Riemann zeta function: Integral representations (formula 10.01.07.0015)

Zeta

$Mathematica Notation$

Zeta Functions and Polylogarithms

Zeta[s]

Integral representations

Multiple integral representations

http://functions.wolfram.com/10.01.07.0015.01

Input Form

Zeta[n] == Integrate[1/(1 - Product[Subscript[t, k], {k, 1, n}]), {Subscript[t, 1], -Infinity, Infinity}, {Subscript[t, 2], -Infinity, Infinity}, â€¦, {Subscript[t, n], 0, Infinity}] /; Subscript[m, 1] == Subscript[m, 2] == â€¦ == Subscript[m, n] == 1 && n - 1 âˆˆ Integers && n - 1 > 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Zeta", "[", "n", "]"]], "\[Equal]", RowBox[List[SubsuperscriptBox["\[Integral]", InterpretationBox["0", DirectedInfinity[-1]], InterpretationBox[SubscriptBox["m", "1"], DirectedInfinity[1]]], RowBox[List[SubsuperscriptBox["\[Integral]", InterpretationBox["0", DirectedInfinity[-1]], InterpretationBox[SubscriptBox["m", "2"], DirectedInfinity[1]]], RowBox[List["\[Ellipsis]", RowBox[List[SubsuperscriptBox["\[Integral]", "0", InterpretationBox[SubscriptBox["m", "n"], DirectedInfinity[1]]], RowBox[List[FractionBox["1", RowBox[List["1", "-", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], SubscriptBox["t", "k"]]]]]], RowBox[List["\[DifferentialD]", SubscriptBox["t", "1"]]], RowBox[List["\[DifferentialD]", SubscriptBox["t", "2"]]], " ", "\[Ellipsis]", RowBox[List["\[DifferentialD]", SubscriptBox["t", "n"]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["m", "1"], "\[Equal]", SubscriptBox["m", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["m", "n"], "\[Equal]", "1"]], "\[And]", RowBox[List[RowBox[List["n", "-", "1"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["n", "-", "1"]], ">", "0"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["n", Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </msubsup> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </msubsup> <mrow> <mo> … </mo> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <msub> <mi> m </mi> <mi> n </mi> </msub> </msubsup> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> t </mi> <mi> k </mi> </msub> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <msub> <mi> t </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <msub> <mi> t </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mo> … </mo> </mrow> </msub> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <msub> <mi> t </mi> <mi> n </mi> </msub> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> ⩵ </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> ⩵ </mo> <mo> … </mo> <mo> ⩵ </mo> <msub> <mi> m </mi> <mi> n </mi> </msub> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["n", Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </msubsup> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </msubsup> <mrow> <mo> … </mo> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <msub> <mi> m </mi> <mi> n </mi> </msub> </msubsup> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> t </mi> <mi> k </mi> </msub> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <msub> <mi> t </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <msub> <mi> t </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mo> … </mo> </mrow> </msub> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <msub> <mi> t </mi> <mi> n </mi> </msub> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> ⩵ </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> ⩵ </mo> <mo> … </mo> <mo> ⩵ </mo> <msub> <mi> m </mi> <mi> n </mi> </msub> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

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

2001-10-29

Zeta[s,a]