Riemann zeta function: Integration (formula 10.01.21.0003)

Zeta

$Mathematica Notation$

Zeta Functions and Polylogarithms

Zeta[s]

Integration

Definite integration

http://functions.wolfram.com/10.01.21.0003.01

Input Form

Integrate[\[LeftBracketingBar] ((1 - 2^(1 - (\[Sigma] + I t))) Zeta[\[Sigma] + I t])/(\[Sigma] + I t) \[RightBracketingBar], {t, -Infinity, Infinity}] == (Pi/\[Sigma]) (1 - 2^(1 - 2 \[Sigma])) Zeta[2 \[Sigma]] /; Element[\[Sigma], Reals] && \[Sigma] > 0 && Element[t, Reals]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], "\[Infinity]"], RowBox[List[RowBox[List["\[LeftBracketingBar]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["(", RowBox[List["\[Sigma]", "+", RowBox[List["\[ImaginaryI]", " ", "t"]]]], ")"]]]]]]], ")"]], RowBox[List["Zeta", "[", RowBox[List["\[Sigma]", "+", RowBox[List["\[ImaginaryI]", " ", "t"]]]], "]"]]]], RowBox[List["\[Sigma]", "+", RowBox[List["\[ImaginaryI]", " ", "t"]]]]], "\[RightBracketingBar]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "\[Sigma]"], RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", "\[Sigma]"]]]]]]], ")"]], RowBox[List["Zeta", "[", RowBox[List["2", "\[Sigma]"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["\[Sigma]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[Sigma]", ">", "0"]], "\[And]", RowBox[List["t", "\[Element]", "Reals"]]]]]]]]

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> <msubsup> <mo> ∫ </mo> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> t </mi> </mrow> <mo> + </mo> <mi> σ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> t </mi> </mrow> <mo> + </mo> <mi> σ </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "t"]], "+", "\[Sigma]"]], Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> t </mi> </mrow> <mo> + </mo> <mi> σ </mi> </mrow> </mfrac> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> σ </mi> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> σ </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["2", " ", "\[Sigma]"]], Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mi> σ </mi> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> σ </mi> <mo> ∈ </mo> <msup> <mi> ℝ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> t </mi> <mo> ∈ </mo> <mi> ℝ </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <abs /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> t </ci> </apply> <ci> σ </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> t </ci> </apply> <ci> σ </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> t </ci> </apply> <ci> σ </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <pi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> σ </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> σ </ci> </apply> </apply> <apply> <power /> <ci> σ </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> σ </ci> <apply> <ci> SuperPlus </ci> <ci> ℝ </ci> </apply> </apply> <apply> <in /> <ci> t </ci> <ci> ℝ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], "\[Infinity]"], RowBox[List[RowBox[List["\[LeftBracketingBar]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["(", RowBox[List["\[Sigma]_", "+", RowBox[List["\[ImaginaryI]", " ", "t_"]]]], ")"]]]]]]], ")"]], " ", RowBox[List["Zeta", "[", RowBox[List["\[Sigma]_", "+", RowBox[List["\[ImaginaryI]", " ", "t_"]]]], "]"]]]], RowBox[List["\[Sigma]_", "+", RowBox[List["\[ImaginaryI]", " ", "t_"]]]]], "\[RightBracketingBar]"]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Sigma]"]]]]]]], ")"]], " ", RowBox[List["Zeta", "[", RowBox[List["2", " ", "\[Sigma]"]], "]"]]]], "\[Sigma]"], "/;", RowBox[List[RowBox[List["\[Sigma]", "\[Element]", "Reals"]], "&&", RowBox[List["\[Sigma]", ">", "0"]], "&&", RowBox[List["t", "\[Element]", "Reals"]]]]]]]]]]

References

A. Ivić, "Some Identities for the Riemann Zeta Function", math.NT/0305219, (2003) http://arXiv.org/abs/math.NT/0305219

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

2003-08-21

Zeta[s,a]