], "]"]], " ", RowBox[List["DivisorSigma", "[", RowBox[List["s_", ",", RowBox[List["n_", "-", "k"]]]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["DivisorSigma", "[", RowBox[List[RowBox[List["r", "+", "s", "+", "1"]], ",", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["r", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", "1"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["r", "+", "1"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["s", "+", "1"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["r", "+", "s", "+", "2"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["r", "+", "s", "+", "2"]], "]"]]]]], "+", FractionBox[RowBox[List["n", " ", RowBox[List["DivisorSigma", "[", RowBox[List[RowBox[List["r", "+", "s", "-", "1"]], ",", "n"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["1", "-", "r"]], "]"]], "+", RowBox[List["Zeta", "[", RowBox[List["1", "-", "s"]], "]"]]]], ")"]]]], RowBox[List["r", "+", "s"]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["DivisorSigma", "[", RowBox[List["s", ",", "n"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["-", "r"]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["DivisorSigma", "[", RowBox[List["r", ",", "n"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["-", "s"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["r", "\[Equal]", "1"]], "&&", RowBox[List["s", "\[Equal]", "3"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["r", "\[Equal]", "1"]], "&&", RowBox[List["s", "\[Equal]", "5"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["r", "\[Equal]", "1"]], "&&", RowBox[List["s", "\[Equal]", "7"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["r", "\[Equal]", "1"]], "&&", RowBox[List["s", "\[Equal]", "11"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["r", "\[Equal]", "3"]], "&&", RowBox[List["s", "\[Equal]", "3"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["r", "\[Equal]", "3"]], "&&", RowBox[List["s", "\[Equal]", "5"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["r", "\[Equal]", "3"]], "&&", RowBox[List["s", "\[Equal]", "9"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["r", "\[Equal]", "5"]], "&&", RowBox[List["s", "\[Equal]", "7"]]]], ")"]]]]]]]]]]

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

2001-10-29